Physics 481
Midterm Exam
Spring 2006
The total number of points for this exam is 40. You have 50 minutes to complete the exam.
Do all problems. Show all your work so that I can see how you arrived at the answer.
You may use Griffith ”Introduction to Quantum Mechanics” and the lecture notes to look
up any relevant formulas.
1. Consider the simple harmonic oscillator potential.
(a) (8 points) Which of the trial wave function groups given below, could be used
to estimate an upper limit for the ground state energy? Which would be the
best choice? Could any of the wave functions be used to estimate an upper
limit for the first excited state energy? If yes, which one would be best suited?
Explain your answers. (α is a variational parameter and Ai are normalization
constants.)
i.
ii.
iii.
iv.
v.
vi.
ψ1 (x) = A1
ψ2 (x) = A2
ψ3 (x) = A3
ψ4 (x) = A4
ψ5 (x) = A5
ψ6 (x) = A6
vii. ψ7 (x) = A7
eαx
eα|x|
2
eαx
2
x eαx
(x + α) (x − α) for |x| < |α|, elsewhere 0
sin(πx/α) for |x| < |α|, elsewhere 0
sin(αx)
x
(b) (10 points) Use the trial function ψ5 to estimate an upper limit for the ground
state energy of the simple harmonic
oscillator potential. The value of the
q
normalization constant A5 is 15/(16α5 ).
(c) (8 points) You put two non-interacting identical spin-1/2 particles in the simple
harmonic oscillator potential and measure the total energy of the system to be
3h̄ω. Give all possible two-particle states.
2. Consider a particle with mass m in the potential
(
V (x) =
β|x| for |x| < V0 /β
V0
elsewhere,
where β and V0 are positive constants.
(a) (4 points) Sketch the potential.
(b) (10 points) Give approximate expressions for the bound state energy levels.
Good luck !
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