Physics 324 Problem Set # 2 Due 10/13/03
1. Problem 2.5 in your textbook.
2. For the infinite square well potential (V(x) = 0 for 0 ≤x≤aand infinite elsewhere),
a particle is prepared in the state ψ(x, 0) at time t= 0 where:
ψ(x, 0) = Bx for 0 ≤x≤a; and ψ(x, 0) = 0 elsewhere.
Note, this is an unphysical wavefunction because it must change suddenly from a value
of Ba at x=ato a value of 0 at x=a. (Fourier analysis nonetheless allows us to express
such a function, or a square wave for example, as a sum of well behaved functions (sines and
cosines).
(a) Find the normalization constant, B.
(b) Find the expectation value of the particle’s position, hx(0)iat time t=0.
(c) We can express ψ(x, 0) as a sum of energy eigenstates, because the energy eigen-
states form a complete orthonormal set:
ψ(x, 0) =
∞
X
n=1
cnψn(x)
where ψn(x) are the solutions to the time independent Schrodinger equation for the
infinite square well potential (energy eigenstates) as given in your text.
Derive an expression for cn, the amplitude of energy eigenstate, n, in the expansion for
ψ(x, 0).
(d) Write an expression for ψ(x, t), that is, an expression for the wave function at all
times t≥0.
3. A particle in an infinite square well potential (same potential as Problem #2)is
prepared in a state:
ψ(x, t)=A[ψ1(x)e−iE1t/¯h+i√2ψ4(x)e−iE4t/¯h]
where ψi(x) is the solution to the time independent Schrodinger equation for energy Ei.
(a) Use the orthonormality of the ψi’s to compute the normalization constant, A.
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