HPS 2523 | History of Quantum Mechanics | Fall 2012 |

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Warm Up Exercises

The equipartition theorem of classical statistical physics assigns (1/2)kT of energy to each degree of freedom of a thermal system at equilibrium at temperature T, where k is Boltzmann's constant. It applies to systems whose Hamiltonian H is quadratic in the canonical phase space coordinates p_{1}, p_{2}, p_{3},..., q_{1}, q _{2}, q_{3},..., which is true of virtually all classical systems. The number of degrees of freedom is just the number of canonical phase space coordinates.

1. Classicaly, ordinary matter (gases, liquids, solids) have finitely many degrees of freedom; a radiation field has infinitely many. Why is this a problem?

2. A crystalline lattice, such as is used classically to model solids, is a system to which the equipartition theorem applies. What does the theorem entail for the specific heats of solids? Why is this a problem?

Bohr-Sommerfeld quantization applies to systems whose motion in their classical phase space is periodic. Not all motions are allowed, but just those that meet the condition:

nh = ∫p_{i} dq_{i}

where p_{i} and q_{i} are the i-th canonical coordinates of the phase space, h is Planck's constant, n is an integer (n=1,2,3, ...) and the integration is carried out over one complete cycle. There is one condition for each value of i.

Answer 1. OR 2. according to which takes your fancy.

1. A one-dimensional harmonic oscillator consists of a mass m moving under Hooke's law according to md^{2}x/dt^{2} = -kx, where x is the position at time t. It has a Hamiltonian H = (1/2)mv^{2} + (1/2)kx^{2}, where v = dx/dt.

a. Represent this system's phase space using canonical coordinates p and q.

b. Apply the Bohr-Sommerfeld condition to it. Which energies are allowed by the condition?

2. A one-dimensional rotor is a mass m that spins about a fixed central point on a rigid radial arm of length R. No external forces act on the rotor, so that its classically allowed motion is just rotation at any constant angular speed.

a. Represent this system's phase space using canonical coordinates p and q.

b. Apply the Bohr-Sommerfeld condition to it. Which angular momenta are allowed by the condition?

Locate expressions for the Rayleigh-Jeans law, the Wien law and the Planck law. Under which conditions does the Planck law revert to

(a) The Wien law; (b) The Rayleigh-Jeans law.

In Bohr's (1913) "On the Constitution of Atoms and Molecules," locate equation 4 for ν = frequency of radiation emitted when an electron drops from quantum number τ_{1} to τ_{2}. Consider the case in which τ_{1} is very large and τ_{1} = τ_{2}+ 1. Show that in this case, the frequency of emitted radiation approximately equals the orbital frequency of the electron, as given by Bohr's equation 2 for ω.

A charge e of mass m moves with initial velocity **v** in a constant magnetic field **H**. Show that:

(a) the charge will move uniformly in a circle;

(b) the frequency of rotation is eH/2πmc, which is independent of the speed v and orbit radius r; and

(c) the motion would have a reversed direction if the field had a reversed sign.

Assume that the force on the charge is **f** = (e/c)(**v**x**H**), where Gaussian units are used. Note that the condition for equilibrium for a circular orbit is that the centripetal force equals the centrifugal force mv^{2}/r, where r is the radius of the orbit. Neglect energy losses due to radiation.

1. We are used to seeing good reasons for adopting Einstein's light quantum hypothesis. At the time of BKS, what was a good reason for NOT adopting the light quantum hypothesis?

2. BKS propose to reduce the validity of the law of conservation of energy to a statistical regularity. In the early 1920s, in recent memory, another closely related law suffered a similar fate. What was it?

What is the Rydberg-Ritz rule of combination of atomic spectroscopy? How is it used in Heisenberg's paper? (Hint: It does not appear by name in Heisenberg's paper.)

1. The equation of motion of a one-dimensional harmonic oscillator of mass m, spring constant k is m d^{2}x/dt^{2} = -kx. Show this is equivalent to the equation of motion d^{2}x/dt^{2} + ω_{0}^{2}x = 0 and that the resonant frequency of the resulting motion is ω_{0} = (k/m)^{1/2}.

2. If this oscillator is acted on by an oscillating external force A sin ωt, then its equation of motion becomes

d

^{2}x/dt^{2}+ ω_{0}^{2}x = A sin ωt

Show that this equation is solved for the case of x(0) = dx(0)/dt = 0 by

x(t) = [ A/(ω

^{2}-ω_{0}^{2}) ] . [ (ω/ω_{0})sinω_{0}t - sinωt ]

3. Qualitatively, describe what happens to this motion as the frequency of the external force ω approaches the resonant frequency ω_{0} of the oscillator.

4. (Bonus question for those with time to kill.) Solve the equation of motion for the case of ω = ω_{0}.

Answer to 4. Found either by solving directly or by taking the limit of ω = ω_{0} in the above expression.

x(t) = [ A/(2ω

_{0}^{2}) ] . [ sinω_{0}t - ω_{0}t cosω_{0}t ]

Note the qualitative change in the behavior when ω = ω_{0}. When ω is close to ω_{0}, no matter how close, the maximum amplitude of the oscillation is finite over all time and is A/(ω^{2}-ω_{0}^{2}). However when ω = ω_{0}, the amplitude oscillates up to a bound that grows indefinitely with time. For large t, that bound is ω_{0}t.

1. Find one experiment to which the literature on dispersion of the early 1920s was responding. Describe it briefly in one or two sentences.

2. Have a question ready for Tony Duncan.

In modern quantum theory, what is the difference between the "Schroedinger picture" and the "Heisenberg picture"?

If f and g are two quantities defined on a classical phase space with canonical coordinates p_{i} and q_{i}, what is their Poisson bracket?.

Use Hamilton's equations of motion to relate the time rate of change of some quantity f with its Poisson bracket, formed with the Hamiltonian H.

Consider an inertial frame of reference with space and time coordinates (x', t'). A long row of people are laid along the x' axis, each holding a white flag. At the time t'=0, they all momentarily raise and lower their flags in perfect synchrony.

Redescribe this process in a second inertial frame of reference (x,t) moving at v with respect to the first frame. Show that what was a coordinated raising of the flags in the primed frame, now manifests as a pulse moving at speed c^{2}/v in the x direction. Use the following Lorentz transformation equations:

x' = γ(x-vt)
t' = γ(t-vx/c^{2})
γ=1/(1-v^{2}/c^{2})^{1/2}

Hint: This is an easy question. If it takes more than two or three lines, you are overthinking it.

Bosonic ladder operators arise in a space with basis states |0>, |1>, |2>, |3>, ... corresponding to lowest excitation/no particle, first excitation/one particle, second excitation/two particles, ... The ladder operators move us up and down the rungs of this ladder:

a^{†}|n> = (n+1)^{1/2}|n+1> a|n> = n^{1/2}|n-1>

where the n's are ordinary numbers.

1. Show that N = a^{†}a is a number operator that counts particle number as its eigenvalues: N|n> = n|n>.

2. Show that [a^{†},a] = I.

3. Show that [N,a^{†}] = a^{†} and [N,a] = -a.

Reflect on the material we covered this term. Identify an item you found especially interesting, surprising, noteworthy, unexpected or remarkable.