Biased Double-well Potential: Bistability, Bifurcation and Hysteresis

Bistability, bifurcation and hysteresis are ubiquitous phenomena that arise from nonlinear dynamics and have considerable importance for technology applications.  For instance, the hysteresis associated with the flipping of magnetic domains under magnetic fields is the central mechanism for magnetic memory, and bistability is a key feature of switching technology.

… one of the most commonly encountered bifurcations is called a saddle-node bifurcation, which is the bifurcation that occurs in the biased double-well potential.

One of the simplest models for bistability and hysteresis is the one-dimensional double-well potential biased by a changing linear potential.  An example of a double-well potential with a bias is

where the parameter c is a control parameter (bias) that can be adjusted or that changes slowly in time c(t).  The net double-well potentials for several values of the control parameter c are shown in Fig. 1.   With no bias, there are two degenerate energy minima.  As c is made negative, the left well has the lowest energy, and as c is made positive the right well has the lowest energy.

The dynamics of this potential energy profile can be understood by imagining a small ball that responds to the local forces exerted by the potential.  For large negative values of c the ball will have its minimum energy in the left well.  As c is increased, the energy of the left well increases, and rises above the energy of the right well.  If the ball began in the left well, even when the left well has a higher energy than the right, there is a potential barrier that the ball cannot overcome and it remains on the left.  This local minimum is a stable equilibrium, but it is called “metastable” because it is not a global minimum of the system.  Metastability is the origin of hysteresis.

Fig. 1 A biased double-well potential in one dimension. The thresholds to destroy the local metastable minima are c = +/-1.05. For values beyond threshold, only a single minimum exists with no barrier. Hysteresis is caused by the mass being stuck in the metastable (upper) minimum because it has insufficient energy to overcome the potential barrier, until the barrier disappears at threshold and the ball rolls all the way down to the bottom to the new location. When the bias is slowly reversed, the new location becomes metastable, until the ball can overcome the barrier and roll down to its original minimum, etc.

           Once sufficient bias is applied that the local minimum disappears, the ball will roll downhill to the new minimum on the right, and in the presence of dissipation, it will come to rest in the new minimum.  The bias can then be slowly lowered, reversing this process. Because of the potential barrier, the bias must change sign and be strong enough to remove the stability of the now metastable fixed point with the ball on the right, allowing the ball to roll back down to its original location on the left.  This “overshoot” defines the extent of the hysteresis. The fact that there are two minima, and that one is metastable with a barrier between the two, produces “bistability”, meaning that there are two stable fixed points for the same control parameter.

           For illustration, assume a mass obeys the flow equation

including a damping term, where the force is the negative gradient of the potential energy.  The bias parameter c can be time dependent, beginning beyond the negative threshold and slowly increasing until it exceeds the positive threshold, and then reversing and decreasing again.  The position of the mass is locally a damped oscillator until a threshold is passed, and then the mass falls into the global minimum, as shown in Fig. 2. As the bias is reversed, it remains in the metastable minimum on the right until the control parameter passes threshold, and then the mass drops into the left minimum that is now a global minimum.

Fig. 2 Hysteresis diagram. The mass begins in the left well. As the parameter c increases, the mass remains in the well, even though it is no longer the global minimum when c becomes positive. When c passes the positive threshold (around 1.05 for this example), the mass falls into the right well, with damped oscillation. Then the control parameter c is decreased slowly until the negative threshold is passed, and the mass switches to the left well with damped oscillations. The difference between the “switch up” and “switch down” values of the control parameter represents the “hysteresis” of the this system.

The sudden switching of the biased double-well potential represents what is known as a “bifurcation”. A bifurcation is a sudden change in the qualitative behavior of a system caused by a small change in a control variable. Usually, a bifurcation occurs when the number of attractors of a system changes. There is a fairly large menagerie of different types of bifurcations, but one of the most commonly encountered bifurcations is called a saddle-node bifurcation, which is the bifurcation that occurs in the biased double-well potential. In fact, there are two saddle-node bifurcations.

Bifurcations are easily portrayed by creating a joint space between phase space and the one (or more) control parameters that induce the bifurcation. The phase space of the double well is two dimensional (position, velocity) with three fixed points, but the change in the number of fixed points can be captured by taking a projection of the phase space onto a lower-dimensional manifold. In this case, the projection is simply along the x-axis. Therefore a “co-dimensional phase space” can be constructed with the x-axis as one dimension and the control parameter as the other. This is illustrated in Fig. 3. The cubic curve traces out the solutions to the fixed-point equation

For a given value of the control parameter c there are either three solutions or one solution. The values of c where the number of solutions changes discontinuously is the bifurcation point c*. Two examples of the potential function are shown on the right for c = +1 and c = -0.5 showing the locations of the three fixed points.

Fig. 3 The co-dimension phase space combines the one-dimensional dynamics along the position x with the control parameter. For a given value of c, there are three or one solution for the fixed point. When there are three solutions, two are stable (the double minima) and one is unstable (the saddle). As the magnitude of the bias increases, one stable node annihilates with the unstable node (a minimum and the saddle merge) and the dynamics “switch” to the other minimum.

The threshold value in this example is c* = 1.05. When |c| < c* the two stable fixed points are the two minima of the double-well potential, and the unstable fixed point is the saddle between them. When |c| > c* then the single stable fixed point is the single minimum of the potential function. The saddle-node bifurcation takes its name from the fact (illustrated here) that the unstable fixed point is a saddle, and at the bifurcation the saddle point annihilates with one of the stable fixed points.

The following Python code illustrates the behavior of a biased double-well potential, with damping, in which the control parameter changes slowly with a sinusoidal time dependence.

Python Code

#!/usr/bin/env python3
# -*- coding: utf-8 -*-
Created on Wed Apr 17 15:53:42 2019

@author: nolte

import numpy as np
from scipy import integrate
from scipy import signal
from matplotlib import pyplot as plt

T = 400
Amp = 3.5

def solve_flow(y0,c0,lim = [-3,3,-3,3]):

    def flow_deriv(x_y, t, c0):
        #"""Compute the time-derivative of a Medio system."""
        x, y = x_y

        return [y,-0.5*y - x**3 + 2*x + x*(2*np.pi/T)*Amp*np.cos(2*np.pi*t/T) + Amp*np.sin(2*np.pi*t/T)]

    tsettle = np.linspace(0,T,101)   
    yinit = y0;
    x_tsettle = integrate.odeint(flow_deriv,yinit,tsettle,args=(T,))
    y0 = x_tsettle[100,:]
    t = np.linspace(0, 1.5*T, 2001)
    x_t = integrate.odeint(flow_deriv, y0, t, args=(T,))
    c  = Amp*np.sin(2*np.pi*t/T)
    return t, x_t, c

eps = 0.0001

for loop in range(0,100):
    c = -1.2 + 2.4*loop/100 + eps;
    coeff = [-1, 0, 2, c]
    y = np.roots(coeff)
    xtmp = np.real(y[0])
    ytmp = np.real(y[1])
    X[loop] = np.min([xtmp,ytmp])
    Y[loop] = np.max([xtmp,ytmp])
    Z[loop]= np.real(y[2])

lines = plt.plot(xc,X,xc,Y,xc,Z)
plt.setp(lines, linewidth=0.5)

y0 = [1.9, 0]
c0 = -2.

t, x_t, c = solve_flow(y0,c0)
y1 = x_t[:,0]
y2 = x_t[:,1]

lines = plt.plot(t,y1)
plt.setp(lines, linewidth=0.5)
plt.ylabel('X Position')

lines = plt.plot(c,y1)
plt.setp(lines, linewidth=0.5)
plt.ylabel('X Position')
plt.xlabel('Control Parameter')
plt.title('Hysteresis Figure')

Further Reading:

D. D. Nolte, Introduction to Modern Dynamics: Chaos, Networks, Space and Time (2nd Edition) (Oxford University Press, 2019)

The Pendulum Lab

Freeman Dyson’s Quantum Odyssey

In the fall semester of 1947, a brilliant young British mathematician arrived at Cornell University to begin a yearlong fellowship paid by the British Commonwealth.  Freeman Dyson (1923 –) had received an undergraduate degree in mathematics from Cambridge University and was considered to be one of their brightest graduates.  With strong recommendations, he arrived to work with Hans Bethe on quantum electrodynamics.  He made rapid progress on a relativistic model of the Lamb shift, inadvertently intimidating many of his fellow graduate students with his mathematical prowess.  On the other hand, someone who intimidated him, was Richard Feynman.

Initially, Dyson considered Feynman to be a bit of a buffoon and slacker, but he started to notice that Feynman could calculate QED problems in a few lines that took him pages.

Freeman Dyson at Princeton in 1972.

I think like most science/geek types, my first introduction to the unfettered mind of Freeman Dyson was through the science fiction novel Ringworld by Larry Niven. The Dyson ring, or Dyson sphere, was conceived by Dyson when he was thinking about the ultimate fate of civilizations and their increasing need for energy. The greatest source of energy on a stellar scale is of course a star, and Dyson envisioned an advanced civilization capturing all that emitted stellar energy by building a solar collector with a radius the size of a planetary orbit. He published the paper “Search for Artificial Stellar Sources of Infra-Red Radiation” in the prestigious magazine Science in 1960. The practicality of such a scheme has to be seriously questioned, but it is a classic example of how easily he thinks outside the box, taking simple principles and extrapolating them to extreme consequences until the box looks like a speck of dust. I got a first-hand chance to see his way of thinking when he gave a physics colloquium at Cornell University in 1980 when I was an undergraduate there. Hans Bethe still had his office at that time in the Newman laboratory. I remember walking by and looking into his office getting a glance of him editing a paper at his desk. The topic of Dyson’s talk was the fate of life in the long-term evolution of the universe. His arguments were so simple they could not be refuted, yet the consequences for the way life would need to evolve in extreme time was unimaginable … it was a bazaar and mind blowing experience for me as an undergrad … and and example of the strange worlds that can be imagined through simple physics principles.

Initially, as Dyson settled into his life at Cornell under Bethe, he considered Feynman to be a bit of a buffoon and slacker, but he started to notice that Feynman could calculate QED problems in a few lines that took him pages.  Dyson paid closer attention to Feynman, eventually spending more of his time with him than Bethe, and realized that Feynman had invented an entirely new way of calculating quantum effects that used cartoons as a form of book keeping to reduce the complexity of many calculations.  Dyson still did not fully understand how Feynman was doing it, but knew that Feynman’s approach was giving all the right answers.  Around that time, he also began to read about Schwinger’s field-theory approach to QED, following Schwinger’s approach as far as he could, but always coming away with the feeling that it was too complicated and required too much math—even for him! 

Road Trip Across America

That summer, Dyson had time to explore America for the first time because Bethe had gone on an extended trip to Europe.  It turned out that Feynman was driving his car to New Mexico to patch things up with an old flame from his Los Alamos days, so Dyson was happy to tag along.  For days, as they drove across the US, they talked about life and physics and QED.  Dyson had Feynman all to himself and began to see daylight in Feynman’s approach, and to understand that it might be consistent with Schwinger’s and Tomonaga’s field theory approach.  After leaving Feynman in New Mexico, he travelled to the University of Michigan where Schwinger gave a short course on QED, and he was able to dig deeper, talking with him frequently between lectures. 

At the end of the summer, it had been arranged that he would spend the second year of his fellowship at the Institute for Advanced Study in Princeton where Oppenheimer was the new head.  As a final lark before beginning that new phase of his studies he spent a week at Berkeley.  The visit there was uneventful, and he did not find the same kind of open camaraderie that he had found with Bethe in the Newman Laboratory at Cornell, but it left him time to think.  And the more he thought about Schwinger and Feynman, the more convinced he became that the two were equivalent.  On the long bus ride back east from Berkeley, as he half dozed and half looked out the window, he had an epiphany.  He saw all at once how to draw the map from one to the other.  What was more, he realized that many of Feynman’s techniques were much simpler than Schwinger’s, which would significantly simplify lengthy calculations.  By the time he arrived in Chicago, he was ready to write it all down, and by the time he arrived in Princeton, he was ready to publish.  It took him only a few weeks to do it, working with an intensity that he had never experienced before.  When he was done, he sent the paper off to the Physical Review[1].

Dyson knew that he had achieved something significant even though he was essentially just a second-year graduate student, at least from the point of view of the American post-graduate system.  Cambridge was a little different, and Dyson’s degree there was more than the standard bachelor’s degree here.  Nonetheless, he was now under the auspices of the Institute for Advanced Study, where Einstein had his office, and he had sent off an unsupervised manuscript for publication without any imprimatur from the powers at be.  The specific power that mattered most was Oppenheimer, who arrived a few days after Dyson had submitted his manuscript.  When he greeted Oppenheimer, he was excited and pleased to hand him a copy.  Oppenheimer, on the other hand, was neither excited nor pleased to receive it.  Oppenheimer had formed a particularly bad opinion of Feynman’s form of QED at the conference held in the Poconos (to read about Feynman’s disaster at the Poconos conference, see my blog) half-a-year earlier and did not think that this brash young grad student could save it.  Dyson, on his part, was taken aback.  No one who has ever met Dyson would ever call him brash, but in this case he fought for a higher cause, writing a bold memo to Oppenheimer—that terrifying giant of a personality—outlining the importance of the Feynman theory.

Battle for the Heart of Quantum Field Theory 

Oppenheimer decided to give Dyson a chance, and arranged for a series of seminars where Dyson could present the story to the assembled theory group at the Institute, but Dyson could make little headway.  Every time he began to make progress, Oppenheimer would bring it crashing to a halt with scathing questions and criticisms.  This went on for weeks, until Bethe visited from Cornell.  Bethe by then was working with the Feynman formalism himself.  As Bethe lectured in front of Oppenheimer, he seeded his talk with statements such as “surely they had all seen this from Dyson”, and Dyson took the opportunity to pipe up that he had not been allowed to get that far.  After Bethe left, Oppenheimer relented, arranging for Dyson to give three seminars in one week.  The seminars each went on for hours, but finally Dyson got to the end of it.  The audience shuffled out of the seminar room with no energy left for discussions or arguments.  Later that day, Dyson found a note in his box from Oppenheimer saying “Nolo Contendre”—Dyson had won!

With that victory under his belt, Dyson was in a position to communicate the new methods to a small army of postdocs at the Institute, supervising their progress on many outstanding problems in quantum electrodynamics that had resisted calculations using the complicated Schwinger-Tomonaga theory.  Feynman, by this time, had finally published two substantial papers on his approach[2], which added to the foundation that Dyson was building at Princeton.  Although Feynman continued to work for a year or two on QED problems, the center of gravity for these problems shifted solidly to the Institute for Advanced Study and to Dyson.  The army of postdocs that Dyson supervised helped establish the use of Feynman diagrams in QED, calculating ever higher-order corrections to electromagnetic interactions.  These same postdocs were among the first batch of wartime-trained theorists to move into faculty positions across the US, bringing the method of Feynman diagrams with them, adding to the rapid dissemination of Feynman diagrams into many aspects of theoretical physics that extend far beyond QED [3].

As a graduate student at Berkeley in the 1980’s I ran across a very simple-looking equation called “the Dyson equation” in our graduate textbook on relativistic quantum mechanics by Sakurai. The Dyson equation is the extraordinarily simple expression of an infinite series of Feynman diagrams that describes how an electron interacts with itself through the emission of virtual photons that link to virtual electron-positron pairs. This process leads to the propagator Green’s function for the electron and is the starting point for including the simple electron in more complex particle interactions.

The Dyson equation for the single-electron Green’s function represented as an infinite series of Feynman diagrams.

I had no feel for the use of the Dyson equation, barely limping through relativistic quantum mechanics, until a few years later when I was working at Lawrence Berkeley Lab with Mirek Hamera, a visiting scientist from Warwaw Poland who introduced me to the Haldane-Anderson model that applied to a project I was working on for my PhD. Using the theory, with Dyson’s equation at its heart, we were able to show that tightly bound electrons on transition-metal impurities in semiconductors acted as internal reference levels that allowed us to measure internal properties of semiconductors that had never been accessible before. A few years later, I used Dyson’s equation again when I was working on small precipitates of arsenic in the semiconductor GaAs, using the theory to describe an accordion-like ladder of electron states that can occur within the semiconductor bandgap when a nano-sphere takes on multiple charges [4].

The Coulomb ladder of deep energy states of a nano-sphere in GaAs calculated using self-energy principles first studied by Dyson.

I last saw Dyson when he gave the Hubert James Memorial Lecture at Purdue University in 1996. The title of his talk was “How the Dinosaurs Might Have Been Saved: Detection and Deflection of Earth-Impacting Bodies”. As always, his talk was wild and wide ranging, using the simplest possible physics to derive the most dire consequences of our continued existence on this planet.

[1] Dyson, F. J. (1949). “THE RADIATION THEORIES OF TOMONAGA, SCHWINGER, AND FEYNMAN.” Physical Review 75(3): 486-502.

[2] Feynman, R. P. (1949). “THE THEORY OF POSITRONS.” Physical Review 76(6): 749-759.  Feynman, R. P. (1949). “SPACE-TIME APPROACH TO QUANTUM ELECTRODYNAMICS.” Physical Review 76(6): 769-789.

[3] Kaiser, D., K. Ito and K. Hall (2004). “Spreading the tools of theory: Feynman diagrams in the USA, Japan, and the Soviet Union.” Social Studies of Science 34(6): 879-922.

[4] Nolte, D. D. (1998). “Mesoscopic Point-like Defects in Semiconductors.” Phys. Rev. B58(12): pg. 7994

Georg Duffing’s Equation

Although coal and steam launched the industrial revolution, gasoline and controlled explosions have sustained it for over a century.  After early precursors, the internal combustion engine that we recognize today came to life in 1876 from the German engineers Otto and Daimler with later variations by Benz and Diesel.  In the early 20th century, the gasoline engine was replacing coal and oil in virtually all mobile conveyances and had become a major industry attracting the top mechanical engineering talent.  One of those talents was the German engineer Georg Duffing (1861 – 1944) whose unlikely side interest in the quantum mechanics revolution brought him to Berlin to hear lectures by Max Planck, where he launched his own revolution in nonlinear oscillators.

The publication of this highly academic book by a nonacademic would establish Duffing as the originator of one of the most iconic oscillators in modern dynamics.

An Academic Non-Academic

Georg Duffing was born in 1861 in the German town of Waldshut on the border with Switzerland north of Zurich.  Within a year the family moved to Mannheim near Heidelberg where Georg received a good education in mathematics as well as music.  His mathematical interests attracted him to engineering, and he built a reputation that led to an invitation to work at Westinghouse in the United States in 1910.  When he returned to Germany he set himself up as a consultant and inventor with the freedom to move where he wished.  In early 1913 he wished to move to Berlin where Max Planck was lecturing on the new quantum mechanics at the University.  He was always searching for new knowledge, and sitting in on Planck’s lectures must have made him feel like he was witnessing the beginnings of a new era.            

At that time Duffing was interested in problems related to brakes, gears and engines.  In particular, he had become fascinated by vibrations that often were the limiting factors in engine performance.  He stripped the problem of engine vibration down to its simplest form, and he began a careful and systematic study of nonlinear oscillations.  While in Berlin, he had became acquainted with Prof. Meyer at the University who had a mechanical engineering laboratory.  Meyer let Duffing perform his experiments in the lab on the weekends, sometime accompanied by his eldest daughter.  By 1917 he had compiled a systematic investigation of various nonlinear effects in oscillators and had written a manuscript that collected all of this theoretical and experimental work.  He extended this into a small book that he published with Vieweg & Sohn in 1918 to be purchased for a price of 5 Deutsch Marks [1].   The publication of this highly academic book by a nonacademic would establish Duffing as the originator of one of the most iconic oscillators in modern dynamics.

Fig. 1 Cover of Duffing’s 1918 publication on nonlinear oscillators.

Duffing’s Nonlinear Oscillator

The mathematical and technical focus of Duffing’s book was low-order nonlinear corrections to the linear harmonic oscillator.  In one case, he considered a spring that either became stiffer or softer as it stretched.  This happens when a cubic term is added to the usual linear Hooke’s law.  In another case, he considered a spring that was stiffer in one direction than another, making the stiffness asymmetric.  This happens when a quadratic term is added.  These terms are shown in Fig. 2 from Duffing’s book.  The top equation is a free oscillation, and the bottom equation has a harmonic forcing function.  These were the central equations that Duffing explored, plus the addition of damping that he considered in a later chapter as shown in Fig. 3. The book lays out systematically, chapter by chapter, approximate and series solutions to the nonlinear equations, and in special cases described analytically exact solutions (such as for the nonlinear pendulum).

Fig. 2 Duffing’s equations without damping for free oscillation and driven oscillation with quadratic (producing an asymmetric potential) and cubic (producing stiffening or softening) corrections to the spring force.
Fig. 3 Inclusion of damping in the case with cubic corrections to the spring force.

Duffing was a practical engineer as well as a mathematical one, and he built experimental systems to test his solutions.  An engineering drawing of his experimental test apparatus is shown in Fig. 4. The small test pendulum is at S in the figure. The large pendulum at B is the drive pendulum, chosen to be much heavier than the test pendulum so that it can deliver a steady harmonic force through spring F1 to the test system. The cubic nonlinearity of the test system was controlled through the choice of the length of the test pendulum, and the quadratic nonlinearity (the asymmetry) was controlled by allowing the equilibrium angle to be shifted from vertical. The relative strength of the quadratic and cubic terms was adjusted by changing the position of the mass at G. Duffing derived expressions for all the coefficients of the equations in Fig. 1 in terms of experimentally-controlled variables. Using this apparatus, Duffing verified to good accuracy his solutions for various special cases.

Fig. 4 Duffing’s experimental system he used to explore and verify his equations and solutions.

           Duffing’s book is a masterpiece of careful systematic investigation, beginning in general terms, and then breaking the problem down into its special cases, finding solutions for each one with accurate experimental verifications. These attributes established the importance of this little booklet in the history of science and technology, but because it was written in German, most of the early citations were by German scientists.  The first use of Duffing’s name associated to the nonlinear oscillator problem occurred in 1928 [2], as was the first reference to him in a work in English in a book by Timoshenko [3].  The first use of the phrase “Duffing Equation” specifically to describe an oscillator with a linear and cubic restoring force was in 1942 in a series of lectures presented at Brown University [4], and this nomenclature had become established by the end of that decade [5].  Although Duffing had spent considerable attention in his book to the quadratic term for an asymmetric oscillator, the term “Duffing Equation” now refers to the stiffening and softening problem rather than to the asymmetric problem.

Fig. 5 The Duffing equation is generally expressed as a harmonic oscillator (first three terms plus the harmonic drive) modified by a cubic nonlinearity and driven harmonically.

Duffing Rediscovered

Nonlinear oscillations remained mainly in the realm of engineering for nearly half a century, until a broad spectrum of physical scientists began to discover deep secrets hiding behind the simple equations.  In 1963 Edward Lorenz (1917 – 2008) of MIT published a paper that showed how simple nonlinearities in three equations describing the atmosphere could produce a deterministic behavior that appeared to be completely chaotic.  News of this paper spread as researchers in many seemingly unrelated fields began to see similar signatures in chemical reactions, turbulence, electric circuits and mechanical oscillators.  By 1972 when Lorenz was invited to give a talk on the “Butterfly Effect” the science of chaos was emerging as new frontier in physics, and in 1975 it was given its name “chaos theory” by James Yorke (1941 – ).  By 1976 it had become one of the hottest new areas of science. 

        Through the period of the emergence of chaos theory, the Duffing oscillator was known to be one of the archetypical nonlinear oscillators.  A particularly attractive aspect of the general Duffing equations is the possibility of studying a “double-well” potential.  This happens when the “alpha” in the equation in Fig. 5 is negative and the “beta” is positive.  The double-well potential has a long history in physics, both classical and modern, because it represents a “two-state” system.  For a fixed “beta” the potential energy as a function of “alpha” is shown in Fig. 6.  The bifurcation cascades of the double-well Duffing equation was investigated by Phillip Holmes (1945 – ) in 1976 [6], and the properties of the strange attractor were demonstrated in 1978 [7] by Yoshisuke Ueda (1936 – ).  Holmes, and others, continued to do detailed work on the chaotic properties of the Duffing oscillator, helping to make it one of the most iconic systems of chaos theory.

Fig. 6 Potential energy of the Duffing Oscillator. The position variable is x, and changing alpha is along the other axis. For positive beta and alpha the potential is a quartic. For positive beta and negative alpha the potential is a double well.

Python Code for the Duffing Oscillator

This Python code uses the simple ODE solver on the driven-damped Duffing double-well oscillator to display the configuration-space trajectories and the Poincaré map of the strange attractor.

#!/usr/bin/env python3
# -*- coding: utf-8 -*-
Created on Wed May 21 06:03:32 2018
@author: nolte
import numpy as np
import matplotlib as mpl
from mpl_toolkits.mplot3d import Axes3D
from scipy import integrate
from matplotlib import pyplot as plt
from matplotlib import cm
import time
import os


# model_case 1 = Pendulum
# model_case 2 = Double Well
print(' ')

alpha = -1       # -1
beta = 1         # 1
delta = 0.3       # 0.3
gam = 0.15    # 0.15
w = 1
def flow_deriv(x_y_z,tspan):
    x, y, z = x_y_z
    a = y
    b = delta*np.cos(w*tspan) - alpha*x - beta*x**3 - gam*y
    c = w
T = 2*np.pi/w

px1 = np.random.rand(1)
xp1 = np.random.rand(1)
w1 = 0

x_y_z = [xp1, px1, w1]

# Settle-down Solve for the trajectories
t = np.linspace(0, 2000, 40000)
x_t = integrate.odeint(flow_deriv, x_y_z, t)
x0 = x_t[39999,0:3]

tspan = np.linspace(1,20000,400000)
x_t = integrate.odeint(flow_deriv, x0, tspan)
siztmp = np.shape(x_t)
siz = siztmp[0]

y1 = x_t[:,0]
y2 = x_t[:,1]
y3 = x_t[:,2]
lines = plt.plot(y1[1:2000],y2[1:2000],'ko',ms=1)
plt.setp(lines, linewidth=0.5)

for cloop in range(0,3):

#phase = np.random.rand(1)*np.pi;
    phase = np.pi*cloop/3

    repnum = 5000
    px = np.zeros(shape=(2*repnum,))
    xvar = np.zeros(shape=(2*repnum,))
    cnt = -1
    testwt = np.mod(tspan-phase,T)-0.5*T;
    last = testwt[1]
    for loop in range(2,siz):
        if (last < 0)and(testwt[loop] > 0):
            cnt = cnt+1
            del1 = -testwt[loop-1]/(testwt[loop] - testwt[loop-1])
            px[cnt] = (y2[loop]-y2[loop-1])*del1 + y2[loop-1]
            xvar[cnt] = (y1[loop]-y1[loop-1])*del1 + y1[loop-1]
            last = testwt[loop]
            last = testwt[loop]
    if cloop == 0:
        lines = plt.plot(xvar,px,'bo',ms=1)
    elif cloop == 1:
        lines = plt.plot(xvar,px,'go',ms=1)
        lines = plt.plot(xvar,px,'ro',ms=1)

Fig. 7 Strange attractor of the double-well Duffing equation for three selected phases.

[1] G. Duffing, Erzwungene Schwingungen bei veranderlicher Eigenfrequenz und ihre technische Bedeutung, Vieweg & Sohn, Braunschweig, 1918.

[2] Lachmann, K. “Duffing’s vibration problem.” Mathematische Annalen 99: 479-492. (1928)

[3] S. Timoshenko, Vibration Problems in Engineering, D. Van Nostrand Company, Inc.,New York, 1928.

[4] K.O. Friedrichs, P. Le Corbeiller, N. Levinson, J.J. Stoker, Lectures on Non-Linear Mechanics delivered at Brown University, New York, 1942.

[5] Kovacic, I. and M. J. Brennan, Eds. The Duffing Equation: Nonlinear Oscillators and their Behavior. Chichester, United Kingdom, Wiley. (2011)

[6] Holmes, P. J. and D. A. Rand. “Bifurcations of Duffings Equation – Application of Catastrophe Theory.” Journal of Sound and Vibration 44(2): 237-253. (1976)

[7] Ueda, Y. “Randomly Transitional Phenomena in the System Governed by Duffings Equation.” Journal of Statistical Physics 20(2): 181-196. (1979)

Feynman and the Dawn of QED

In the years immediately following the Japanese surrender at the end of WWII, before the horror and paranoia of global nuclear war had time to sink into the psyche of the nation, atomic scientists were the rock stars of their times.  Not only had they helped end the war with a decisive stroke, they were also the geniuses who were going to lead the US and the World into a bright new future of possibilities.  To help kick off the new era, the powers in Washington proposed to hold a US meeting modeled on the European Solvay Congresses.  The invitees would be a select group of the leading atomic physicists: invitation only!  The conference was held at the Rams Head Inn on Shelter Island, at the far end of Long Island, New York in June of 1947.  The two dozen scientists arrived in a motorcade with police escort and national press coverage.  Richard Feynman was one of the select invitees, although he had done little fundamental work beyond his doctoral thesis with Wheeler.  This would be his first real chance to expound on his path integral formulation of quantum mechanics.  It was also his first conference where he was with all the big guns.  Oppenheimer and Bethe were there as well as Wheeler and Kramers, von Neumann and Pauling.  It was an august crowd and auspicious occasion.

Shelter Island and the Foundations of Quantum Mechanics

            The topic that had been selected for the conference was Foundations of Quantum Mechanics, which at that time meant quantum electrodynamics, known as QED, a theory that was at the forefront of theoretical physics, but mired in theoretical difficulties.  Specifically, it was waist deep in infinities that cropped up in calculations that went beyond the lowest order.  The theorists could do back-of-the-envelope calculations with ease and arrive quickly at rough numbers that closely matched experiment, but as soon as they tried to be more accurate, results diverged, mainly because of the self-energy of the electron, which was the problem that Wheeler and Feynman had started on at the beginning of his doctoral studies [1].  As long as experiments had only limited resolution, the calculations were often good enough.  But at the Shelter Island conference, Willis Lamb, a theorist-turned-experimentalist from Columbia University, announced the highest resolution atomic spectroscopy of atomic hydrogen ever attained, and there was a deep surprise in the experimental results.

An obvious photo-op at Shelter Island with, left to right: W. Lamb, Abraham Pais, John Wheeler (holding paper), Richard P. Feynman (holding pen), Herman Feschbach and Julian Schwinger.

            Hydrogen, of course, is the simplest of all atoms.  This was the atom that launched Bohr’s model, inspired Heisenberg’s matrix mechanics and proved Schrödinger’s wave mechanics.  Deviations from the classical Bohr levels, measured experimentally, were the testing grounds for Dirac’s relativistic quantum theory that had enjoyed unparalleled success until Lamb’s presentation at Shelter Island.  Lamb showed there was an exceedingly small energy splitting of about 200 parts in a billion that amounted to a wavelength of 28 cm in the microwave region of the electromagnetic spectrum.  This splitting was not predicted, nor could it be described, by the formerly successful relativistic Dirac theory of the electron. 

            The audience was abuzz with excitement.  Here was a very accurate measurement that stood ready for the theorists to test their theories on.  In the discussions, Oppenheimer guessed that the splitting was likely caused by electromagnetic interactions related to the self energy of the electron.  Victor Weisskopf of MIT with Julian Schwinger of Harvard suggested that, although the total energy calculations of each level might be infinite,  the difference in energy DE should be finite.  After all, in spectroscopy it is only the energy difference that is measured experimentally.  Absolute energies are not accessible directly to experiment.  The trick was how to subtract one infinity from another in a consistent way to get a finite answer.  Many of the discussions in the hallways, as well as many of the presentations, revolved around this question.  For instance, Kramers suggested that there should be two masses in the electron theory—one is the observed electron mass seen in experiments, and the second is a type of internal or bare mass of the electron to be used in perturbation calculations. 

            On the train ride up state after the Shelter Island Conference, Hans Bethe took out his pen and a sheaf of paper and started scribbling down ideas about how to use mass renormalization, subtracting infinity from infinity in a precise and consistent way to get finite answers in the QED calculations.  He made surprising progress, and by the time the train pulled into the station at Schenectady he had achieved a finite calculation in reasonable agreement with Lamb’s shift.  Oppenheimer had been right that the Lamb shift was electromagnetic in origin, and the suggestion by Weisskopf and Schwinger that the energy difference would be finite was indeed the correct approach.  Bethe was thrilled with his own progress and quickly wrote up a paper draft and sent a copy in letters to Oppenheimer and Weisskopf [2].  Oppenheimer’s reply was gracious, but Weisskopf initially bristled because he also had tried the calculations after the conference, but had failed where Bethe had succeeded.  On the other hand, both pointed out to Bethe that his calculation was non-relativistic, and that a relativistic calculation was still needed.

When Bethe returned to Cornell, he told Feynman about the success of his calculations but that a relativistic version was still missing. Feynman told him on the spot that he knew how to do it and that he would have it the next day. Feynman’s optimism was based on the new approach to relativistic quantum electrodynamics that he had been developing with the aid of his newly-invented “Feynman Diagrams”. Despite his optimism, he hit a snag that evening as he tried to calculate the self-energy of the electron. When he met with Bethe the next day, they both tried to to reconcile the calculations with Feynman’s new approach, but they failed to find a path through the calculations that made sense. Somewhat miffed, because he knew that his approach should work, Feynman got down to work in a way that he had usually avoided (he had always liked finding the “easy” path through tough problems). Over several intense months, he began to see how it all would work out.

           At the same time that Feynman was making progress on his work, word arrived at Cornell of progress being made by Julian Schwinger at Harvard.  Schwinger was a mathematical prodigy like Feynman, and also like Feynman had grown up in New York city, but they came from very different neighborhoods and had very different styles.  Schwinger was a formalist who pursued everything with precision and mathematical rigor.  He lectured calmly without notes in flawless presentations.  Feynman, on the other hand, did his physics by feel.  He made intuitive guesses and checked afterwards if they were right, testing ideas through trial and error.  His lectures ranged widely, with great energy, without structure, following wherever the ideas might lead.  This difference in approach and style between Schwinger and Feynman would have embarrassing consequences at the upcoming sequel to the Shelter Island conference that was to be held in late March 1948 at a resort in the Pocono Mountains in Pennsylvania.

The Conference in the Poconos

           The Pocono conference was poised to be for the theorists Schwinger and Feynman what the Shelter Island had been for the experimentalists Rabi and Lamb—a chance to drop bombshells.  There was a palpable buzz leading up to the conference with advance word coming from Schwinger about his successful calculation of the g-factor of the electron and the Lamb shift.  In addition to the attendees who had been at Shelter Island, the Pocono conference was attended by Bohr and Dirac—two of the giants who had invented quantum mechanics.  Schwinger began his presentation first.  He had developed a rigorous mathematical method to remove the infinities from QED, enabling him to make detailed calculations of the QED corrections—a significant achievement—but the method was terribly complicated and tedious.  His presentation went on for many hours in his carefully crafted style, without notes, delivered like a speech.  Even so, the audience grew restless, and whenever Schwinger tried to justify his work on physical grounds, Bohr would speak up, and arguments among the attendees would ensue, after which Schwinger would say that all would become clear at the end.  Finally, he came to the end, where only Fermi and Bethe had followed him.  The rest of the audience was in a daze.

            Feynman was nervous.  It had seemed to him that Schwinger’s talk had gone badly, despite Schwinger’s careful preparation.  Furthermore, the audience was spent and not in a mood to hear anything challenging.  Bethe suggested that if Feynman stuck to the math instead of the physics, then the audience might not interrupt so much.  So Feynman restructured his talk in the short break before he was to begin.  Unfortunately, Feynman’s strength was in physical intuition, and although he was no slouch at math, he was guided by visualization and by trial and error.  Many of the steps in his method worked (he knew this because they gave the correct answers and because he could “feel” they were correct), but he did not have all the mathematical justifications.  What he did have was a completely new way of thinking about quantum electromagnetic interactions and a new way of making calculations that were far simpler and faster than Schwinger’s.  The challenge was that he relied on space-time graphs in which “unphysical” things were allowed to occur, and in fact were required to occur, as part of the sum over many histories of his path integrals.  For instance, a key element in the approach was allowing electrons to travel backwards in time as positrons.  In addition, a process in which the electron and positron annihilate into a single photon, and then the photon decays into an electron-positron pair, is not allowed by mass and energy conservation, but this is a possible history that must add to the sum.  As long as the time between the photon emission and decay is short enough to satisfy Heisenberg’s uncertainty principle, there is no violation of physics.

Feynman’s first published “Feynman Diagram” in the Physical Review (1948) [3] (Photograph reprinted from “Galileo Unbound” (D. Nolte, Oxford University Press, 2018)

            None of this was familiar to the audience, and the talk quickly derailed.  Dirac pestered him with questions that he tried to deflect, but Dirac persisted like a raven pecking at dead meat.  A question was raised about the Pauli exclusion principle, about whether an orbital could have three electrons instead of the required two, and Feynman said that it could (all histories were possible and had to be summed over), an answer that dismayed the audience.  Finally, as Feynman was drawing another of his space-time graphs showing electrons as lines, Bohr rose to his feet and asked whether Feynman had forgotten Heisenberg’s uncertainty principle that made it impossible to even talk about an electron trajectory.  It was hopeless.  Bohr had not understood that the diagrams were a shorthand notation not to be taken literally.  The audience gave up and so did Feynman.  The talk just fizzled out.  It was a disaster.

           At the close of the Pocono conference, Schwinger was the hero, and his version of QED appeared to be the right approach [4].  Oppenheimer, the reigning king of physics, former head of the successful Manhattan Project and newly selected to head the prestigious Institute for Advanced Study at Princeton, had been thoroughly impressed by Schwinger and thoroughly disappointed by Feynman.  When Oppenheimer returned to Princeton, a letter was waiting for him in the mail from a colleague he knew in Japan by the name of Sin-Itiro Tomonaga [5].  In the letter, Tomonaga described work he had completed, unbeknownst to anyone in the US or Europe, on a renormalized QED.  His results and approach were similar to Schwinger’s but had been accomplished independently in a virtual vacuum that surrounded Japan after the end of the war.  His results cemented the Schwinger-Tomonaga approach to QED, further elevating them above the odd-ball Feynman scratchings.  Oppenheimer immediately circulated the news of Tomonaga’s success to all the attendees of the Pocono conference.  It appeared that Feynman was destined to be a footnote, but the prevailing winds were about to change as Feynman retreated to Cornell. In defeat, Feynman found the motivation to establish his simplified yet powerful version of quantum electrodynamics. He published his approach in 1948, a method that surpassed Schwinger and Tomonaga in conceptual clarity and ease of calculation. This work was to catapult Feynman to the pinnacles of fame, becoming the physicist next to Einstein whose name was most recognizable, in that later half of the twentieth century, to the man in the street (helped by a series of books that mythologized his exploits [6]).

[1] See Chapter 8 “On the Quantum Footpath”, Galileo Unbound (Oxford, 2018)

[2] Schweber, S. S. QED and the men who made it : Dyson, Feynman, Schwinger, and Tomonaga. Princeton, N.J. :, Princeton University Press. (1994)

[3] Feynman, R. P. “Space-time Approach to Quantum Electrodynamics.” Physical Review 76(6): 769-789. (1949)

[4] Schwinger, J. “ON QUANTUM-ELECTRODYNAMICS AND THE MAGNETIC MOMENT OF THE ELECTRON.” Physical Review 73(4): 416-417. (1948)

[5] Tomonaga, S. “ON INFINITE FIELD REACTIONS IN QUANTUM FIELD THEORY.” Physical Review 74(2): 224-225. (1948)

[6] Surely You’re Joking, Mr. Feynman!: Adventures of a Curious Character, Richard Feynman, Ralph Leighton (contributor), Edward Hutchings (editor), 1985, W W Norton,

Dirac: From Quantum Field Theory to Antimatter

Paul Adrian Maurice Dirac (1902 – 1984) was given the moniker of “the strangest man” by Niels Bohr while he was reminiscing about the many great scientists with whom he had worked over the years [1].  It is a moniker that resonates with the innumerable “Dirac stories” that abound in the mythology of the hallways of physics departments around the world.  Dirac was awkward, shy, a loner, rarely said anything, was completely literal, had not the slightest comprehension of art or poetry, nor any clear understanding of human interpersonal interaction.  Dirac was also brilliant, providing the theoretical foundation for the central paradigm of modern physics—quantum field theory.  The discovery of the Higgs boson in 2012, a human achievement that capped nearly a century of scientific endeavor, rests solidly on the theory of quantum fields that permeate space.  The Higgs particle, when it pops into existence at the Large Hadron Collider in Geneva, is a singular quantum excitation of the Higgs field, a field that usually resides in a vacuum state, frothing with quantum fluctuations that imbue all particles—and you and me—with mass.  The Higgs field is Dirac’s legacy.

… all of a sudden he had a new equation with four-dimensional space-time symmetry.

Copenhagen and Bohr

Although Dirac as a young scientist was initially enthralled with relativity theory, he was working under Ralph Fowler (1889 – 1944) in the physics department at Cambridge in 1923 when he had the chance to read advanced proofs of Heisenberg’s matrix mechanics paper.  This chance event launched him on his own trajectory in quantum theory.  After Dirac was awarded his doctorate from Cambridge in 1926, he received a stipend that sent him to work with Niels Bohr (1885 – 1962) in Copenhagen—ground zero of the new physics. During his time there, Dirac became famous for taking long walks across Copenhagen as he played about with things in his mind, performing mental juggling of abstract symbols, envisioning how they would permute and act.  His attention was focused on the electromagnetic field and how it interacted with the quantized states of atoms.  Although the electromagnetic field was the classical field of light, it was also the quantum field of Einstein’s photon, and he wondered how the quantized harmonic oscillators of the electromagnetic field could be generated by quantum wavefunctions acting as operators.  But acting on what?  He decided that, to generate a photon, the wavefunction must operate on a state that had no photons—the ground state of the electromagnetic field known as the vacuum state.

            In late 1926, nearing the end of his stay in Copenhagen with Bohr, Dirac put these thoughts into their appropriate mathematical form and began work on two successive manuscripts.  The first manuscript contained the theoretical details of the non-commuting electromagnetic field operators.  He called the process of generating photons out of the vacuum “second quantization”.  This phrase is a bit of a misnomer, because there is no specific “first quantization” per se, although he was probably thinking of the quantized energy levels of Schrödinger and Heisenberg.  In second quantization, the classical field of electromagnetism is converted to an operator that generates quanta of the associated quantum field out of the vacuum (and also annihilates photons back into the vacuum).  The creation operators can be applied again and again to build up an N-photon state containing N photons that obey Bose-Einstein statistics, as they must, as required by their integer spin, agreeing with Planck’s blackbody radiation. 

            Dirac then went further to show how an interaction of the quantized electromagnetic field with quantized energy levels involved the annihilation and creation of photons as they promoted electrons to higher atomic energy levels, or demoted them through stimulated emission.  Very significantly, Dirac’s new theory explained the spontaneous emission of light from an excited electron level as a direct physical process that creates a photon carrying away the energy as the electron falls to a lower energy level.  Spontaneous emission had been explained first by Einstein more than ten years earlier when he derived the famous A and B coefficients, but Einstein’s arguments were based on the principle of detailed balance, which is a thermodynamic argument.  It is impressive that Einstein’s deep understanding of thermodynamics and statistical mechanics could allow him to derive the necessity of both spontaneous and stimulated emission, but the physical mechanism for these processes was inferred rather than derived. Dirac, in late 1926, had produced the first direct theory of photon exchange with matter.  This was the birth of quantum electrodynamics, known as QED, and the birth of quantum field theory [2].

Fig. 1 Paul Dirac in his early days.

Göttingen and Born

            Dirac’s next stop on his postodctoral fellowship was in Göttingen to work with Max Born (1882 – 1970) and the large group of theoreticians and mathematicians who were like electrons in a cloud orbiting around the nucleus represented by the new quantum theory.  Göttingen was second only to Copenhagen as the Mecca for quantum theorists.  Hilbert was there and von Neumann too, as well as the brash American J. Robert Oppenheimer (1904 – 1967) who was finishing his PhD with Born.  Dirac and Oppenheimer struck up an awkward friendship.  Oppenheimer was considered arrogant by many others in the group, but he was in awe of Dirac who arrived with his manuscript on quantum electrodynamics ready for submission.  Oppenheimer struggled at first to understand Dirac’s new approach to quantizing fields, but he quickly grasped the importance, as did Pascual Jordan (1902 – 1980), who was also in Göttingen.

            Jordan had already worked on ideas very close to Dirac’s on the quantization of fields.  He and Dirac seemed to be going down the same path, independently arriving at very similar conclusions around the same time.  In fact, Jordan was often a step ahead of Dirac, tending to publish just before Dirac, as with non-commuting matrices, transformation theory and the relationship of canonical transformations to second quantization.  However, Dirac’s paper on quantum electrodynamics was a masterpiece in clarity and comprehensiveness, launching a new field in a way that Jordan had not yet achieved with his own work.  But because of the closeness of Jordan’s thinking to Dirac’s, he was able to see immediately how to extend Dirac’s approach.  Within the year, he published a series of papers that established the formalism of quantum electrodynamics as well as quantum field theory.  With Pauli, he systematized the operators for creation and annihilation of photons [3].  With Wigner, he developed second quantization for de Broglie matter waves, defining creation and annihilation operators that obeyed the Pauli exclusion principle of electrons[4].  Jordan was on a roll, forging ahead of Dirac on extensions of quantum electrodynamics and field theory, but Dirac was about to eclipse Jordan once and for all.

St. John’s at Cambridge

            At the end of the Spring semester in 1927, Dirac was offered a position as a fellow of St. John’s College at Cambridge, which he accepted, returning to England to begin his life as a college professor.  During the summer and into the Fall, Dirac returned to his first passion in physics, relativity, which had yet to be successfully incorporated into quantum physics.  Oskar Klein and Walter Gordon had made initial attempts at formulating relativistic quantum theory, but they could not correctly incorporate the spin properties of the electron, and their wave equation had the bad habit of producing negative probabilities.  Probabilities went negative because the Klein-Gordon equation had two time derivatives instead of one.  The reason it had two (while the non-relativistic Schrödinger equation has only one) is because space-time symmetry required the double space derivative of the Schrödinger equation to be paired with a double time derivative.  Dirac, with creative insight, realized that the problem could be flipped by requiring the single time derivative to be paired with a single space derivative.  The problem was that a single space derivative did not seem to make any sense [5].

St. John’s College at Cambridge

            As Dirac puzzled how to get an equation with only single derivatives, he was playing around with Pauli spin matrices and hit on a simple identity that related the spin matrices to the electron momentum.  At first he could not get the identity to apply to four-dimensional relativistic momenta using the usual 2×2 spin matrices.  Then he realized that four-dimensional space-time could be captured if he expanded Pauli’s 2×2 spin matrices to 4×4 spin matrices, and all of a sudden he had a new equation with four-dimensional space-time symmetry with single derivatives on space and time.  As a test of his new equation, he calculated fine details of the experimentally-measured hydrogen spectrum, known as the fine structure, which had resisted theoretical explanation, and he derived answers in close agreement with experiment.  He also showed that the electron had spin-1/2, and he calculated its magnetic moment.  He finished his manuscript at the end of the Fall semester in 1927, and the paper was published in early 1928[6].  His relativistic quantum wave equation was an instant sensation, becoming known for all time as “the Dirac Equation”.  He had succeeded at finding a correct and long-sought relativistic quantum theory where many before had failed.  It was a crowning achievment, placing Dirac firmly in the firmament of the quantum theorists.

Fig. 1 The relativistic Dirac equation. The wavefunction is a four-component spinor. The gamma-del product is a 4×4 matrix operator. The time and space derivatives are both first-order operators.


            In the process of ridding the Klein-Gordon equation of negative probability, which Dirac found abhorent, his new equation created an infinite number of negative energy states, which he did not find abhorent.  It is perhaps a matter of taste what one theoriest is willing to accept over another, and for Dirac, negative energies were better than negative probabilities.  Even so, one needed to deal with an infinite number of negative energy states in quantum theory, because they are available to quantum transitions.  In 1929 and 1930, as Dirac was writing his famous textbook on quantum theory, he became intrigued by the similarity between the positive and negative electron states of the vacuum and the energy levels of valence electrons on atoms.  An electron in a state outside a filled electron shell behaves very much like a single-electron atom, like sodium and lithium with their single valence electrons.  Conversely, an atomic shell that has one electron less than a full complement can be described as having a “hole” that behaves “as if” it were a positive particle.  It is like a bubble in water.  As water sinks, the bubble rises to the top of the water level.  For electrons, if all the electrons go one way in an electric field, then the hole goes the opposite direction, like a positive charge. 

            Dirac took this analogy of nearly-filled atomic shells and applied it to the vacuum states of the electron, viewing the filled negative energy states like the filled electron shells of atoms.  If there is a missing electron, a hole in this infinite sea, then it would behave as if it had positive charge.  Initially, Dirac speculated that the “hole” was the proton, and he even wrote a paper on that possibility.  But Oppenheimer pointed out that the idea was inconsistent with observations, especially the inability of the electron and proton to annihilate, and that the ground state of the infinite electron sea must be completely filled. Hermann Weyl further pointed out that the electron-proton theory did not have the correct symmetry, and Dirac had to rethink.  In early 1931 he hit on an audacious solution to the puzzle.  What if the hole in the infinite negative energy sea did not just behave like a positive particle, but actually was a positive particle, a new particle that Dirac dubbed the “anti-electron”?  The anti-electron would have the same mass as the electron, but would have positive charge. He suggested that such particles might be generated in high-energy collisions in vacuum, and he finished his paper with the suggestion that there also could be an anti-proton with the mass of the proton but with negative charge.  In this singular paper, titled “Quantized Singularities of the Electromagnetic Field” published in 1931, Dirac predicted the existence of antimatter.  A year later the positron was discovered by Carl David Anderson at Cal Tech.  Anderson had originally called the particle the positive electron, but a journal editor of the Physical Review changed it to positron, and the new name stuck.

Fig. 3 An electron-positron pair is created by the absorption of a photon (gamma ray). Positrons have negative energy and can be viewed as a hole in a sea of filled electron states. (Momentum conservation is satisfied if a near-by heavy particle takes up the recoil momentum.)

            The prediction and subsequent experimental validation of antmatter stands out in the history of physics in the 20th Century.  In previous centuries, theory was performed mainly in the service of experiment, explaining interesting new observed phenomena either as consequences of known physics, or creating new physics to explain the observations.  Quantum theory, revolutionary as a way of understanding nature, was developed to explain spectroscopic observations of atoms and molecules and gases.  Similarly, the precession of the perihelion of Mercury was a well-known phenomenon when Einstein used his newly developed general relativity to explain it.  As a counter example, Einstein’s prediction of the deflection of light by the Sun was something new that emerged from theory.  This is one reason why Einstein became so famous after Eddington’s expedition to observe the deflection of apparent star locations during the total eclipse.  Einstein had predicted something that had never been seen before.  Dirac’s prediction of the existence of antimatter similarly is a triumph of rational thought, following the mathematical representation of reality to an inevitable conclusion that cannot be ignored, no matter how wild and initially unimaginable it is.  Dirac went on to receive the Nobel prize in Physics in 1933, sharing the prize that year with Schrödinger (Heisenberg won it the previous year in 1932).

[1] Framelo, “The Strangest Man: The Hidden Life of Paul Dirac” (Basic Books, 2011)

[2] Dirac, P. A. M. (1927). “The quantum theory of the emission and absorption of radiation.” Proceedings of the Royal Society of London Series A114(767): 243-265.;  Dirac, P. A. M. (1927). “The quantum theory of dispersion.” Proceedings of the Royal Society of London Series A114(769): 710-728.

[3] Jordan, P. and W. Pauli, Jr. (1928). “To quantum electrodynamics of free charge fields.” Zeitschrift Fur Physik 47(3-4): 151-173.

[4] Jordan, P. and E. Wigner (1928). “About the Pauli’s equivalence prohibited.” Zeitschrift Fur Physik 47(9-10): 631-651.

[5] This is because two space derivatives measure the curvative of the wavefunction which is related to the kinetic energy of the electron.

[6] Dirac, P. A. M. (1928). “The quantum theory of the electron.” Proceedings of the Royal Society of London Series A 117(778): 610-624.;  Dirac, P. A. M. (1928). “The quantum theory of the electron – Part II.” Proceedings of the Royal Society of London Series A118(779): 351-361.

Physics and the Zen of Motorcycle Maintenance

When I arrived at Berkeley in 1981 to start graduate school in physics, the single action I took that secured my future as a physicist, more than spending scores of sleepless nights studying quantum mechanics by Schiff or electromagnetism by Jackson —was buying a motorcycle!  Why motorcycle maintenance should be the Tao of Physics was beyond me at the time—but Zen is transcendent.


The Quantum Sadistics

In my first semester of grad school I made two close friends, Keith Swenson and Kent Owen, as we stayed up all night working on impossible problem sets and hand-grading a thousand midterms for an introductory physics class that we were TAs for.  The camaraderie was made tighter when Keith and Kent bought motorcycles and I quickly followed suit, buying my first wheels –– a 1972 Suzuki GT550.    It was an old bike, but in good shape and ready to ride, so the three of us began touring around the San Francisco Bay Area together on weekend rides.  We went out to Mt. Tam, or up to Vallejo, or around the North and South Bay.  Kent thought this was a very cool way for physics grads to spend their time and he came up with a name for our gang –– the “Quantum Sadistics”!  He even made a logo for our “colors” that was an eye shedding a tear drop shaped like the dagger of a quantum raising operator.

At the end of the first year, Keith left the program, not sure he was the right material for a physics degree, and moved to San Diego to head up the software arm of a start-up company that he had founder’s shares in.  Kent and I continued at Berkeley, but soon got too busy to keep up the weekend rides.  My Suzuki was my only set of wheels, so I tooled around with it, keeping it running when it really didn’t want to go any further.  I had to pull its head and dive deep into it to adjust the rockers.  It stayed together enough for a trip all the way down Highway 1 to San Diego to visit Keith and back, and a trip all the way up Highway 1 to Seattle to visit my grandparents and back, having ridden the full length of the Pacific Coast from Tijuana to Vancouver.  Motorcycle maintenance was always part of the process.

Andrew Lange

After a few semesters as a TA for the large lecture courses in physics, it was time to try something real and I noticed a job opening posted on a bulletin board.  It was for a temporary research position in Prof. Paul Richard’s group.  I had TA-ed for him once, but knew nothing of his research, and the interview wasn’t even with him, but with a graduate student named Andrew Lange.  I met with Andrew in a ground-floor lab on the south side of Birge Hall.  He was soft-spoken and congenial, with round architect glasses, fine sandy hair and had about him a hint of something exotic.  He was encouraging in his reactions to my answers.  Then he asked if I had a motorcycle.  I wasn’t sure if he already knew, or whether it was a test of some kind, so I said that I did.  “Do you work on it?”, he asked.  I remember my response.  “Not really,” I said.  In my mind I was no mechanic.  Adjusting the overhead rockers was nothing too difficult.  It wasn’t like I had pulled the pistons.

“It’s important to work on your motorcycle.”

For some reason, he didn’t seem to like my answer.  He probed further.  “Do you change the tires or the oil?”.  I admitted that I did, and on further questioning, he slowly dragged out my story of pulling the head and adjusting the cams.  He seemed to relax, like he had gotten to the bottom of something.  He then gave me some advice, focusing on me with a strange intensity and stressing very carefully, “It’s important to work on your motorcycle.”

I got the job and joined Paul Richards research group.  It was a heady time.  Andrew was designing a rocket-borne far-infrared spectrometer that would launch on a sounding rocket from Nagoya, Japan.  The spectrometer was to make the most detailed measurements ever of the cosmic microwave background (CMB) radiation during a five-minute free fall at the edge of space, before plunging into the Pacific Ocean.  But the spectrometer was missing a set of key optical elements known as far-infrared dichroic beam splitters.  Without these beam splitters, the spectrometer was just a small chunk of machined aluminum.  It became my job to create these beam splitters.  The problem was that no one knew how to do it.  So with Andrew’s help, I scanned the literature, and we settled on a design related to results from the Ulrich group in Germany.

Our spectral range was different than previous cases, so I created a new methodology using small mylar sheets, patterned with photolithography, evaporating thin films of aluminum on both sides of the mylar.  My first photomasks were made using an amazingly archaic technology known as rubylith that had been used in the 70’s to fabricate low-level integrated circuits.  Andrew showed me how to cut the fine strips of red plastic tape at a large scale that was then photo-reduced for contract printing.  I modeled the beam splitters with equivalent circuits to predict the bandpass spectra, and learned about Kramers-Kronig transforms to explain an additional phase shift that appeared in the interferometric tests of the devices.  These were among the first metamaterials ever created (although this was before that word existed), with an engineered magnetic response for millimeter waves.  I fabricated the devices in the silicon fab on the top floor of the electrical engineering building on the Berkeley campus.  It was one of the first university-based VLSI fabs in the country, with high-class clean rooms and us in bunny suits.  But I was doing everything but silicon, modifying all their carefully controlled processes in the photolithography bay.  I made and characterized a full set of 5 of these high-tech beam splitters–right before I was ejected from the lab and banned.  My processes were incompatible with the VLSI activities of the rest of the students.  Fortunately, I had completed the devices, with a little extra material to spare.

I rode my motorcycle with Andrew and his friends around the Bay Area and up to Napa and the wine country.  One memorable weekend Paul had all his grad students come up to his property in Mendocino County to log trees.  Of course, we rode up on our bikes.  Paul’s land was high on a coastal mountain next to the small winery owned by Charles Kittel (the famous Kittel of “Solid State Physics”).  The weekend was rustic.  The long-abandoned hippie-shack on the property was uninhabitable so we roughed it.  After two days of hauling and stacking logs, I took a long way home riding along dark roads under tall redwoods.

Andrew moved his operation to the University of Nagoya, Japan, six months before the launch date.  The spectrometer checked out perfectly.  As launch day approached, it was mounted into the nose cone of the sounding rocket, continuing to pass all calibration tests.  On the day of launch, we held our breath back in Berkeley.  There was a 12 hour time difference, then we received the report.  The launch was textbook perfect, but at the critical moment when the explosive nose-cone bolts were supposed to blow, they failed.  The cone stayed firmly in place, and the spectrometer telemetered back perfect measurements of the inside of the rocket all the way down until it crashed into the Pacific, and the last 9 months of my life sank into the depths of the Marianas Trench.  I read the writing on the thin aluminum wall, and the following week I was interviewing for a new job up at Lawrence Berkeley Laboratory, the DOE national lab high on the hill overlooking the Berkeley campus.

Eugene Haller

The  instrument I used in Paul Richard’s lab to characterize my state-of-the-art dichroic beamsplitters was a far-infrared Fourier-transform spectrometer that Paul had built using a section of 1-foot-diameter glass sewer pipe.  Bob McMurray, a graduate student working with Prof. Eugene Haller on the hill, was a routine user of this makeshift spectrometer, and I had been looking over Bob’s shoulder at the interesting data he was taking on shallow defect centers in semiconductors.   The work sounded fascinating, and as Andrew’s Japanese sounding rocket settled deeper into the ocean floor, I arranged to meet with Eugene Haller in his office at LBL.

I was always clueless about interviews.  I never thought about them ahead of time, and never knew what I needed to say.  On the other hand, I always had a clear idea of what I wanted to accomplish.  I think this gave me a certain solid confidence that may have come through.  So I had no idea what Eugene was getting at as we began the discussion.  He asked me some questions about my project with Paul, which I am sure I answered with lots of details about Kramers-Kronig and the like.  Then came the question strangely reminiscent of when I first met Andrew Lange:  Did I work on my car?  Actually, I didn’t have a car, I had a motorcycle, and said so.  Well then, did I work on my motorcycle?  He had that same strange intensity that Andrew had when he asked me roughly the same question.  He looked like a prosecuting attorney waiting for the suspect to incriminate himself.  Once again, I described pulling the head and adjusting the rockers and cams.

Eugene leaned back in his chair and relaxed.  He began talking in the future tense about the project I would be working on.  It was a new project for the new Center for Advanced Materials at LBL, for which he was the new director.  The science revolved around semiconductors and especially a promising new material known as GaAs.  He never actually said I had the job … all of a sudden it just seemed to be assumed.  When the interview was over, he simply asked me to give him an answer in a few days if I would come up and join his group.

I didn’t know it at the time, by Eugene had a beautiful vintage Talbot roadster that was his baby.  One of his loves was working on his car.  He was a real motor head and knew everything about the mechanics.  He was also an avid short-wave radio enthusiast and knew as much about vacuum tubes as he did about transistors.  Working on cars (or motorcycles) was a guaranteed ticket into his group.  At a recent gathering of his former students and colleagues for his memorial, similar stories circulated about that question:  Did you work on your car?  The answer to this one question mattered more than any answer you gave about physics.

I joined Eugene Haller’s research group at LBL in March of 1984 and received my PhD on topics of semiconductor physics in 1988.  My association with his group opened the door to a post-doc position at AT&T Bell Labs and then to a faculty position at Purdue University where I currently work on the physics of oncology in medicine and have launched two biotech companies—all triggered by the simple purchase of a motorcycle.

Andrew Lange’s career was particularly stellar.  He joined the faculty of Cal Tech, and I was amazed to read in Science magazine in 2004 or 2005, in a section called “Nobel Watch”, that he was a candidate for the Nobel Prize for his work on BoomerAng that had launched and monitored a high-altitude balloon as it circled the South Pole taking unprecedented data on the CMB that constrained the amount of dark matter in the universe.  Around that same time I invited Paul Richards to Purdue to give our weekly physics colloquium to talk about his own work on MAXIMA. There was definitely a buzz going around that the BoomerAng and MAXIMA collaborations were being talked about in Nobel circles. The next year, the Nobel Prize of 2006 was indeed awarded for work on the Cosmic Microwave Background, but to Mather and Smoot for their earlier work on the COBE satellite.

Then, in January 2010, I was shocked to read in the New York Times that Andrew, that vibrant sharp-eyed brilliant physicist, was found lifeless in a hotel room, dead from asphyxiation.  The police ruled it a suicide.  Apparently few had known of his life-long struggle with depression, and it had finally overwhelmed him.  Perhaps he had sold his motorcycle by then.  But I wonder—if he had pulled out his wrenches and gotten to work on its engine, whether he might have been enveloped by the zen of motorcycle maintenance and the crisis would have passed him by.  As Andrew had told me so many years ago, and I wish I could have reminded him, “It’s important to work on your motorcycle.”

Physicists in Revolution: 1848

The opening episode of Victoria on Masterpiece Theatre (PBS) this season finds the queen confronting widespread unrest among her subjects who are pressing for more freedoms and more say in government. Louis-Phillipe, former King of France, has been deposed in the February Revolution of 1848 in Paris and his presence at the Royal Palace does not help the situation.

In 1848 a wave of spontaneous revolution swept across Europe.  It was not a single revolution of many parts, but many separate revolutions with similar goals.  Two essential disruptions of life occurred in the early 1800’s.  The first was the partitioning of Europe at the Congress of Vienna from 1814 to 1815, presided over by Prince Metternich of Austria, that had carved up Napoleon’s conquests and sought to establish a stable order based on the old ideal of absolute monarchy.  In the process, nationalities were separated or suppressed.  The second was the industrialization of Europe in the early 1800’s that created economic upheaval, with masses of working poor fleeing effective serfdom in the fields and flocking to the cities.  Wages fell, food became scarce, legions of the poor and starving bloomed.  Because of these influences, European society had become unstable, supercooled beyond a phase transition and waiting for a seed or catalyst to crystalize the continent into a new state of matter. 

When the wave came, physicists across Europe were caught in the upheaval.  Some were caught up in the fervor and turned their attention to national service, some lost their standing and their positions during the inevitable reactionary backlash, others got the opportunities of their careers.  It was difficult for anyone to be untouched by the 1848 revolutions, and physicist were no exception.

The Spontaneous Fire of Revolution

The extraodinary wave of revolution was sparked by a small rebellion in Sicily in January 1848 that sought to overturn the ruling Bourbons.  It was a small rebellion of little direct consequence to Europe, but it succeeded in establishing a liberal democracy in an independent state that stood as a symbol of what could be achieved by a determined populace.  The people of Paris took notice, and in the sudden and unanticipated February Revolution, the French constitutional monarchy under Louis-Phillipe was overthrown in a few days and replaced by the French Second Republic.  The shock of Louis-Phillipe’s fall reverberated across Europe, feared by those in power and welcomed by those who sought a new world order.  Nationalism, liberalism, socialism and communism were on the rise, and the opportunity to change the world seemed to have arrived.  The Five Days of Milan in Italy, the March Revolution of the German states, the Polish rebellion against Prussia, and the Young Irelander Rebellion in Ireland were all consequences of the unstable conditions and the unprecidented opportunities for the people to enact change.  None of these uprisings were coordinated by any central group.  It was a spontaneous consequence of similar preconditions that existed across nearly all the states of Europe.

Arago and the February Revolution in Paris

The French were no newcomers to street rebellions.  Paris had a history of armed conflict between citizens manning barricades and the superior forces of the powers at be.  The unforgettable scene in Les Misérables of Marius at the barricade and Jean Valjean’s rescue through the sewers of Paris was based on the 1832 June Rebellion in Paris.  Yet this event was merely an echo of the much larger rebellion of 1830 that had toppled the unpopular monarchy of Charles X, followed by the ascension of the Bourgeois Monarch Louis Phillipe at the start of the July Monarchy.  Eighteen years later, Louis Phillipe was still on the throne and the masses were ready again for a change.  Alexis de Tocqueville saw the change coming and remarked, “We are sleeping together in a volcano. … A wind of revolution blows, the storm is on the horizon.”  The storm would sweep up a generation of participants, including the French physicist Francois Arago (1786 – 1853).

Lamartine in front of the Town Hall of Paris on 25 February 1848 (Image by Henri Félix Emmanuel Philippoteaux in public domain).

Arago is one of the under-appreciated French physicists of the 1800’s.  This may be because so many of his peers have become icons in the history of physics: Fourier, Fresnel, Poisson, Laplace, Malus, Biot and Foucault.  The one place where his name appears—the Spot of Arago—was not exclusively his discovery, but rather was an experimental demonstration of an effect derived by Poisson using Fresnel’s new theory of diffraction.  Poisson derived the phenomenon as a means to show the absurdity of Fresnel’s undulatory theory of light, but Arago’s experimental demonstration turned the tables on Poisson and the emissionists (followers of Newton’s particulate theory of light).  Yet Arago played a role behind the scenes as a supporter and motivator of some of the most important discoveries in optics.  In particular, it was Arago’s encouragement and support of the (at that time) unknown Fresnel, that helped establish the Fresnel theory of diffraction and the wave nature of light.  Together, Arago and Fresnel established the transverse nature of the light wave, and Arago is also the little-known discoverer of optical rotation.  As a young scientist, he attempted to measure the drift of the ether, which was a null experiment that foreshadowed the epochal experiments of Michelson and Morley 80 years later.  In his later years, Arago proposed the methodology for measuring the speed of light in both stationary and moving materials, which became the basis for the important measurements of the speed of light by Fizeau and Foucault (who also attempted to measure ether drift).

In addition to his duties as the director of the National Observatory and as the perpetual secretary of the Academie des Sciences (replacing Fourier), he entered politics in 1830 when he was elected as a member of the chamber of deputies.  At the fall of Louis-Phillipe in the February Revolution of 1848, he was appointed as a member of the steering committee of the newly formed government of the French Second Republic, and he was named head of the Marine and Colonies as well as the head of the Department of War.  Although he was a staunch republican and supporter of the people, his position put him in direct conflict with the later stages of the revolutions of 1848. 

The population of Paris became disenchanted with the conservative trends in the Second Republic.  In June of 1848 barricades were again erected in the streets of Paris, this time in opposition to the Republic.  Forces were drawn up on both sides, although many of the Republican forces defected to the insurgents, and attempts were made to mediate the conflict.  At the barricade on the rue Soufflot near the Pantheon, Arago himself approached the barricades to implore defenders to disperse.  It is a measure of the respect Arago held with the people when they replied, “Monsieur Arago, we are full of respect for you, but you have no right to reproach us.  You have never been hungry.  You don’t know what poverty is.” [1] When Arago finally withdrew, he feared that death and carnage were inevitable.  They came at noon on June 23 when the barricade at Porte Saint-Denis was attacked by the National Guards.  This started a general onslaught of all the barricades by Republican forces that left 1,500 workers dead in the streets and more than 11,000 arrested.  Arago resigned from the steering committee on June 24, although he continued to work in the government until the coup d’Etat by Louis Napolean, the nephew of Napoleon Bonaparte, in 1852 when he became Napoleon III, Emperor of the Second French Empire. Louis Napoleon demanded that all government workers take an oath of allegiance to him, but Arago refused.  Yet such was the respect that Arago commanded that Louis Napoleon let him continue unmolested as the astronomer of the Bureau des Longitudes.

Riemann and Jacobi and the March Revolution in Berlin

The February Revolution of Paris was followed a month later by the March Revolutions of the German States.  The center of the German-speaking world at that time was Vienna, and a demonstration by students broke out in Vienna on March 13. Emperor Ferdinand, following the advice of Metternich, called out the army who fired on the crowd, killing several protestors.  Throngs rallied to the protest and arms were distributed, readying for a fight.  Rather than risk unreserved bloodshed, the emperor dismissed Metternich who went into exile to London (following closely the footsteps of the French Louis-Phillipe).  Within the week, the revolutionary fervor had spread to Berlin where a student uprising marched on the royal palace of King Frederick Wilhelm IV on March 18.  They were met by 20,000 troops. 

The March 1848 revolution in Berlin (Image in the public domain).

Not all university students were liberals and revolutionaries, and there were numerous student groups that formed to support the King.  One of the students in one of these loyalist groups was a shy mathematician who joined a loyalist student militia to protect the King.  Bernhard Riemann (1826 – 1866) had come to the University of Berlin after spending a short time in the Mathematics department at the University in Göttingen.  Despite the presence of Gauss there, the mathematics department was not considered strong (this would change dramatically in about 50 years when Göttingen became the center of German mathematics with the arrival of Felix Klein, Karl Schwarzschild and Hermann Minkowski).  At Berlin, Riemann attended lectures by Steiner, Jacobi, Dirichlet and Eisenstein. 

On the night of the uprising, a nervous Riemann found himself among a group of students, few more than 20 years old, guarding the quarters of the King, not knowing what would unfold.  They spent a sleepless night that dawned on the chaos and carnage at the barricades at Alexander Platz with hundreds of citizens dead.  King Wilhelm was caught off guard by the events, and he assured the citizens that he would reorganize the government and yield to the demonstrator’s demands for parliamentary elections, a constitution, and freedom of the press.  Two days later the king attended a mass funeral for the fallen, attended by his generals and ministers who wore the german revolutionary tricolor of black, red and gold.  This ploy worked, and the unrest in Berlin died away before the king was forced to abdicate.  This must have relieved Riemann immensely, because this entire episode was entirely outside his usual meek and mild character.  Yet the character of all the unrelated 1848 revolutions had one thing in common: a sharp division among the populace between the liberals and the conservatives.  As Riemann had elected to join with the loyalists, one of his professors picked the other side.

Carl Gustav Jacob Jacobi (1804 – 1851) had been born in Potsdam and had obtained his first faculty position at the University of Königsberg where he was soon ranked among the top mathematicians in Europe.  However, in his early thirties he was stricken with diabetes, and the harsh winters of Königsberg became to difficult to bear.  He returned to the milder climate of Berlin to a faculty position at the university when the wave of revolution swept over the city.  Jacobi was a liberal thinker and was caught up in the movement, attending meetings at the Constitution Club.  Once the danger to Wilhelm IV had passed, the reactionary forces took their revenge, and Jacobi’s teaching stipend was suspended.  When he threatened to move to the University of Vienna, the royalists relented, so Jacobi too was able to weather the storm. 

The surprising footnote to this story is that Jacobi delivered lectures on a course on the application of differential equations to mechanics in the winter semester of 1847 – 1848 right in the midst of the political turmoil.  His participation in the extraordinary political events of that time apparently did not hamper him from giving one of the most extraordinary sets of lectures in mathematical physics.  Jacobi’s lectures of 1848 were the greatest advance in mathematical physics since Euler had reinterpreted Newton a hundred years earlier.  This is where Jacobi expanded on the work of Hamilton, establishing what is today called the Hamilton-Jacobi theory of dynamics.  He also derived and proved, using Liouville’s theorem of 1838, that the volume of phase space was an invariant in a conservative dynamical system [2].  It is tempting to imagine Jacobi returning home late at night, after rousing discussions of revolution at the Constitution Club, to set to work on his own revolutionary theories in physics.

Doppler and the Hungarian Revolution

Among all the states of Europe, the revolutions of 1848 posed the greatest threat to the Austrian Empire, which was a beaurocratic state entangling scores of diverse nationalities sprawled across the largest state of Europe.  The Austrian Empire was the remnant of the Holy Roman Empire that had succumbed to the Napoleonic invasion.  The lands that were controlled by Austria, after Metternich engineered the Congress of Vienna, included Poles, Ukranians, Romanians, Germans, Czechs, Slovaks, Hungarians, Slovenes, Serbs, Albanians and more.  Holding this diverse array of peoples together was already a challenge, and the revolutions of 1848 carried with them strong feelings of nationalism.  The revolutions spreading across Europe were the perfect catalyst to set off the Hungarian Revolution that grew into a war for independence, and the fierce fighting across Hungary could not be avoided even by cloistered physicists.

Christian Doppler (1803 – 1853) had moved in 1847 from Prague (where he had proposed what came to be called the Doppler effect in 1842 to the Royal Bohemian Society of Sciences) to the Academy of Mines and Forests in Schemnitz (modern Banská Štiavnica in Slovakia, but then part of the Kingdom of Hungary) with more pay and less work.  His health had been failing, and the strenuous duties at Prague had taken their toll.  If the goal of this move to an obscure school far from the center of Austrian power had been to lead a peaceful life, Doppler’s plans were sorely upset.

The news of the protests in Vienna arrived in Schemnitz on the 17th of March, and student demonstrations commenced immediately.  Amidst the uncertainty, Doppler requested a leave of absence from the summer semester and returned to Vienna.  It is not clear why he went there, whether to be near the center of excitement, or to take advantage of the free time to pursue his own researches.  While in Vienna he read a treatise before the Academy on galvano-electric effects.  He returned to Schemnitz in the Fall to relative peace, until the 12th of December, when the Hungarians rejected to acknowledge the new Emperor Franz Josef in Vienna, replacing his Uncle Ferdinand who was forced to abdicate, and the Hungarian war for independence began.

Görgey’s troops crossing the Sturec pass. Their ability to evade the Austrian pursuit was legendary (Image by Keiss Károly in the public domain).

One of Doppler’s former students from his days in Prague was appointed to command the newly formed Hungarian army.  General Arthur Görgey (1818 – 1916) moved to take possession of the northern mining towns (present day Slovakia) and occupied Schemnitz.  When Görgey learned that his old teacher was in the town he sent word to Doppler to meet him at his headquarters.  Meeting with a revolutionary and rebel could have marked Doppler as a traitor in Vienna, but he decided to meet him anyway, taking along one of his colleagues as a “witness” that the discussion were purely academic.  This meeting opens an interesting unsolved question in the history of physics. 

Around this time Doppler was interested in the dynamical properties of the pendulum for cases when the suspension wire was exceptionally long.  Experiments on such extreme pendula could provide insight into changes in gravity with height as well as the effects of the motion of the Earth.  For instance, Coriolis had published his paper on forces in rotating frames many years earlier in 1835.  Because Schemnitz was a mining town, there was ample access to deep mine shafts in which to set up a pendulum with a very long wire.  This is where the story becomes murky.  Within the family of Doppler’s descendants there are stories of Doppler setting up such an experiment, and even a night time visit to the Doppler house by Görgey.  The pendulum was thought to be one of the topics discussed by Doppler and Görgey at their first meeting, and Görgey (from his life as a scientist prior to becoming a revolution general) had arrived to help with the experiment [3]

This story is significant for two reasons.  First, it would be astounding to think of General Görgey taking a break from the revolution to do some physics for fun.  Görgey has not been graced by history with a benevolent reputation.  He was known as a hard and sometimes vicious leader, and towards the end of the short-lived Hungarian Revolution he displaced the President Kossuth to become the dictator of Hungary.  The second reason, which is important for the history of physics, is that if Doppler had performed this experiment in 1848, it would have preceded the famous experiment by Foucault by more than two years.  However, the paper published by Doppler around this time on the dynamics of the pendulum did not mention the experiment, and it remains an open question in the history of physics whether Doppler may have had priority over Foucault.

The Austrian Imperial Army laid siege to Schemnitz and commenced a short bombardment that displaced Görgey and his troops from the town.  Even as Schemnitz was being liberated, a letter arrived informing Doppler that his old mentor Stampfer at the University of Vienna was retiring and that he had been chosen to be his replacement.  The March Revolution had led to the abdication of the previous Austrian emperor and his replacement by the more liberal-minded Franz Josef who was interested in restructuring the educational system in the Austrian empire.  On the advice of Doppler’s supporters who were in the new government, the Institute of Physics was formed and Doppler was named as its first director.  He arrived in the spring of 1850 to take up his new post.

The Legacy of 1848

Despite the early successes and optimism of the revolutions of 1848, reactionary forces were quick to reverse many of the advances made for universal suffrage, constitutional government, freedom of the press, and freedom of expression.  In most cases, monarchs either retained power or soon returned.  Even the reviled Metternich returned to Vienna from exile in London in 1851.  Yet as is so often the case, once a door has been opened it is difficult to shut it again.  The pressure for reforms continued long after the revolutions faded away, and by 1870 many of the specific demands of the people had been instituted by most of the European states.  Russia was an exception, which may explain why the inevitable Russian Revolution half a century later was so severe.            

The revolutions of 1848 cannot be said to have had a long-lasting impact on the progress of physics, although they certainly had a direct impact on the lives of selected physicists.  The most lasting effect of the revolutions on science was the restructuring of educational systems, not only in Austria, but in many of the European states.  This was perhaps one of the first times when the social and economic benefits of science education to the national welfare was understood and implemented across Europe, although a similar recognition had occurred earlier during the French Revolution, for instance leading to the founding of the Ecole Polytechnique.  The most important, though subtle, effect of the revolutions of 1848 on society was the shift away from autocratic rule to democracy, and the freeing of expression and thought from rigid bounds.  The coming revolution in physics at the turn of the next century may have been helped a little by the revolutionary spirit that still echoed from 1848.

[1] pg. 201, Mike Rapport, “1848: Year of Revolution” (Basic Books, 2008)

[2] D. D. Nolte, The Tangled Tale of Phase Space, Chap. 6 in Galileo Unbound (Oxford University Press, 2018)

[3] Schuster, P. Moving the stars : Christian Doppler, his life, his works and principle, and the world after. Pöllauberg, Austria, Living Edition. (2005)