QUANTUM PHYSICS I
PROBLEM SET 3
due October 10, 2007
A. (Griffiths, 2.4) Uncertainty on the square well states
Calculate hxi,hx2i,hpiand hp2ifor the nth stationary state of the infinite square well. Check that the uncertainty
principle is satisfied. Which state comes closer to the uncertainty limit ?
B. (Griffiths, 2.5 and 2.6, sort of ...) When is the wave function phase relevant ?
A particle on an infinite square well has an inital wave function that is an equal superposition of the two first states:
Ψ(x, 0) = A(ψ1(x) + ψ2(x)).(1)
i) Normalize Ψ(x, 0).
ii) Find Ψ(x, t) and |Ψ(x, t)|2. To simplify the result, let ω=π2~/(2ma2).
iii) Compute hxiand notice it is oscillatory. What is the amplitude and angular frequency of this oscillation ?
Although the overall phase of the wave function is of no physical significance (it cancels out whenever you calculate
a measurable quantity), the relative phase of the coefficients in eq. (1) does matter. For example, suppose we take
Ψ(x, 0) = A(ψ1(x) + eiψψ2(x)) (2)
instead of eq. (1). Find
iv) Ψ(x, t),
v) |Ψ(x, t)|2and
vi) hxi
and compare with the φ= 0 case.
C. (Griffths 2.22) The gaussian wave packet
A free particle has the initial wave function
Ψ(x, 0) = Ae−ax2,(3)
where Aand aare constants (and ais real and positive).
a) Normalize Ψ(x, 0).
b) Find Ψ(x, t) Griffths gives away the answer so I might as well do the same:
Ψ(x, t) = 2a
π1/4e−ax2/(1+(2i~at/m))
p1 + (2i~at/m .(4)
c) Find |Ψ(x, t)|2and express your answer in terms of
w=ra
1 + (2~at/m)2.(5)
Sketch |Ψ|2as a function of x at t= 0 andfor some large t. Qualitatively, what happens to |Ψ|2as time goes on ?
d) Find hxi,hpi,hx2i,hp2i, σxand σp. Partial answer: hp2i=a~2.
e) Does the uncertainty principle hold ? At what time does the the system come closest to the uncertainty limit ?
Do you have a physical understanding of why the uncertainty evolves in time the way it does ?
