Assignment 9, solutions

Problem 1:

The wave function of a system in the ground state is given as

ψ(r,t) = [exp(-iωt)/(πa03)½] exp(-r/a0).

(a)  Sketch the probability density in coordinate space as function of r/a0.
(b)  Find the momentum space wave function Φ(p,t). 
Hint:  Use spherical coordinates for evaluation of the integral transform.
(c)  Find the probability density function in momentum space.  Sketch it as a function of pa0/ħ.

Solution:

Problem 2:

An exotic atom consists of a Helium nucleus (Z = 2) and an electron and an antiproton p(bar) both in n = 2 states.  Take the mass of the p(bar) to be 2000 electron masses and that of the helium nucleus to be 8000 me.  For an electron in the n = 1 state of hydrogen E = -13.6 eV.
(a)  How much energy is required to remove the electron from this atom?
(b)  How much energy is required to remove the p(bar) from this atom?
(c)  Assume both the p(bar) and the electron are in 2p states.  Then each can de-excite to their ground state.  It is observed that radiation always accompanies those transitions when the electron jumps first, but when the p(bar) jumps first there is often no photon emitted.  Explain!

Solution:

Problem 3:

The potential energy of the nuclei of a diatomic molecule as a function of their separation r is given by
U(r) = -2D[a0/r - a02/r2].
Here D is a constant with units of energy.
Approximate this potential energy function near its minimum by a harmonic oscillator potential energy function and determine the vibrational energy levels of the molecule with zero angular momentum.

Solution:

Problem 4:

The Rydberg constant, RH = 109737.568525/cm is one of the most accurately known fundamental constants.
(a)  Find the wave number of the Balmer alpha line (n = 3 to n' = 2) in  atomic hydrogen.  Neglect fine structure.
(b)  Is the Balmer alpha line in atomic deuterium shifted towards the blue or towards the red compared to normal hydrogen?
(c)  Calculate the shift in wave number between deuterium and hydrogen.

Solution:

Problem 5:

An electron in the hydrogen atom occupies the combined position and spin state

R21(r)[√⅓ Y10(θ,φ)χ+ + √⅔ Y11(θ,φ)χ-].

(a)  If you measure L2, what value(s) might you get, and with what probability(ies)?
(b)  If you measure Lz, what value(s) might you get, and with what probability(ies)?
(c)  If you measure S2, what value(s) might you get, and with what probability(ies)?
(d)  If you measure Sz, what value(s) might you get, and with what probability(ies)?
(e)  If you measured the position of the electron, what is the probability density for finding the electron at r, θ, φ in terms of the variables given above.
(f)  If you measured both Sz and the distance of the electron from the proton, what is the probability per unit length for finding the particle with spin up a distance r from the proton in terms of the variables given above?

Useful integral:  ∫0πsinθ dθ∫0dφ |Ylm(θ,φ))|2 = 1.

Solution: