PHY4604–Introduction to Quantum Mechanics
Fall 2004
Practice Test 1
October 4, 2004
These problems are similar but not identical to the actual test. One or two parts will actually show up.
1.
Short answer.
• Explain the photoelectric effect
• Explain the significance of ¯ in quantum mechanics, and give an example of a place where it shows up.
• Discuss the uncertainty principle briefly
• Explain the difference between the 2 versions of Schr¨odinger’s equation i ¯
∂ψ
∂t
= −
2 h 2 m
∂ 2
∂x
ψ
2
+ V ψ and
− h 2
2 m
∂ 2 ψ
∂x 2
+ V ψ = Eψ
• What are the units of P ( x, t ), the probability density in 1 dimension?
Justify your answer.
• Calculate the commutator [ p x
, x 2 ]
• Calculate the expression for the Bohr levels of the hydrogen atom from the
Bohr-Ehrenfest quantization condition.
2. Consider a wave packet defined by
ψ ( x ) =
Z
∞
−∞ dkf ( k ) e i ( kx − ωt ) with ω = ¯ 2 / 2 m and f ( k ) given by f ( k ) =


0 k < − ∆ k/ 2 a − ∆ k/ 2 < k < ∆ k/ 2
0 ∆ k/ 2 < k
(a) Find the form of ψ ( x ) at t = 0.
1
(1)
(2)

(b) Find the value of a for which ψ ( x ) is properly normalized.
(c) How is this related to the choice of a for which
Z
∞
−∞ dk | f ( k ) | 2 = 1?
(3)
(d) Show that for a reasonable definition of ∆ x , the size of the packet given by your answer in a), ∆ k ∆ x > 1 .
3. A particle in an infinite square well (of width a ) has as its initial wave function an equal mixture of the first two stationary states:
Ψ( x, 0) = C [ ψ
1
( x ) + ψ
2
( x )]
(a) Normalise Ψ( x, 0). (That is, find C .)
(b) Find Ψ( x, t ) and | Ψ( x, t ) | 2 . Express the latter in terms of sin and cos using e iθ = cos θ + i sin θ . Use ω = π 2 ¯ 2 ma 2 .
(c) Compute < x > . Notice that it oscillates in time. What is the frequency of the oscillation? What is the amplitude?
(d) Compute < p > .
(e) Find the expectation value of the Hamiltonian operator, H , in terms of E
1 and E
2
.
2