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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 onedimensional 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
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) = H_{0}  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
where
t = t/N , and the order of the operators must be kept
since x and H_{0} 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^{ H0t}xe^{  H0t}(0)
 


 (0)e^{ H0t}xdt'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
and x times the ground state is proportional to the first excited state,
where
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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Previous: 6 Application to the