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Time evolution of the ladder operators

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The evolution operator U(t,0) of a one-dimensional harmonic oscillator is written:
U(t,0) = e^(-iHt/h)
with:
H = hw(a^t*a + 1/2)

Consider the operators:
a(t) = U^t(t,0) a U(t,0)
a^t(t) = U^t(t,0) a^t U(t,0)

By calculating their action on the eigenkets | phi(n) > of H, find the expression for a(t) and a^t(t) in terms of a and a^t.

See attached file for question in proper mathematical format.

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The solution shows how to calculate the time evolution of the ladder operators of the simple harmonic oscillator.

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