 Solution

ID: past-exam-of-the-theoretical-physics-ii-course-of-the-university-of-cambridge/2017/2/a/solution

Complete the square in the potential energy. With

x_{F} = F / (M ω^{2})

H = \frac{p ^{2}}{2 M} + \frac{M ω ^{2}}{2} (x - x_{F})^{2} - \frac{F ^{2}}{2 M ω ^{2}} .

The quantum harmonic oscillator is translated without changing its level spacing. If

φ_{n} (x)

is its normalized unforced energy eigenstate, then

ψ_{n}^{(F)} (x) = φ_{n} (x - x_{F}), E_{n}^{(F)} = ℏ ω (n + \frac{1}{2}) - \frac{F ^{2}}{2 M ω ^{2}}, n = 0, 1, 2, \dots .

Equivalently,

∣ n; F ⟩ = e^{- i x_{F} p /ℏ} ∣ n ⟩

. The ground state position is centered at

x_{F}

ℏ/ (2 M ω)

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