Problem Set 2: Old Quantum Theory Physics 550

Physics 550 Problem Set 2: Old Quantum Theory

Name:

Instructions / Notes / Suggestions:

• Each problem is worth five points.

• In order to receive credit, you must show your work.

• Circle your final answer.

• Unless otherwise specified, answers should be given in terms of the variables in the

problem and/or physical constants and/or cartesian unit vectors (eˆx, eˆx, eˆz) or radial unit vector ˆr.

• Staple your work. Problem Set 2: Old Quantum Theory Physics 550

Problem #1:

Consider a simple harmonic oscillator, with the equation of motion:

mx¨ + kx = 0

Using Old Quantum Theory, find an expression for its energy levels. Problem Set 2: Old Quantum Theory Physics 550

Problem #2:

As an extension of Problem #1: Consider a two-dimensional harmonic oscillator.

Assume that the oscillator can be considered as two independent one-dimensional oscillators (in the x and y directions). The equations of motion are then:

mx¨ + kxx =0

my¨ + kyy =0

(a) Using Old Quantum Theory, find its energy levels.

Now assume that the oscillator is isotropic (kx = ky). (b) Find the energy levels of the oscillator. (c) Determine the degeneracy of the nth energy level. Problem Set 2: Old Quantum Theory Physics 550

Problem #3:

Consider an electron orbiting a nucleus with a circular orbit of radius R.

(a) Using Old Quantum Theory, find the energy levels of the electron. (b) Determine the emission (or absorption) frequencies of the atom. Compare your answer to the Rydberg formula. Problem Set 2: Old Quantum Theory Physics 550

Problem #4:

Consider the Rydberg formula:  1 1  ν = K 2 − 2 n1 n2 where ν is the frequency of emitted radiation, K = cR, where R is the Rydberg constant

and c is the speed of light, and n1 < n2 are quantum numbers corresponding to transitions for the emitted (or absorbed) radiation.

Show that in the limit of large quantum numbers, the frequency ν corresponds to classical theory.