Download 10. Quantum Mechanics Part II

Quantum Mechanics: The Other
Great Revolution of the 20th
Century – Part II
Michael Bass, Professor Emeritus
CREOL, The College of Optics and
Photonics
University of Central Florida
© M. Bass
Quantum Mechanics


The revolution that was Quantum Mechanics
provided mathematical models that yielded
the features we observed.
But (and these are big “buts”) there were
problems such as:





We had wave-particle duality.
We had uncertainty.
What about correspondence,
complimentarity, and
the statistical interpretation?
© M. Bass
Models of the atom




In the late 1800s Helmholtz, Kelvin and others proposed
mechanical and hydrodynamic models of the atom.
When J. J. Thompson discovered the electron these became
meaningless.
In 1901 Jean Baptiste Perrin (Nobel Prize in 1926 “…
sedimentation equilibrium”) in Paris proposed a planetary
model with a positively charged nucleus and negatively
electrons circulating around it. He described them as
circulating as “petites planetes”.
Thompson proposed a model with a positively charged nucleus
and the electron oscillating at its center.

This was consistent with the electron-on-a-spring model of Drude,
Lorentz, Planck and Voigt.

Remember this gave dispersion, absorption and reflection pretty well.
© M. Bass
The victory of the planetary
model

Then Rutherford’s scattering
experiments demonstrated that
Thompson’s model had to be wrong.



There was a nucleus and it was
massive and positively charged.
Electrons circulated around it
somehow.
The “somehow” was up to Bohr to
describe.
© M. Bass
Bohr’s achievement


As discussed in Part I he stated the obvious and showed that quantization worked.
In June 1913 he wrote in “On the constitution of atoms and molecules”:

“In the investigation of the configuration of the electrons in the atom we
immediately meet with the difficulty that a ring, if only the strength of the
central charge and the number of electrons in the ring are given, can rotate
with an infinitely great number of different times of rotation (he meant angular
velocities), according to the assumed different radius of the ring; there seems
to be nothing at all, from mechanical consideration, to discriminate between
the different radii and times of vibration (he meant angular velocities). In the
further investigation we shall therefore introduce and make use of a hypothesis
from which we can determine the quantities in question. This hypothesis is
that for any stable ring (any ring occurring in the natural atoms)
there will be a definite ratio between the kinetic energy of an
electron in the ring and the time of rotation (angular velocity). This
hypothesis, for which no attempt at a mechanical foundation will be
given (as it seems hopeless), is chosen as the only one which seems
to offer the possibility of an explanation of the whole group of
experimental results, which gather about and seems to confirm
conceptions of the mechanisms of the radiation as the ones proposed
by Planck and Einstein”
© M. Bass
Bohr’s brilliance


Through stilted and odd grammar the
brilliance comes through.
 The Rutherford model of the atom
could not be reconciled with Newtonian
mechanics and Maxwell’s
electrodynamics.
E = hn
Atoms have stable states and when
the electrons stay in a stable state no
radiation is emitted but when the state of
the atom changes from one of these
states to another a quantum of
electromagnetic energy is released (or
absorbed) having energy equal to the
energy difference between the two states.
E = En
E = hn
E = En-1
hn = En – En-1
© M. Bass
The 1913 paper

This was the main event. In July 1913 Bohr published his
model.

He stated:






Energy is not emitted continuously but only when electrons change from
one stationary state to another.
While ordinary dynamics holds for systems in the stationary states it does
not while the systems pass from one such state to another.
The frequency of the radiation emitted when such a change takes place is
the energy difference divided by Planck’s constant.
For a simple system of an electron rotating around a positively charged
nucleus the stationary states are determined by requiring that the ratio of
the total energy and the frequency of revolution of the electron is an
integer multiple of h/2p.
The lowest energy state of an atomic system is the state when the angular
momentum of every electron is just h/2p.
He used these ideas to derive Rydberg’s constant from e, m, c and h.
© M. Bass
Discussions



In a seminar in Zurich in September 1913 the assembled
physicists generally accepted the results but had
philosophical difficulties.
 After all, this impudent Dane said there were times when
mechanics didn’t apply.
Max von Laue (Nobel Prize 1914 for x-ray diffraction) stated
emphatically that “this is all nonsense”.
Einstein rose from the audience (remember he was still only
at Prague – not Berlin and not yet a Nobel laureate) and
with some irony said,
 “Very remarkable. There must be something to it. I do
not believe that the derivation of the absolute value of
the Rydberg constant is purely fortuitous.”
© M. Bass
Public notice

James Jeans at a meeting of the British Association
for the Advancement of Science called Bohr’s model




“a most ingenious and suggestive, and I think we must add,
convincing explanation of the spectral series”
In reporting on Jeans’ comments the Times of
London referred to “Dr. Bohr’s ingenious explanation
of the hydrogen spectrum.”
Nature called it “convincing and brilliant…a simple,
plausible and easily amenable to mathematical
treatment model.”
In 1922 Bohr would receive the Nobel prize for his
“investigation of the structure of atoms and the
radiation emanating from them”

This is 1 year after Einstein received his Nobel prize for “services to
theoretical physics and his discovery of the photoelectric effect”
© M. Bass
Philosophical problems





Bohr himself regarded it as merely a preliminary and
hypothetical way of representing experimental facts.
This sort of plausibility argument is often made in
Physics but then the actual science must be done to
connect plausibility to actuality.
Such models point the way.
The conflicts between the quantum theoretic structures
and classical conceptions had to be resolved.
 Stationary states were not classical yet they had to
exist!!!
The resolution, in Bohr’s view, was to be in the
CORRESPONDENCE PRINCIPLE
© M. Bass
Correspondence




As early as 1906 Planck had shown that when h
approached 0 quantum theory converged to classical
physics.
 That is when the Planck distribution goes over to the
Rayleigh-Jeans law.
The general idea and a statement of the
Correspondence Principle is that quantum theory
must contain classical physics as a limit.
Clearly Planck would arrive at the same conclusion for a
finite value for h and very low frequency.
Bohr seized on this to formulate the Correspondence
Principle.
© M. Bass
Limits


In Bohr’s model low frequencies are cases
where the change in quantum number is
small compared to the quantum numbers
themselves.
In such case the results should approach
classical predictions.


When the
quantum
numbers are
large the energy
differences are
small and the
states close to
one another.
Almost a
continuum.
The frequencies of the emitted radiation would be
small and Planck had shown this approached
classical behavior.
Without this as a limit, quantum theory would
be incomplete.
© M. Bass
Why such worries


Classical theory allowed calculating not only
frequencies but intensities and polarizations.
If quantum theory was to be really valuable it
must do the same.



Since Bohr expressly disclaimed knowledge of the
mechanism of transition between stationary states his
model couldn’t, by itself, serve as a rational basis to
find intensities and polarizations.
Correspondence provided a way out.
The problem was how to do it.
© M. Bass
Einstein’s A and B coefficients



In 1916 Einstein published the crucial paper in which he
showed that by assuming stimulated transitions and
spontaneous transitions between states and Bohr’s
radiation frequency condition you would get
Planck’s radiation law for systems in equilibrium.
The key matter here is that Einstein states that an atom
(he actually used the word molecule) can pass from
one state to an energetically lower state “without
excitation by an external cause”.
Einstein himself pointed out that from this he was led in
an “amazingly simple and general way to Planck’s law.”
© M. Bass
Problems and the seeds of
disagreement



Einstein’s statistical approach proved the basis of
what would become the modern interpretation of
quantum mechanics.
It also provided the basis for Einstein’s later
difficulties with quantum mechanics.
 Recall the famous Einsteinism “God does not play
dice with the universe”
Bohr, however, saw in the renunciation of the causal
structure of transitions the way out!
 There were events that could only be
described by their probability of occurrence.
 Bohr and Einstein had to agree to disagree.
© M. Bass
By dropping causality

Bohr and colleagues could

Connect polarization of radiation with changes in
azimuthal quantum number.


Kramers (Bohr’s student) could write a thesis entitled
“Intensities of spectral lines” in which he finds:


+1 or -1 for circularly polarized light and 0 for light polarized
parallel to the axis of the system.
The relative intensities of the fine structure and Stark shifted
lines of H.
All of this because of Correspondence.

It was the philosophical underpinning of quantum
mechanics – it gave the conceptual construct under which
quantum mechanics could flourish.
© M. Bass
A problem?




In using Correspondence Bohr was resorting to using
classical physics to establish quantum physics.
This is inherently inconsistent as the assumptions of
quantum mechanics conflict with the classical theory Bohr
was using to justify quantum theory.
Finally, Bohr realized that the Correspondence Principle
must be regarded as “purely a law of quantum theory and
can not diminish the contrast between the underlying
assumptions of quantum and classical theories.”
It became the basis of the Copenhagen Interpretation
of quantum mechanics.
 That is, quantum transitions are statistically not
causally determined.
© M. Bass
A deeper meaning

Arnold Sommerfeld showed that quantum mechanics
is ideally suited to treatment using Hamilton’s
formulation of mechanics.



You need Correspondence to even think of doing this.
pk dqk  nk h
Sommerfeld showed that

This is the first clue that would lead to the
uncertainty principle.

It shows the relationship of two conjugate variables of
motion and the quantum principle (that is h is involved).
© M. Bass
More data keeps coming in

The Stern-Gerlach experiments in 1922 showed that when
a beam of atoms (H, Na, K, Cd, Th, Z, Cu, Ag, and Au) was
passed through a magnetic field the beam split into two
beams.
B



The degree of deflection of each was such that the
magnetic moment of the atoms was found to be easily
within 10% of the Bohr magneton.
What made the atoms know to go into their particular
beamlets?
Explanations were proposed but eventually found to violate
such things as energy conservation or requiring systems
only able to emit quantized radiation.
© M. Bass
Wolfgang Pauli




Wolfgang Pauli, an Austrian, received his Ph. D. working with
Sommerfeld in Munich and went to Gottingen as an assistant to
Max Born.
He attended a lecture by Bohr on the meaning of spherically
symmetric shells in the atomic model.
Pauli became obsessed as to why all electrons for an atom in its
ground state were not bound in the innermost shell as Bohr
seemed to claim.
In the fall of 1922 Pauli accepted Bohr’s invitation to
Copenhagen to assist him in a German version of his works.
 They collaborated!! and in time Pauli became convinced of
the “two-valuedness” of the electron.
 It gave him a way to count the number of electrons in the
stationary states.
© M. Bass
The Exclusion Principle




No two electrons in one system may have exactly the same set of
quantum numbers.
This idea enabled Pauli to account for the periodic table – no mean
feat.
But what was the “two-handedness”.
Samuel Goudsmit and George Uhlenbeck; spin; summer of 1925.
 “Zimple, dere vas nutting elze”
 They were very worried that they might be laughed at.
 After all Bohr, Heisenberg, Pauli and others never mentioned it.
 They showed it to Ehrenfest who loved the simple visualization it
gave
 He told them about Compton’s idea of a spinning electron to
explain the natural unit of magnetism.
 Either their idea was brilliant or it was nonsense.
 They should publish it.
© M. Bass
Lorentz’s objection






After giving Ehrenfest their article for Naturwissenshaften,
at his suggestion, they described it to Lorentz.
A week later Lorentz sent a carefully written paper showing
that the concept of a spinning electron led to so much
magnetic energy that it would be more massive than the
proton.
Thus, he, Lorentz, concluded that the spinning electron was
nonsense.
Goudsmit and Uhlenbeck were devastated.
Then Ehrenfest told them Lorentz was completely wrong –
he spoke more colorfully telling them they were too young
to see the “dumbness” of Lorentz’s paper.
Their paper appeared on November 20, 1925.
© M. Bass
The spinning electron



The spinning electron had two states, + or – ½.
In 1927 Pauli succeeded in formulating, in nonrelativistic quantum mechanics, using spin
matrices, a consistent theory of such an electron.
Paul Adrian Maurice Dirac would do it accounting
for Einstein’s relativity.

Shared the Nobel Prize in 1933 with Erwin Schrodinger
but for different work.
© M. Bass
Still more disturbing data

In 1927, Clinton J. Davisson and Lester H. Germer in the USA
were trying to study low energy scattering of electrons from
pure Ni.





They got the Nobel Prize in 1937 for this work.
As luck would have it their vacuum system had a leak and the
nickel became oxidized. Since their budget was limited they
tried to clean the sample by exposing it to a flow of heated
hydrogen gas.
Then when they did the experiment they found the electrons
scattered into specific angles that looked just like a diffraction
pattern of waves passing through the finely spaced layers in the
Ni crystals they had produced in the cleaning process.
The inescapable conclusion – electrons, though considered
particles, showed wavelike properties.
If waves were particle-like we now had
evidence that particles were wave-like.
© M. Bass
Prince Louis-Victor de Broglie





His genius was to have predicted particle waves four
years earlier.
This enabled him to understand the quantized
angular momentum of Bohr’s model to result from
constructive interference of the particle waves.
In 1929 he won the Nobel prize for this insight.
The quantum world was even stranger than first
thought.
It would get more so.
© M. Bass
The growing synthesis




Planck had shown waves were particle
like.
Einstein had shown energy and mass
were equivalent.
de Broglie had shown that particles
were wave like.
The distinction was blurring in the
quantum world.
© M. Bass
Some thoughts

“Anyone who is not shocked by
quantum theory does not understand it”


Niels Bohr, 1927
“Nobody understands quantum theory”

Richard Feynman, 1967
© M. Bass
Previews of Coming
Attractions

Next time in Quantum Mechanics, Part III we
will:





Look at the Heisenberg and Schrodinger
formulations.
Interpretations.
Uncertainty and its meaning.
Why the Schrodinger wave equation method was
accepted more easily than Heisenberg’s matrix
mechanics.
Heisenberg and the Nazi atomic bomb project.
© M. Bass