Harmonic potentials, raising and lowering operators

Problem:

The orthonormal set of wave functions for the stationary states of the harmonic oscillator with U(x) = ½mωx2 is
n(η) = Nn Hn(η) exp(-½η2)}, with η = (mω/ħ)½x.
The Hermite polynomials Hn(η) satisfy the recurrence relations
ηHn(η)  = nHn-1(η) + ½Hn+1(η)  and dHn(η)/dη = 2nHn-1(η).
The normalization constants Nn are given by
N0 = π¼, Nn+1 = (2(n+1))Nn.

(a)  Show that the matrix elements of X can be expressed as
<m|X|n> = (nħ/(2mω))½δm,n-1 + ((n+1)ħ/(2mω))½δm,n+1.
(b)  Derive a similar expression for the matrix elements of X2.
(c)  The ladder operators for the harmonic oscillator have the properties
a|n> = √(n+1) |n+1>,   a|n> = √(n) |n-1>.
Derive the expressions for a and a in terms of η and d/dη.

Hint: Start by investigating the action of the operator  d/dη on Ψn(η).

Solution:

Problem:

(a)  Construct the normalized, linear combination of one-dimensional harmonic-oscillator states of the form |ψ> = c00> + c11>, with c0 and c1 real, such that the expectation value of the position operator X is maximized.  Here |Φ0> and |Φ1> refer to the ground state and the first excited state, respectively.
(b)  For the above state evaluate the expectation values of the momentum and parity operators. 
Useful formulas:

a|n> = √(n)|n-1>,  a|n> = √(n+1)|n+1>.
X = (ħ/(2mω))½(a + a),  P = i(mħω/2)½(a - a).
<x|0> = (mω/(πħ))¼exp(-½mωx2/ħ).
-∞exp(-a2x2)dx = √π/(2a)  (a > 0).

Solution:

Problem:

Consider a particle in a harmonic oscillator potential with Hamiltonian H = p2/(2m) + ½mωx2.
Its state vector at  t = 0 is |ψ(0)> = exp(-|α|2/2)∑0n/(√n!))|n>,
where the |n> are the orthonormal eigenstates of H.
(a)  Show that |ψ(0)> is an eigenstate of the lowering operator a and find the eigenvalue.
(b)  If the energy of the particle is measured at t = 0, what values of E can be found and with what probability?
(c)  H = ħω(aa + 1).  Find <H> and ΔH at t = 0.
(d)  Find |ψ(t)> and show that it is still an eigenstate of a.  Find the eigenvalue.
(e)  Find <x>(t).  Why is the state |ψ(0)> called a quasi classical state?

Solution:

Problem:

A one-dimensional harmonic oscillator has mass m and angular frequency ω.  Denoting the momentum by p and the coordinate by x, we can define the operators
a = αx + iβp,  a = αx – iβp,  where α =√(mω/(2ħ)),  β =1/√(2mωħ).
(a)  Find [a, a].
(b)  Find the Hamiltonian in terms of a and a.
(c)  Let |{|n>} denote the eigenstates of the Hamiltonian with eigenvalues ħω(n + ½). Given a|n> = (n + 1)½|n + 1>,  a|n> = n½|n - 1>, find the expectation value of x4 for the system in the state |n>.
Show your work!

Solution:

Problem:

For a one-dimensional simple harmonic oscillator we may define raising and lowering operators
a = (mω/(2ħ))½(X + iP/(mω)),  a = (mω/(2ħ))½(X - iP/(mω)),
with properties
a|n> = √(n) |n - 1>,  n ≠ 0, a|n> = 0, b = 0, and
a|n> = √(n + 1) |n + 1>.
(a)  Show by direct calculation that the ground state of the simple harmonic oscillator satisfies
(ΔX)2(ΔP)2 = ¼|<[x,p]>|2,
and hence is a minimum uncertainty state.
(b)  Consider a coherent state of a one-dimensional oscillator, |b>, defined as |b> = exp(-|c|2/2)∑0(cn/(√n!))|n>.
Show that |b> is an eigenstate of the lowering operator a and find the eigenvalue.
(c)  Show that |Ψ(t)> = U(t,0)|b> is an eigenstate of a and find the eigenvalue,
(d)  Show that |Ψ(t)> = U(t,0)|b> is also a minimum uncertainty state.

Solution:

Problem:

No potential is harmonic for arbitrarily large displacements from the origin.  Eventually nonlinearities set in.  But, as long as the displacement is small, so that the lowest order quadratic term dominates, we can treat the potential as harmonic.  However, there is zero point motion of the quantum state.  If the extent of the ground state is large compared to the range over which the potential is quadratic, the spectrum in no way looks like that of a SHO.

Consider the potentials V(x) = V0sin2kx.  Find the characteristic oscillation frequency near an equilibrium point.  Determine the extent of the ground state for the corresponding SHO.  By comparing this to the characteristic scale over which the potential is harmonic, estimate the number of levels for which the energy spectrum looks harmonic (i.e. equally spaced).

Solution: