Physics 324 Problem Set # 3 Due 10/27/03
1. A harmonic oscillator consists of a 1 g mass on a spring. Its oscillation frequency is 2 Hz and its
amplitude of oscillation is 1 cm. Approximately, what is the maginitude of the energy quantum number
associated with this classical oscillator?
2. From Eqns. 2.48 and 2.54 in your textbook, we find that the normalized wavefunction for the lowest
energy state of a harmonic oscillator potential can be written as:
ψ0(x, t)=mω
π¯h1/4
e−mωx2/2¯he−iωt/2
(a) Using Eqns. 2.51 and 2.54 in your textbook, find an expression for the normalized wavefunction
of the first excited state of the harmonic oscillator, ψ1(x, t).
(b) What are the expectation values of position and momentum, hxiand hpi, for the two states ψ0(x, t)
and ψ1(x, t) ? (You may use symmetry arguments to reduce the work required.)
(c) Compute hx2iand hp2ifor state ψ0(x, t).
(d) Use the results from parts (b) and (c) to verify the uncertainty principle for state ψ0(x, t).
(e) Using the results that hKEi=hp2/2miand hV(x)i=hmω2x2/2i, compute hKE +V(x)ifor state
ψ0(x, t). Is the result what you expect?
3. An alternate way of expressing the energy eigenstates (solutions to the time independent Schrodinger
equation) for the harmonic oscillator potential comes from the analytic approach, summarized in Eqn. 2.69
in your text:
ψn(x)=mω
π¯h1/41
p(2nn!) Hn(ζ)e−ζ2/2where ζ=rmω
¯hx
and Hn(ζ) are Hermite polynomials which satisfy the following recursion relation:
Hn+1(ζ)=2ζHn(ζ)−2nHn−1(ζ)
(a) Using the results that H0(ζ) = 1 and H1(ζ)=2ζ, use the recursion relation to compute Hn(ζ) for
n= 2 through n=6.
(b) A particle in a harmonic oscillator potential is prepared in the initial state:
ψ(x, 0) = Ahψ0(x)+2ψ1(x)+ψ2(x)i
where ψn(x) is the normalized solution to the time independent Schrodinger equation for the harmonic
oscillator potential for the nth energy level. Without explicitly evaluating any integrals, find A.
(c) Without explicitly evaluating any integrals, find the expectation value for the energy, hˆ
Hi, for the
state ψ(x, t) in part (b).
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