Quantum Mechanics 1

• First step toward the Quantum, Light in a Box. Take a box, in thermal equilibrium at a certain temperature, hot enough so that we can observe the light coming out of a little hole. Think oven.

– Along x, p waves, p=1,2,3…– Along y, q waves, q=1,2,3...– Along z, r waves, r=1,2,3…

• these are the “normal modes” of Maxwell wave equation. All have same average thermal energy for given temperature.

Quantum Mechanics 2

• Thermal equilibrium has quite a long history: thermodynamics and statistical theory of gases etc. Hard to escape that each degree of freedom must have the same average energy.

• We look at the inside of the box through the little hole as we heat the oven,

• dull red, cherry red, bright red, yellow, white-hot, bluish, etc.

• but p, q, and r go up to infinity, more shorter and shorter wavelengths, or wavenumbers kx

ky kz each go to infinity.• The number of k’s increases like the volume of

a sphere, so they crowd to big values, predict most light is very blue.

Quantum Mechanics 3

• We said “very blue” = violet comes after blue in the spectrum, so the prediction that the spectrum is crowded toward the short wavelengths was called the “ultraviolet catastrophe”

• Of course, there can’t be a catastrophe: something must save us! Once again, it is Maxwell vs. some other sacred belief. Last time it was Maxwell vs. Newton. Newton lost. Now it is Maxwell vs. equipartition.

Quantum Mechanics 4

Quantum Mechanics 5

Quantum Mechanics 6

• Planck thought this was a weird condition. It was soon seen (by Einstein) that this explains lots of other things too. The specific heat of a gas of hydrogen molecules, H2 is rather strange (= amount of energy to increase temperature by one degree)

• explained by needing kT at least enough to equal the energy of translation, rotation, vibration of the molecule

Quantum Mechanics 7

• Photoelectric effect: “photons” are real!---Einstein’s first home-run. (“Photon” is a term to describe the quantum of light.) The experiment, in a vacuum illuminated by a selected wavelength of light illuminating a clean metal surface:

Quantum Mechanics 8

• What would we expect? We are kicking electrons out of the metal. They didn’t just fall out, so they must be bound. Model binding by a spring, which we have to break to get the electron out. The electric field of the light wiggles the electron, pumping in energy until we break the spring. Expect:

1 there is a threshold in light intensity to get electrons and the kinetic energy goes up with light intensity

2 any wavelength will do, low frequencies more effective

3 it takes a long time (seconds to hours) to free the electrons

Quantum Mechanics 9

• What is seen:1 electrons come instantly when light is turned

on2 there is no threshold in light intensity, and

kinetic energy of the electrons independent of light intensity

3 there is a threshold in the frequency of the light to get any electrons, only the short wavelengths are effective, depending on the choice of metal (its work function W)

• Big, big discrepancy. Einstein saw all would become clear if the quantum of light was really a particle, the photon, with energy

E=hf

Quantum Mechanics 10

• Typical experiment results

Quantum Mechanics 11

• The Einstein theory

• It gives us everything we need, with precision agreement with the black body Planck fit.

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Quantum Mechanics 12

• Quick revisit of how we know that light is a wave (after all, Newton had decided that light was particles moving with c): Thomas Young (1801) was skillful enough to see interference from two slits:

• a recent improvement is a fast switch that can send light from one slit to a detector,

• after it has gone through the slit

Quantum Mechanics 13

• Compton said, let’s use a wave-type device to choose light (waves) of a given wave length and then scatter them from an electron, a billiard ball-type experiment, and then make another wave-type measurement to see if we can predict their new wavelength, if they still have one:

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Quantum Mechanics 14

• For the electron, the lightest particle around to

scatter from,

Compare to 500 nm for light, so not much effect

for visible light. We need light of wavelength,

say 0.02 nm, to get a big effect. Light of this

wavelength we call “X-rays” discovered by

Roentgen in 1895. They can be nicely

diffracted by the planes of atoms in a crystal,

and that is how Compton supplied the wave-

type measurement before and after scattering.

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Quantum Mechanics 15

• For just now, we can summarize the wave-particle puzzle by saying that when we do a wave-type experiment, light behaves like a wave. When we do a particle-type experiment, light behaves like a particle. When the two are combined, as in the Compton scattering experiment, the behavior of the light switches back and forth according to the demands made on it by the actual apparatus used. The key new point here is the fact that the measuring apparatus seems to affect the physical system, instead of just recording some pure reality.

Quantum Mechanics 16

• A look at atoms, with the quantum in mind.• Surely there is charge in atoms, and the

nature of the electron was clarified by J.J. Thompson, measured e/m and Milliken, measured e: electrons are light.

• Excited atoms give off definite frequencies of light, so something is ringing like a bell. It will be the light parts that are moving, the electrons. What is the structure? Plum Pudding model

Quantum Mechanics 17

• Breakthrough! Rutherford calculated the small deflection an alpha particle (Helium nucleus from a radioactive element) passing through a very thin foil of a heavy element (gold)

• The angles are expected to be small, because the positive charge is spread out so the field is relatively small, and while the negative charge is assumed to be concentrated on the electrons, the mass of the electron is 8000 times smaller than that of the Helium, so it can’t push the alpha around much at all. Rutherford designed an experiment to look at these small angles.

Quantum Mechanics 18

• In Rutherford’s words “It was as incredible as if you fired a 15-inch shell at a piece of tissue paper and it came back and hit you.”

• He could calculate that the only way this could happen was if all the positive charge of the atom was concentrated in a clump no more than 1/10,000 the diameter of the atom. He called this the nucleus, and started to study it.

• Meanwhile, this leaves us with puzzles about the structure of the atoms. What keeps the electrons from falling in to all that positive charge? “Planetary motion” you would think, but moving electrons radiate about 1% of their energy per turn. (Maxwell)

Quantum Mechanics 19

• Bohr saw that the quantum concept could be the answer, and he knew too, that he didn’t know enough to make a true model, so he tried something provisional, rather a new style. His propositions:

1. Use Newton’s Laws to calculate dynamics2. Electrons can only exist is certain special

orbits called stationary states.

3. The angular momentum L of the electrons moving around the nucleus is quantized in units of h/2 defined as the symbol pronounced “h bar,” or L= n .

4. When the atom makes a transition, between two states different by E, E=f the famous “quantum jumps.”

Quantum Mechanics 20

• The calculation is simple:

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Quantum Mechanics 21

• The energy of the atom is the sum of the kinetic energy and the potential energy:

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