Quantum Mechanics - Infinite Potential Well

See Problem 51

The eigenfunctions satisfy the condition

₀∫^aψ_n*(x) ψ_l (x) dx = δ_nl

δ_nl = 1 if n = l, otherwise δ_nl = 0. This is a statement that the eigenfunctions are

A. Solutions to the Schrodinger equation 
B. Orthonormal
C. Bounded 
D. Linearly dependent
E. Symmetric

(GR9277 #52)

Solution:

Orthonormality = orthogonal and normal

₀∫^aψ_n*(x) ψ_l (x) dx = 〈ψ_n|ψ_l〉= δ_nl

Orthogonal,  δ_nl = 0
Normal, δ_nl = 1 if n = l

Answer: B

Quantum Mechanics- Wave Function

The states and are orthonormal. For what value of x are the states and given above orthogonal?

A. 10
B. 5
C. 0
D. −5
E. −10

(GR0177 #28)

Solution:

States are orthogonal when their dot-product (bra-ket) is 0.
orthonormal → .



Answer: E

Quantum Mechanics - Expectation Value

The state   is linear combination of three orthonormal eigenstates of the operator corresponding to eigenvalues -1,1 and 2. What is the expectation value of    for this state?

A. 2/3
B. √(7/6)
C. 1
D. 4/3
E. (√3+2√2-1)/√6

(GR0177 #29)

Solution:

The expectation value:



Answer: C