Physics 471 Problem Set 3 Fall 2003
9. (Griffiths 2.6) A particle in the infinite square well has as its initial wave function an even
mixture of two stationary states:
Ψ(x, 0) = A[ψ1(x)+ψ2(x)] .
(a) Normalize Ψ(x, 0). (That is, find Ausing the orthogonality of ψ1and ψ2.)
(b) Find Ψ(x, t)and|Ψ(x, t)|2. (Express the latter in terms of sinusoidal functions of time,
eliminating the exponentials with the help of Euler’s formula: eiθ =cosθ+isin θ.)
(c) Compute x. Notice that is oscillates in time. What is the frequency of oscillation?
What is the amplitude of oscillation?
(d) Compute p.
(e) Find the expectation value of H. How does it compare with E1and E2.
(f) A classical particle in the well would bounce back and forth between the walls. If its
energy is equal to the expectation value you found in (e), what is the frequency of
the classical motion? How does it compare with the quantum frequency you found
in (c)?
10. (Griffiths 2.8) A particle in the infinite square well has the initial wave function
Ψ(x, 0) = Ax(a−x).
(a) Normalize Ψ(x, 0). Graph it. Which stationary state does it most closely resemble?
On that basis, estimate the expectation value of the energy.
(b) Compute x,pand Hat t= 0. (How does Hcompare with your estimate in
(a))?
11. For the particle described by the initial wave function in the previous problem, we showed
that
Ψ(x, t)=
∞
n=0
Cnψ2n+1(x)e−iE2n+1t/¯h,
where
Cn=8√15
(2n+1)
3π3and E2n+1 =¯h2(2n+1)
2π2
2ma2.
(a) Evaluate Cnfor n=0,1,2.
(b) What is the probability that a measurement of the particle’s energy will give ¯h2π2/2ma2?
(c) What is the probability that a measurement of the energy will give a value different
from ¯h2π2/2ma2?
12. Determine the expansion coefficients Cnfor the initial square well wave function
Ψ(x, 0) = Ax if 0 ≤x≤a/2
A(a−x)ifa/2≤x≤a
Remember to normalize Ψ(x, 0). (Ans. C2n+1 =(−1)n4√6/[(2n+1)
2π2], C2n=0.)