Harmonic potentials, raising and lowering operators

Problem:

The orthonormal set of wave functions for the stationary states of the harmonic oscillator with U(x) = ½mω²x² is
{Φ_n(η) = N_n H_n(η) exp(-½η²)}, with η = (mω/ħ)^½x.
The Hermite polynomials H_n(η) satisfy the recurrence relations
ηH_n(η)  = nH_n-1(η) + ½H_n+1(η)  and dH_n(η)/dη = 2nH_n-1(η).
The normalization constants N_n are given by
N₀ = π^¼, N_n+1 = (2(n+1))^-½N_n.

(a)  Show that the matrix elements of X can be expressed as
<m|X|n> = (nħ/(2mω))^½δ_m,n-1 + ((n+1)ħ/(2mω))^½δ_m,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 Φ_n(η).

Solution:

• Concepts:
  The eigenfunctions of the harmonic oscillator, recurrence relations.
• Reasoning:
  To evaluate the matrix elements, we have to evaluate integrals.  This is done with the help of recurrence relations and normalization conditions.
• Details of the calculation:
  (a)  <m|X|n> =  (ħ/(mω))^½<m|η|n>, <m|n> = δ_m,n.
  <m|η|n> = ∫_-∞^∞Φ_m*(η)ηΦ_n(η)dη = N_mN_n∫_-∞^∞H_m(η)ηH_n(η) exp(-η²) dη
  (use recurrence relation)
  <m|η|n> = N_mN_nn∫_-∞^∞H_m(η)H_n-1(η) exp(-η²) dη + N_mN_n½∫_-∞^∞H_m(η)H_n+1(η) exp(-η²) dη
  = N_mN_n(N_mN_n-1)^-1 n∫_-∞^∞Φ_m(η)Φ_n-1(η) dη + N_mN_n(N_mN_n+1)^-1 ½∫_-∞^∞Φ_m(η)Φ_n+1(η)dη
  = (N_n/N_n-1)nδ_m,n-1 + ½ N_n/N_n+1)δ_m,n+1 = (n/2)^½δ_m,n-1 + ½(2(n+1))^½δ_m,n-1.
  <m|X|n> = (nħ/(nmω))^½δ_m,n-1 + ((n+1)ħ/(2mω))^½δ_m,n+1.

  (b)  <m|X²|n> =  (ħ/(mω))<m|η²|n>.
  <m|η²|n> = ∫_-∞^∞Φ_m*(η)η²Φ_n(η)dη = N_mN_n∫_-∞^∞(mH_m-1(η) + ½H_m+1(η))(nH_n-1(η) + ½H_n+1(η)) exp(-η²) dη
  = N_mN_n[(mn/(N_m-1N_n-1))δ_m-1,n-1 + ½n/(N_m+1N_n-1))δ_m+1,n-1 + ½m/(N_m-1N_n+1))δ_m-1,n+1 + ¼/(N_m+1N_n+1))δ_m+1,n+1
  = ½(mn)^½δ_m,n + ½(n(m+1))^½δ_m+2,n + ½(m(n+1))^½δ_m-2,n + ¼(2(m+1)2(n+1))^½δ_m,n
  = (n + ½)δ_m,n + ½(n(n-1))^½δ_m+2,n + ½(m(m-1))^½δ_m-2,n.
  <m|X²|n> = (ħ/(mω))(n + ½)δ_m,n + ½(ħ/(mω))(n(n-1))^½δ_m+2,n + ½(ħ/(mω))(m(m-1))^½δ_m-2,n.

  (c)  dΦ_n(η)/dη = N_n(dH_n(η)/dη)exp(-½η²) - N_nH_n(η) η exp(-½η²)
  = N_n2nH_n-1(η) exp(-½η²) - N_n(nH_n-1(η) + ½H_n+1(η)) exp(-½η²).
  ηΦ_n(η) = N_nH_n(η) η exp(-½η²) = N_n(nH_n-1(η) + ½H_n+1(η)) exp(-½η²).
  dΦ_n(η)/dη + ηΦ_n(η) =  N_n2nH_n-1(η) exp(-½η²) = 2n(N_n/N_n-1)Φ_n-1(η) = √(2n)Φ_n-1(η)
  We have
  a|n> = √(n) |n-1>,
  therefore a = (1/√2)[η + d/dη].
  Similarly,
  dΦ_n(η)/dη - ηΦ_n(η) =  -N_nH_n+1(η) exp(-½η²) = -(N_n/N_n+1)Φ_n+1(η) = -√(2(n+1))Φ_n+1(η).
  We have
  a^†|n> = √(n+1) |n+1>.
  therefore a^† = (1/√2)[η - d/dη].

Problem:

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:

• Concepts:
  Postulates of quantum mechanics, the 1D harmonic oscillator.
• Reasoning:
  The mean value of any observable A is given by <ψ|A|ψ>.  We are asked to maximize the mean value of the observable x, if the particle is in a linear superposition of two eigenstates of the 1D harmonic oscillator.
• Details of the calculation:
  (a)  |ψ> = c₀|Φ₀> + c₁|Φ₁>.  Normalization: |c₀|² + |c₁|² = 1.
  Since c₀ and c₁ are real, we have c₀² + c₁² = 1.
  <X>  = <ψ|X|ψ> = (ħ/(2mω))^½<ψ|(a + a^†)|ψ>
  = (ħ/(2mω))^½[c₀c₁<Φ₀|a|Φ₁> + c₀c₁<Φ₁|a^†|Φ₀>]
  = 2(ħ/(2mω))^½c₀c₁ = 2(ħ/(2mω))^½c₀(1 - c₀²)^½.
  d<X>/dc₀ = 0 yields c₀ = 1/√2.  We then need c₁ = 1/√2 to have a maximum.
  |ψ> = (1/√2)|Φ₀> + (1/√2)|Φ₁>. 
  (We need c₀ and c₁ to have the same sign, obviously choosing both coefficients negative would also be ok.)
  (b)  <P> = ½i(mħω/2)^½[<Φ₀|a|Φ₁> - <Φ₁|a^†|Φ₀>] = 0.
  Denote the parity operator by Π.
  Π|Φ₀> = |Φ₀>, Π|Φ₁> = -|Φ₁>.
  <Π> = ½[<Φ₀|Φ₀> - <Φ₁|Φ₁>] = 0.

Problem:

Consider a particle in a harmonic oscillator potential with Hamiltonian H = p²/(2m) + ½mω²x².
Its state vector at  t = 0 is |ψ(0)> = exp(-|α|²/2)∑₀^∞(αⁿ/(√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(a^†a + 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:

• Concepts:
  The eigenstates of the harmonic oscillator, raising and lowering operators.
• Reasoning?
  The system is in a given linear superposition of eigenstates of the harmonic oscillator.
  We can do the problem by letting a and a^† operate on those eigenstates.
  (a)  a|n> = n^½|n-1>,  a^†|n> = (n+1)^½|n+1>.
• Details of the calculations:
  (a) a|ψ(0)> = exp(-|α|²/2)∑₀^∞(αⁿ/(√n!))a|n> = α exp(-|α|²/2)∑₁^∞(αⁿ(√n)/(√n!))|n-1>
  = α exp(-|α|²/2)∑₁^∞(α^n-1/√((n-1)!))|n-1> = α exp(-|α|²/2)∑₀^∞(αⁿ/(√n!))|n> = α|ψ(0)>.
  The eigenvalue of a is α.
  (b)  The possible values are E_n = (n + ½)ħω.  E_n is found with probability  exp(-|α|²)(α²ⁿ/n!).
  (c)  <H> = ħω<ψ(0)|(a^†a + 1)|ψ(0)> = ħω(<aψ(0)|aψ(0)> + ½) = ħω(|α|² + ½).
  <H²> = ħ²ω²<ψ(0)|(a^†a + 1)²|ψ(0)> = ħ²ω²(|α|⁴ + 2|α|² + ¼).
  ΔH = (<H²> - <H>)^½ = ħω|α|.
  ΔH/<H> = |α|/(|α|² + ½) ≈ 1/|α| << 1 if |α|  >> 1.
  (d)  U(t, 0)|ψ(0)> = exp(-iHt/ħ)|ψ(0)> = exp(-|α|²/2)∑₀^∞(αⁿ/(√n!))exp(-i(n + ½)ωt)|n>
  = exp(-iωt/2) exp(-|α|²/2)∑₀^∞(αⁿ/(√n!))exp(-inωt)|n> = exp(-iωt/2) exp(-|α'|²/2)∑₀^∞(α'ⁿ/(√n!))|n>,
  with α' = α exp(-iωt), |α'| = |α|.
  |ψ(t)> is an eigenstate of a as shown in part (a) with eigenvalue α'.
  (e)  <x(t)> = (ħ/(2mω))^½ <a + a^†>(t) = (ħ/(2mω))^½(<aψ(t)|ψ(t)> + <ψ(t)|aψ(t)>)
   = (ħ/(2mω))^½(α'* + α') = (2ħ/(mω))^½Re(α exp(-iωt)).
  <x>(t) oscillates with frequency ω like the classical coordinate x.  For large |α| the uncertainty in the energy is small compared to the value of the energy.

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> = n^½|n - 1>, find the expectation value of x⁴ for the system in the state |n>.
Show your work!

Solution:

• Concepts:
  The eigenstates of the harmonic oscillator Hamiltonian, the raising and lowering operators, the expectation value of an operator
• Reasoning:
  We are asked to express the harmonic oscillator Hamiltonian in terms of a and a^†.
• Details of the calculation:
  (a)  [p,p] = [x,x] = 0.  [x,p] = iħ.
  [a, a^†] = [(αx + iβp),(αx - iβp)] = iβα[p,x] - iβα [x,p] = 2iħβα = 1.
  (b)  H = p²/(2m)  + mω²x²/2.
  x = (a + a^†)/(2α),  p = -i(a - a^†)/(2β).
  x² = (1/(4α²))(aa + aa^† + a^†a + a^†a^†), p² = (1/(4β²))(-aa + aa^† + a^†a - a^†a^†).
  H = (ħω/4)(-aa + aa^† + a^†a - a^†a^†) + (ħω/4)(aa + aa^† + a^†a + a^†a^†)
  = (ħω/2)(aa^† + a^†a) = ħω(½ + a^†a).
  (c)  x² = (1/(4α²))(aa + aa^† + a^†a + a^†a^†).
  <n|x⁴|n> = (1/(16α⁴)) <n|(a + a^†)⁴|n> has 16 terms, but only those with equal numbers of a and a^† can be non-zero.  Only 6 terms survive.
  <n|aaa^†a^†|n> = (n + 1)(n + 2),  <n|a^†a^†aa|n> = n(n - 1),  <n|aa^†aa^†|n> = (n + 1)², 
  <n|a^†aa^†a|n> = n², <n|aa^†a^†a|n> = n(n + 1), <n|a^†aaa^†|n> = n(n + 1).
  <n|x⁴|n> = (6n² + 6n + 3)/(4α²).

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:

• Concepts:
  Postulates of quantum mechanics, the 1D harmonic oscillator.
• Reasoning:
  The mean value of any observable A is <ψ|A|ψ>. 
  The root-mean -quare deviation is ΔA = (<(A - <A>)²>)^½ = (<A²> - <A>²)^½.
• Details of the calculation:
  (a)  Let  X_s = (mω/ħ)^½X,  P_s =  (mωħ)^-½P.
  a = (2)^-½(X_s + iP_s), a^† = (2)^-½(X_s - iP_s).
  X_s = (2)^-½(a^† + a),  P_S = i(2)^-½(a^† - a),  {X_s,P_s] = i.
  (ΔX)² = <X²> - <X>²,  (ΔP)² = <P²> - <P>².
  For the harmonic oscillator in its ground state <X> = <P> = 0.
  <X_s²> = ½<0|aa + aa^† + a^†a + a^†a^†|0> = ½<0|aa^†|0> = ½.
  <P_s²> = -½<0|aa - aa^† - a^†a + a^†a^†|0> = ½<0|aa^†|0> = ½.
  <X_s²><P_s²> = ¼.  <X²><P²> = ¼ħ².  [X,P] = iħ.  |<[X,P]>|² = ħ².
  Therefore (ΔX)²(ΔP)² = ¼|<[X,P]>|²,
  the ground state is a minimum uncertainty state.
  
  (b)   a|b> = exp(-|c|²/2)∑₀^∞(cⁿ/(√n!))a|n> = c exp(-|c|²/2)∑₁^∞(cⁿ(√n)/(√n!))|n-1>
  = c exp(-|c|²/2)∑₁^∞(c^n-1/√((n-1)!))|n-1> = c exp(-|c|²/2)∑₀^∞(cⁿ/(√n!))|n> = c|b>.
  The eigenvalue of a is c.
  
  (c)  |Ψ(t)> = U(t,0)|b> = exp(-iHt/ħ)|b>
  = exp(-|c|²/2)∑₀^∞(cⁿ/(√n!))exp(-i(n + ½)ωt)|n>
  = exp(-iωt/2) exp(-|c|²/2)∑₀^∞(cⁿ/(√n!))exp(-inωt)|n>
  =  exp(-iωt/2) exp(-|c'|²/2)∑₀^∞(c'ⁿ/(√n!))|n>,
  with c' = c exp(-iωt), |c'| = |c|.
  |ψ(t)> is an eigenstate of a  with eigenvalue c'.
  
  (d)  |ψ(t)> is an eigenstate of the lowering operator.
  <X(t)> = (ħ/(2mω))^½<a^† + a>(t)
  = (ħ/(2mω))^½(<aψ(t)|ψ(t)> + <ψ(t)|aψ(t)>)
  = (ħ/(2mω))^½(c'* + c').
  <X(t)>² = (ħ/(2mω))(c'* + c')².
  <X²(t)> = (ħ/(2mω))<a^†2 + a² + 2a^†a + 1>(t)
  = (ħ/(2mω))((c'* + c')² + 1).
  (ΔX)² = (ħ/(2mω)).
  <P(t)> = i(mωħ/2)^½<a^† - a>(t) = i(mωħ/2)^½(c'* - c').
  <P(t)>² = -(mωħ/2)(c'* - c')².
  <P²(t)> = -(mωħ/2)<a^†2 + a² - 2a^†a - 1>(t)
  = -(mωħ/2)((c'* - c')² - 1).
  (ΔP)² = (mωħ/2).
  ΔXΔP = ħ/2.
  The coherent state is also a minimum uncertainty state.

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:

• Concepts:
  Taylor series expansion about a local potential minimum
• Reasoning:
  For an equilibrium point x₀ we have dV/dx|_x0 = 0, d²V/dx²|_x0 > 0.
  V(Δx) = ½d²V/dx²|_x0*(Δx)²
  Denote d²V/dx²|_x0 = V₀''.  Then the oscillation frequency is ω = (V₀''/m)^½.
• Details of the calculation:
  We have a periodic function with multiple equilibrium points.  We can pick any of them.
  sin kx ~ kx + (1/6)(kx)³,  V₀sin²kx = V₀(k²x² + (1/3)(kx)⁴ + ...) = ½V₀''x² + ...  .
  ω = (2V₀k²/m)^½.
  The value of x when the next term in the expansion of the potential becomes as important as k²x² is x₁ = (3)^½/k.
  Then k²x₁² = (1/3)(kx₁)⁴.
  We need Δx << x₁.  For the harmonic oscillator Δx = <x²>^½
  x = (ħ/(2mω))^½(a + a^†),  a|n> = (n)^½|n-1>,  a^†|n> = (n+1)^½|n+1>.
  x² = (ħ/(2mω))(a + a^†)(a + a^†), <x²> = (ħ/(mω))(n + ½).
  We therefore need nħ/(mω) << 3/k², or n << (18mV₀)^½/(ħk).