Issued Wed, Aug 28 Due Wed, Sept 4

Required reading for this week: Gas. Ch. 1, start Ch. 2.

Here, and throughout the semester, please show your work on all problems. Organize your homework so the grader can follow your solution clearly. Explain (in words) what you're doing and what assumptions you make, whenever it seems appropriate.

1a) Gasiorowicz ("Gas") 1-12a tells you how much energy you find per unit volume inside a black body cavity, in terms of the fundamental constant "a"=7.56*10^-15 erg/(cm^3 K^4). Gas. 1-12b tells you how much ENERGY leaves a black object per unit area, per unit time, and involves a constant called "sigma" = 5.42*10^-5 erg/(cm^2 s K^4). Assuming the number for "a" is correct, use Gas. eqn 1-1 to check if he got "sigma" right. (I get 5.67, not 5.42)

b) Assume the sun is an ideal black body (it's not a bad assumption!) with radius 7.0*10^10 cm, distance to earth 1.5*10^13 cm, and delivering 1.4*10^6 erg/(cm^2 s) to the surface of the earth at high noon. (This last number is called the "solar constant", and is useful e.g. for estimating the required size of solar panels on your house) Estimate the surface temperature of the sun.

(Your answer should be between 1000 - 10,000 K's)

2a) Given Gas. eqn's 1-3 and 1-8, calculate the energy density in a wavelength interval . (Eliminate frequency entirely from your formula, so you have with no 's appearing) Use this result to find the value of , for which the density is a maximum.

Show that your answer for is of the form b/T, and solve for b. (See Gas. problem 1-3 for a hint) (You should find b is between 0.1 and 1. in cgs units) You may want to use Mathematica to solve the nasty transcendental equation, using the FindRoot[eqn == 0, {x,x0}] function. (You can also solve the equation pretty easily on your calculator with iteration, or trial and error if you think about it for a bit)

b) Use your estimate of the sun's surface temperature to calculate from the sun. Roughly what color does this correspond to?

3) Gas. problem 1-5.

4) Gas. problem 1-7

5a) Use the Bohr quantization rules to calculate the energy levels and allowed radii for a harmonic oscillator, for which the energy is

Erratum! The last term in the energy should be

(k r^2/2), not (k^2 r/2)!!

(that is, the force is , where k is a constant, the spring constant of the oscillator). Assume the orbits are circles.

What is the analog of the Rydberg formula? (i.e., what is the formula for the frequency of light emitted when the oscillator drops from level number "ni" to level number "nf", where the n's are the numbers used in quantizing the angular momentum)

b) Show that the correspondence principle is satisfied. (That is, show that the quantum frequency of emitted radiated when dropping from level "n" to level "n-1" is the same as the classical rotation frequency of the "n" 'th level)

In this case, you should discover that the correspondence principle turns out by accident to be satisfied even when n is not especially large!

(Because of Labor Day, no problem from Ch. 2, but start reading it)

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