# Consider the following solution to the hydrogen atom problem

```Physics 422 (Opportunity I)
NAME ___ ___________
February 14, 2001
Make sure your name is on all pages of your test. Show all work clearly!!!! The
question is wrong if all work is not clearly shown. It is your responsibility to make sure
you understand the question before attempting it. Assume ideal conditions unless
otherwise stated. Label things clearly.
Potentially useful information:
1) Consider the following solution to the hydrogen atom problem.
1
 ( r , ,  ) 
64
5
2

Z
2 ao  i
  re e sin 
 ao 
Zr
a) What is the energy of this state?
b) What is the angular momentum quantum number, l associated with this state?
c) What is the expectation of the angular momentum projection?
In other words, &lt; Lz&gt; = ?
d) What is the probability of finding the electron somewhere along the z-axis?
e) What is the probability of finding the electron at a distance grater than the Bohr radius
ao? Just set-up the problem. You do not need to solve it completely.
2) In the case of spin-orbit coupling, show that Jˆ z defined by
Jˆ  Lˆ  Sˆ commutes with L̂ Sˆ .
z
z
z
3) If one starts with the Zeeman effect in hydrogen and increases the strength of the
magnetic field, a result known as the Paschen-Bach effect is realized. In the PaschenBach effect the perturbation produced by the external magnetic field is large enough so
that spin-orbit coupling can be ignored, but smaller than the Coulomb interaction, so the
Paschen-Bach can be treated as a perturbation. The total Hamiltonian can be described by
the following relation.
pˆ 2 e2
eB ˆ
Hˆ 


( Lz  2Sˆz )
2 | rˆ | 2me c
Use perturbation theory to determine the first order energy correction to the ground state.
4) Two identical, noninteracting spin &frac12; particles of mass m are in a one-dimensional
harmonic oscillator for which the Hamiltonian is given by:
pˆ 2 1
pˆ 2 1
Hˆ  1x  m 2 xˆ12  2 x  m 2 xˆ22
2m 2
2m 2
a) What is the ground state energy?
b) What is the energy of the first excited state?
c) One of the excited states can be expressed in the total spin notation:
| n1 , n2 | s, ms  where | n1 , n2  corresponds to the number operator Nˆ  aˆ †aˆ for each
particle and | s, ms  refers to the total spin.
as:
1
 1

|1, 0  
| 0,1   |1, 0 

2
 2

where |1, 0 
1
1
|  z,  z  
|  z,  z  in our old notation.
2
2
What is the eigenvalue of the exchange operator P̂12 for this state?
d) What are the other three excited states, expressed in this notation?
```