Show that all eigenstates of the Hamiltonian of the one-dimensional harmonic oscillator are not degenerate.
A particle of mass m moves in one dimension in the
potential energy function U(x) = ½mω2x2.
(a) Write down the Hamiltonian abd the time-independent Schroedinger equation.
(b) Show that by appropriate choice of a dimension less coordinate z the Schroedinger equation can be written as
½ħω(-d2/dz2 + z2)Ψ = EΨ.
Verify that normalized solutions to this equation are
Ψ0 = (mω/(πħ))1/4exp(-½z2) and Ψ1 = (4mω/(πħ))1/4 z exp(-½z2)
and find the corresponding eigenvalues E0 and E1 .
(c) Suppose that at t = 0 the system was prepared in such a way that its wavefunction was Ψ = (2mω/(πħ))1/4exp(-z2).
What is the probability that a measurement of the energy at t = 0 would yield the value of E0?
What is the probability that a measurement of the energy at t = 0 would yield the value of E1?
(d) Would the system remain in the state Ψ = (2mω/(πħ))1/4exp(-z2) for subsequent times? Explain!
system consists on N independent quantum mechanical harmonic oscillators of
frequency f. The system is in thermal equilibrium with a
reservoir at temperature T.
(a) Calculate the average energy <E> for the system.
(b) Find the heat capacity of the system when kT >> ħω.
Consider a harmonic oscillator with
H = ½P2/m + ½mω2X2 = ħω(a†a + ½).
Assume that at t = 0 the system is in the state |Ψ(0)> such that <Ψ(0)|a|Ψ(0)> = a0.
(a) Show that <X>(t) behaves as a function of time as the coordinate x of a classical harmonic oscillator with amplitude xM = (2ħ/(mω))1/2|a0|.
Such a classical harmonic oscillator has energy E = ½mω2xM2 = ħω|a0|2.
The expectation value of the energy of the quantum-mechanical oscillator is
ħω<Ψ(0)|a†a|Ψ(0)> + ½ħω.
(b) Assume that <a†a>(0) = |a0|2
Show that <a†a>(0) = |a0|2 or <a>(0) = a0 implies that a|Ψ(0)> = a0|Ψ(0)>,
and that a|Ψ(0)> = a0|Ψ(0)> implies <a†a>(o) = |a0|2 or <a>(0) = a0.
The two conditions <a†a>(0) = |a0|2 or <a>(0) = a0 define what is called a quasi classical state.
|Ψ(0)> = ∑ncn|Φn>, H|Φn>
= (n + ½)ħω|Φn>.
For a normalized |Ψ(0)> find the expansion coefficients cn .
(d) Show that ΔXΔP = ħ/2.
For the two-dimensional harmonic oscillator find the two lowest energy eigenstates and write down the corresponding wave functions.