Phys 323, Sample Problem (A. Polychronakos)

A particle of mass m is in a one-dimensional infinite square well that extends from x = –a to x = a.

a) Find the energy eigenfunctions ψn (x) and corresponding eigenvalues En of this particle. (Hint: you may use the results of the book for an infinite square well between x=0 and x=a, appropriately modified!)

b) The parity operator Π is defined as: Π ψ(x) = ψ (–x) for any function ψ(x). Does Π commute with the Hamiltonian H of this particle?

c) Are the energy eigenfunctions ψn (x) also eigenfunctions of Π and, if yes, with what eigenvalue each?

The wavefunction of the particle at some initial time is ψ = C sin |πx/a| , with C a real positive constant. ( ψ = 0 for |x| > a )

d) Normailize the wavefunction by calculating the appropriate value of C.

e) Calculate the expectation value of the energy of this particle.

f) Is the above wavefunction an eigenfunction of Π and, if yes, with what eigenvalue?

g) What is the probability that a measurement of the energy of this particle will yield the value E2 ? (Hint: the result of (c) and (f) may help you.)

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Answers:

a) ψn (x) = (1/√a) sin[nπ(x+a)/(2a)] , En = (nħπ/2a)²/(2m) , n = 1,2,3, ...

b) Yes

c) Yes; eigenvalue = 1 for n = 1,3,5, … and = –1 for n = 2,4,6, …

d) C = 1/√a

e) <H> = (ħπ/a)²/(2m) = E2

f) Yes, eigenvalue = 1

g) Prob (E = E2 ) = 0