Transcript Quantum Mechanics: The Other Great Revolution of the 20th

Quantum Mechanics: The Other
Great Revolution of the 20th
Century – Part III
Michael Bass, Professor Emeritus
CREOL, The College of Optics and
Photonics
University of Central Florida
© M. Bass
Quantum Mechanics
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The formulations of W. Heisenberg and E.
Schrödinger.
How to interpret quantum phenomena and
what was the meaning of the things
Heisenberg and Schrödinger calculated?
Prior to 1925 quantum physics was a
“hodgepodge” of hypotheses, principles,
theorems and recipes.
It was not a logically consistent theory.
© M. Bass
Prior to 1925
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Every single quantum theory problem had to
be solved
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first, in terms of classical physics using the
Correspondence Principle, and then
translated into the language of quantum physics.
First you described such classical things as
position and momentum and then converted
them to their quantum physics analogues.
© M. Bass
1925
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Correspondence had enabled one to retain such
classical ideas as energy, momentum, angular
momentum, spin and so on. Then you had to tailor
the results to fit the experimental data.
Werner Heisenberg recognized that there was an
alternative to this cumbersome approach and
published it in the paper “On a quantum theoretical
interpretation of kinematical and mechanical
relations”
Within months of Heisenberg’s paper, Schrödinger
published his paper with the now famous “wave”
equation for quantum mechanics.
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“On a quantum theoretical interpretation of
kinematical and mechanical relations”
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To formulate a proper quantum theory you must
abandon any classical description of the motion
and instead describe nature by an analysis of
observable magnitudes.
Such observables were optical frequencies and
intensities. (Heisenberg actually referred to
dipole amplitudes.)
He struggled to present a formulation of
quantum phenomena that today we know as the
Matrix Approach. (More on this later.)
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We will see that his struggle was the result of not yet
knowing what a matrix was.
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Heisenberg the person
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Born in 1901 and so he was just 24 years old when he
revolutionized quantum mechanics.
During WWI he left school for farm work as his contribution
to the war effort.
The elements that formed his personal values:
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rebellion against non-ideal values, such as capitalism, materialism,
hypocrisy and moral decadence;
adherence to a close circle of likeminded friends united in a
"harmony of souls";
a love of nature;
and a deeply felt affinity with German culture.
© M. Bass
More history
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He graduated with his Ph. D. in 1923 working for
Arnold Sommerfeld at the University of Munich.
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At his thesis defense he could not answer
experimentalist Wilhelm Wien's questions about:
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"It may be that you know something; it may be that you
know nothing. We shall see."
the resolving power of optical instruments and how a
storage battery works.
Wien only let him pass after Sommerfeld vigorously
defended his pupil.
In 1925 he was working with Max Born at Gottingen
where both he and Pauli attended and were
influenced by Bohr’s lectures.
In 1926 he went to Coppenhagen to become Bohr’s
assistant.
© M. Bass
Under the Nazis
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In 1932 (the same year the Nazi’s came to power in
Germany) Heisenberg won the Nobel prize for his 1925
work.
Must emphasize no one claims Heisenberg was in fact a
Nazi though he participated in the Nazi scientific community
during the period from 1932-1945.
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Other German scientists were Nazis – e.g., Johannes Stark and
Phillip Lenard
Heisenberg stayed in Germany and worked in the Nazi
scientific establishment rising to be the “Oppenheimer” of
the Nazi atomic bomb project.
See the play “Coppenhagen” to try to understand the trip he
made in 1941 to try to recruit Bohr to work with the
Germans and the ambiguity surrounding his circumstances.
© M. Bass
At Farm Hill in 1945
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The Alsos mission led by Samuel Goudsmit scoured Europe
just behind the Allied troops to find German scientists, to
de-brief them, and to secure them from the Soviets.
The nuclear scientists were sent to Farm Hill in
England where they were when the Hiroshima and
Nagasaki bombs were dropped ending the hostilities.
Their rooms were bugged and they were overheard
discussing:
 How impossible it was for US scientists to have done it.
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After all the great Heisenberg had shown that it would require
using heavy water to moderate the reaction.
Besides, weren’t they, the Germans, the greatest scientists of all.
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More things overheard
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We, the German scientists, were so moral and ethical that we
were only working on the heavy water concept to develop
nuclear power plants in fuel short Germany.
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Later, Heisenberg tried to claim that he had been
trying to lead the “maniacs” away from nuclear
weaponry by his wrong calculations and his insistence
on heavy water. (He died in 1976 holding to this fairy tale.)
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They didn’t see the contradiction in this.
In fact, this idea was being suggested by a group of unrepentant Nazi
scientists as a cover story.
There is little to support this idea.
The Nazis would have summarily killed a deceiver and there were rabid
Nazi scientists like Stark who would have turned in the culprit.
I suspect Heisenberg was ashamed of himself and his
colleagues and wanted to improve his place in history.
© M. Bass
Back to 1925
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Heisenberg did not throw out correspondence.
He modified it into the foundations of his
approach.
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Quantum mechancs demands correspondence
as one of its fundamental tenets.
He insisted that all that matters is what can be
observed.
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In this Heisenberg, as was Einstein and Schrödinger,
strongly influenced by the philosophy of Ernst Mach.
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In fact, Heisenberg attributes to Einstein his concern with
the observable and nothing else.
© M. Bass
Matrices
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Heisenberg analyzed the Fourier components of time
dependent quantities and through a strange
multiplication method identified them with such
measurables as energy and dipole moment.
The “strange multiplication” was matrix multiplication
but no one yet knew that.
We will see that Pascual Jordan was the one to explain it
to Born and then Heisenberg.
It was this barrier, the lack of familiarity with matrices,
that prevented most physicists from taking to Heisenberg’s
formulation and led to the widespread pleasure they took
in Schrödinger’s yet to be published wave formulation.
© M. Bass
Matrix Mathematics
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Isn’t it uncanny how mathematics would be
prepared to support the new physics?
It was only in 1907, when Heisenberg was 6,
that M. Bocher published what was to become
the standard text on matrix theory in English
in New York.
It was 1910 before it was translated into
German and that was only one year after G.
Kowalewski’s treatise on determinants.
The second edition of Bocher’s book made it
into German in 1924 just in time for those few
who had read it to connect matrices to
Heisenberg’s ideas.
 a11 a12
a
 21 a22
a31 a32
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a41 a42
a13
a23
a33
a43
a14 
a24 
a34 
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a44 
AB  BA
© M. Bass
Reluctance
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Keep in mind that most physicists in 1925 did not know what
matrices were much less how to manipulate them.
Just consider that Pauli was to reject Born’s offer to “work
towards a logically consistent foundation of matrix
mechanics” because he was (along with others) reluctant to
apply them to theoretical problems.
Of course, this refusal allowed Pauli the time to explore the statistics
of systems of electrons, identify the exclusion principle, and win a
Nobel prize.
Pauli was not alone. Heisenberg’s paper was not reviewed in the
Phisikalishe Berichte, the official abstracts of the German Physical Society.
It was given only one sentence.
Similarly in the Physical Society of London/Institute of Electrical Engineers
Common Science Abstracts and in the American Physical Society
publications.
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© M. Bass
Even Fermi
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Emilio Segre describes Enrico Fermi’s attitude in 1926,
when leaving Born’s group in Gottingen to work in
Leyden,
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“Heisenberg’s great paper on matrix mechanics of 1925 did not
appear sufficiently clear to Fermi, who reached a full
understanding of quantum mechanics only later through
Schrödinger’s wave mechanics. I want to emphasize that this
attitude of Fermi was certainly not due to the mathematical
difficulties and novelty of matrix algebra (for Fermi such
difficulties were minor obstacles) but rather the physical ideas
underlying this paper that were alien to him.”
Born had to search further for someone to put matrix
mathematics on a firm footing.
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Quite by accident he found the someone in Pascual Jordan
while on a train trip.
© M. Bass
The fateful train trip
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Shortly after Heisenberg’s paper appeared there was to be a
meeting in Hanover, Germany.
On the train Born mentioned Heisenberg’s work to a colleague.
Sitting in the same compartment was a very young Pascual
Jordan who was going to the same conference.
Jordan overheard the conversation and, as they left the train,
had the temerity to approach the great professor, Born, and
comment that he, Jordan had some skills in mathematics and
recognized Born’s description as that of matrix manipulation.
Exactly 60 days later, on September 27, 1925 the Born-Jordan
paper was received at Zeitschrift fur Physik.
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It contained the pertinent theorems of matrix math and the proof of the
matrix equation
ih
pq  qp  
2
l
© M. Bass
More Collaboration
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Born and Jordan got together with
Heisenberg and only 2 months later another
paper arrived at Zeitschrift fur Physik.
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This was entitled “On Quantum Mechanics II”
This was a vast generalization of the two
previous papers including
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extension to systems of many degrees of
freedom,
canonical transformations, and
quantum mechanical treatments of timedependent and time-independent perturbations.
© M. Bass
Uncertainty
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xp  
Et  
From these analyses Heisenberg began to realize that
there were problems.
Pairs of canonical variables were connected to Planck’s
constant in such a manner that you could not know
both simultaneously with unlimited accuracy.
If you determined one with great accuracy, you lost
information about the other.
This, the Uncertainty Principle, has become a
fundamental feature of the quantum world and since
the whole universe was once very, very small, a
governing feature of what we can know about
everything.
 Think about it!
© M. Bass
Another conceptual
development
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Start from light, not the mechanics of particles.
 After all light is our main means to perceive the universe.
Light seemed to be both wave and particle like.
 It made its choice depending on how we observed it.
As far back as Hamilton, efforts had been under way to establish the
“mechanics of light” (remember Newton and his corpuscles of light).
De Broglie had shown that the things we thought were particles could
also be considered waves.
Then came Erwin Schrödinger who, followed a deliberate
mathematical path to the theory of wave mechanics.
 A synthesis of the wave-particle duality.
© M. Bass
Erwin Schrödinger
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Schrödinger was schooled in the classical “gymnasium”
learning both Latin and Greek and seriously studying
philosophy.
It took him just 4 years, 1906-1910, to receive his Ph. D. in
Vienna from Fritz Hazornel in theoretical matters, Franz
Exner in experimental work and Wilhelm Wirtinger in
mathematics.
In 1911 Exner took him on as an assistant.
Schrödinger became involved in the controversy between
atomicity and continuity of matter coming down strongly on
the side of atomicity.
 This was in part because Ludwig Boltzmann was his
scientific hero and pointed the way to an atomistic view
of matter.
Along the way he acquired a mastery of eigenvalue problems
that would serve him well later.
© M. Bass
A philosophical viewpoint
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Schrödinger was strongly influenced by the
philosophical writings of Spinoza, Schopenhauer,
Mach and others who formed the realist school of
philosophy.
Like Einstein and Heisenberg, Schrödinger believed
that what was important in physics was what could
be observed (e.g.: measured).
 It didn’t hurt that this theoretician had had an
experimentalist’s aspect of his Ph. D. studies and
an experimentalist’s assistantship.
This concern with observables is clear in his effort to
interpret his wave functions in terms of probability for
measurable event.
© M. Bass
Schrödinger's journey – part 1
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By 1920 he was unhappy in Vienna and went,
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first to work with Max Wien in Jena, and
then took an assistant professorship at the
Technische Hochschule in Stuttgart.
Both of these are engineering schools.
Then he went to Breslau and finally settled at
the University of Zurich in 1921 as the
successor to Peter Debye.
He stayed in Zurich until 1927 during which
time he developed wave mechanics.
© M. Bass
Schrödinger’s journey – part 2
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In 1927 Schrödinger moved to the University of
Berlin as the successor to Max Planck.
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This was the most prestigious position in German
science.
Schrödinger was recognized and honored.
However, five years later, deeply repelled by the
Nazis and their rise to power in Germany,
Schrödinger gave it up.
He left Germany for the Institute for Advanced
Study in Dublin and stayed there until 1956.
He returned to Vienna where he died in 1958.
© M. Bass
Debye’s choice
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When Debye identified Schrödinger as his replacement it
resulted in part from the fact that neither thought they
could understand some seminars given by de Broglie.
Debye asked Schrödinger to present some seminars on
the subject that greatly impressed Debye.
Schrödinger was chosen but he considered the seminars
as the starting point of his work on wave mechanics.
 In fact, he wrote later in his life that he had needed
the “pushing” of Debye before he realized that the
subject was attractive.
© M. Bass
Wave mechanics shaky start
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The seminars gave rise to an effort to generalize de
Broglie’s waves for bound particles.
Schrödinger finally found what he called a “neat” solution
that gave the energy levels as eigenvalues of a certain
operator (the Hamiltonian).
Then he applied this concept to the hydrogen atom and
got the wrong results.
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We now know that his error was due to not including spin.
Of course, you can’t fault him as spin had not yet been
discovered.
Schrödinger was so disappointed that he abandoned his
method as inadequate.
A few months later he returned to it and noticed that if
he treated the electron non relativistically he got
agreement with observation in the non relativistic limit.
© M. Bass
Schrödinger's triumph
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In January 1926 he sent his manuscript containing a
complex diffusion equation that he showed was a
wave equation for a complex function to Annalen der
Physik.
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Between January and June he submitted all four
parts of his wave mechanics including
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This manuscript included his results for the hydrogen atom.
both the time independent and time dependent Schrodinger
equations,
their solutions for the spherical hydrogen atom, and
Schrödinger's comments on the meaning of the wave
function and calculations of measurable quantities.
The rest, as they say, is history.
© M. Bass
Go with what you know
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Physicists immediately realized that they
were familiar with Schrödinger's equation.
They could follow his math and his statistical
interpretation of the meaning of the wave
function fit with the non causal concepts first
put forward by Einstein.
The problem remained, however, how do
you define an acceptable wave function.
In time it became clear that by insisting that
measurables were all that mattered then
only single valued, finite and continuous
wavefunctions could exist wherever the
potential energy was finite.
© M. Bass
Equivalence
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Eventually wave and matrix mechanics were
shown to give the same results and were, in
fact, completely equivalent.
In fact Korel Lanczos, Schrodinger and Pauli
were to do this independently.
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Lanczos was a colleague of Einstein and gave the
singularly most impenetrable lecture I ever heard.
Paul Adrian Maurice Dirac eventually, with
knowledge of spin, was able to develop a
fully relativistic form of wave mechanics.
© M. Bass
Philosophical difficulties
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It is still hard to accept that the most we can know
about something is the probability of its
happening.
Nevertheless, it is beyond doubt that quantum
mechanics works – it is the most precisely tested
theory we have and it works as advertised.
In fact, Richard Feynman and others demonstrated
that quantum concepts can remove singularities
from other theories making quantum phenomena
essential to the foundations of physics.
© M. Bass
Afterthoughts
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By the 1950s quantum mechanics had been shown to
be a fundamental feature of the universe.
It was to be proven in ever more exquisite
experiments over the next near half century.
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Optics played a critical role in this just as in the beginning it
provided the data that demanded quantum phenomena.
We all casually talk of energy levels, photon energy,
transitions between stable states and the like but we
rarely realize the incredible intellectual journey we
took to get here.
© M. Bass