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# 7 Quantum Harmonic Oscillator

Having shown an interconnection between the mathematics of classical mechanics and electromagnetism, let's look at the driven quantum harmonic oscillator too. Let's start with a one-dimensional quantum harmonic oscillator in its ground state at time t = 0 , and apply a force F(t) . The Hamiltonian is therefore

 H(t) = + mx 2 - F(t)x (37)

and checking the classical equations of motion
 = - = - mx + F(t) = = (38)
give Newton's equations of motion. The wave function satisfies Schrödinger's equation

 H(t)|(t) = - |(t) . (39)

For short times, this can be integrated

 |(t + t) = e - H(t)t|(t) . (40)

Let's assume that the force is weak, and let's calculate the expectation value of the operator x as a function of time. We write the Hamiltonian as

 H(t) = H0 - F(t)x (41)

and the short time solution is

 |(t + t) = e - H0t[1 - F(t)xt]|(t) + O([F(t)t]2) . (42)

We can write

 |(t) = (0) (43)

where t = t/N , and the order of the operators must be kept since x and H0 do not commute. Taking the limit that N goes to infinity and keeping only linear order terms, this becomes

 |(t) = e - H0t|(0) - dt'e - H0(t - t')F(t')xe - H0t'|(0) (44)

The expectation value of x to linear order in F is then
 x(t) = (t)|x|(t) = (0)|e H0txe - H0t|(0) - (0)|e H0txdt'e - H0(t - t')F(t')xe - H0t'|(0) + dt'(0)|e H0t'F(t')xe H0(t - t')xe - H0t|(0) . (45)

We now use the boundary condition that |(0) = |0 is the ground state with

 H0|0 = |0 . (46)

and x times the ground state is proportional to the first excited state,

 x|0 = |1 (47)

where

 H0|1 = |1 . (48)

Substituting these results, the expectation value of x for a weak force is
 x(t) = - 2Re dt'e - i(t - t')F(t') = dt'sin([t - t'])F(t') (49)
which agrees with the classical result using the Green's function solution.

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