Past exam of the quantum theory course of the University of Oxford / 2015 / 2 / a

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- Solution a

Solution

The position operator

X

and momentum operator

P

are self-adjoint operators, so the adjoint operator is

a_{+} = P + imω X .

Use the canonical commutation relation with the order of factors preserved:

a_{+} a_{-} a_{-} a_{+} = (P + imω X) (P - imω X) = P^{2} + m^{2} ω^{2} X^{2} + imω (XP - PX) = P^{2} + m^{2} ω^{2} X^{2} - m ℏ ω = 2 m (H - \frac{1}{2} ℏ ω), = P^{2} + m^{2} ω^{2} X^{2} - imω (XP - PX) = P^{2} + m^{2} ω^{2} X^{2} + m ℏ ω = 2 m (H + \frac{1}{2} ℏ ω) .

Subtracting the first product from the second proves

[a_{-}, a_{+}] = 2 m ℏ ω I .

These creation and annihilation operators are unscaled and carry the paper's phase convention; their commutator is not the unit-normalized oscillator commutator.

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