Wadsworth Institute Wave Functions Problems

Consider the one-dimensional normalized wave functions ψ0(x),
ψ1(x) with the properties
ψ0(−x) = ψ0(x) = ψ∗
0 (x), ψ1(x) = N
dψ0
dx
Consider also the linear combination
ψ(x) = c1ψ0(x) + c2ψ1(x)
with |c1|
2 + |c2|
2 = 1. The constants N, c1, c2 are considered as known.
(a) Show that ψ0 and ψ1 are orthogonal and that ψ(x) is normalized.
(b) Compute the expectation values of x and p in the states ψ0, ψ1 and ψ.
(c) Compute the expectation value of the kinetic energy T in the state ψ0 and demonstrate
that
ψ0|T 2
|ψ0 = ψ0|T |ψ0 ψ1|T |ψ1
and that
ψ1|T |ψ1 ≥ ψ|T |ψ ≥ ψ0|T |ψ0
(d) Show that
ψ0|x 2
|ψ0 ψ1|p2
|ψ1 ≥
h¯ 2
4
(e) Calculate the matrix element of the commutator [x 2, p2] in the state ψ.