To be used in future
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5.6 Time-dependent oscillator
A related problem is the time-dependent oscillator, where
The problem can be set out in exactly the same way as above, keeping track of the fact that
is now time-dependent,
The only difference is in the form of the matrix
We get
We can give a very elegant form for the determinant
We now write the difference equation
What we do is to develop the determinant
w.r.t. the last row. The very last entry comes with a plus sign, and the
gets an additional minus sign. Thus we get two terms
The first determinant is
, and the second determinant can be developed by the final column, and is thus equal to
, i.e., we find
From explicit avaluation we find
, and thus we need to use
and
as our starting values for the recurrence relation.
which we can rewrite as (using the explicit form of
)
With the definition
we find that as we take
, we get the second order differential equation
with boundary conditions (this is where we see the reason for scaling
by
)
Thus the path integral takes the form
where the phase factor is the classical action, since the Lagrangian is quadratic (see the following section for more details). This can be evaluated, but requires solution to the differential equation
which clearly depends on the form of
. This then still needs to be substituted into the Lagrangian, and integrated over the duraction of motion.
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