8/18/2019 Phys 481 Midterm Practice

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Additional Practice Midterm Problems

TA: Jeff Schueler February 23, 2016

1.   Angular Momentum Fun  (Note: Part (a) is optional. The rest of the problem is important to your un-derstanding of angular momentum. Part (a) shows a nice way to play around with spherical harmonics,so if you don’t feel like working through the algrabra, take the result as given and begin with part (b).):

Consider the combined orbital and spin state

ψ =N (x− iy +   1√ 

2z)

r  χS 

, where  χs  =

 1−1

.

(a) Show that  ψ  can be re-written as

ψ =  N 

 8π

3  Y −1

1  +

 2π

3  Y 0

1

χS    (1)

(It will be useful to consider that in cartesian coordinates   Y ±11

  = ∓ 

3/(8π)x±iyr   and   Y 01

  = 3/(4π)zr ).

(b) What are the possible values of a measurement of   Lz   of this system and their correspondingprobabilities?

(c) What are the possible values of a measurement of   L2 for this system and their correspondingprobabilities?

(d) What are the possible values of a measurement of   S 2 for this system and their correspondingprobabilities?

(e) What are the possible values of a measurement of   S x   for this system and their corresponding

probabilities?(f) What are the possible values of a measurement of   S z   for this system and their corresponding

probabilities?

2. Consider an isotropic 2D simple harmonic oscillator with unperturbed Hamiltonian

H 0 =  P 2x2m

 +P 2y2m

 + 1

2mω2(X 2 + Y 2).

It can be shown that for the three lowest lying distinct energy levels of  H 0:

E 000 =   ω   (ground state, no degeneracy)

E 010 =  E 0

01 = 2 ω   (1st excited state, two-fold degeneracy)

E 020

 =  E 002

 =  E 011

 = 3 ω   (2nd excited state, three-fold degeneracy.)

(a) Suppose a perturbation H   =  λmω2xy   is added to the system.

i. Find the first order correction to the ground state of this system.(Note:   ψ0g.s. =

 mωπ   exp

−mω2 

 x2

exp−mω

2  y2

).

ii. It can be shown that (to first order) the Hamiltonian of the first excited state of this systemcan be written as

H   = λ ω

2

0 11 0

.

Using degenerate perturbation theory, determine the first order energy corrections to the firstexcited state of this system. What happens to the degeneracy? How do you know?

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8/18/2019 Phys 481 Midterm Practice

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iii. Determine the “good” linear combinations of states  ψ01  and  ψ10.

(b) It can be shown (after lots of tedious algebra) that the perturbed Hamiltonian, in the basis of degenerate second excited states, is of the form

H   =√ 

2

2  λ ω

0 0 10 0 11 1 0

Using degenerate perturbation theory, determine the first order energy corrections to the secondexcited state of this system. What happens to the degeneracies when  λ   is increased? What arethe “good” linear combinations of states that leave the energy of the Hamiltonian unchanged as  λis suppressed?

3. (If you feel like you want more perturbation theory practice) This problem is a variant on problem 7.14in Griffiths. If a photon had a small mass,  mγ  = 0, the Coulomb potential would be replaced by theYukawa potential

V (r) = −   e2

4π0r exp

−cmγ r 

.

Using perturbation theory, evaluate the first order correction to the binding energy of the hydrogenatom in the ground state (Important: You are not given what H’ is in this problem. Expand  V (r) tofirst order and you should get what you desire!). What would happen if you tried to apply second orderperturbation theory to the ground state of hydrogen?

4. Consider a particle moving in a 1D potential,  V (x) = ∞, for  x <  0 and  V (x) =  ax, for  x >   0. Usingthe trial wavefunction  ψ(x) = x exp(−αx), where  α  is your variational parameter, estimate the groundstate energy. [Ans :   E (α) =    

2

2m

3ma2 2

2/3+   3

2a

2 2

3ma

1/3

]

5. (If you have a lot of time on your hands): Consider an anharmonic oscillator Hamiltonian given by

H  = H 0 + aX 3 + bX 4,

where H 0 is the unperturbed SHO Hamiltonian.

(a) Show that the first order correction to the energy is given by

E 1n  = 3

4b

   

mω

2

(2n2 + 2n + 1).

(Notice that the first order correction doesn’t depend on the constant  a. Why do you think thisis?)

(b) Show that the second order correction to the energy is given by

E 2n  = 15

4 b

   

mω

3

(n2 + n + 1130

) a2

 ω −  1

8

   

mω

4

(34n3 + 51n2 + 59n + 21) b2

 ω.

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