Quantum Mechanics: The Other Great Revolution of the 20th Century – Part II

Michael Bass, Professor EmeritusCREOL, The College of Optics and

PhotonicsUniversity of Central Florida

© M. Bass

© 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.

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 = h E = h

E = En-1

E = En

h = 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/2.

The lowest energy state of an atomic system is the state when the angular momentum of every electron is just h/2.

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. 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.

When the quantum numbers are large the energy differences are small and the states close to one another. Almost a continuum.

© 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. +1 or -1 for circularly polarized light and 0 for light

polarized parallel to the axis of the system. Kramers (Bohr’s student) could write a thesis

entitled “Intensities of spectral lines” in which he finds:

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. 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).

hndqp kkk

© 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.

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.

B

© 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 non-

relativistic 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.