Advanced Quantum Mechanics II PHYS 40202

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To be used in future

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5.7 Semi-classical limit

5.7.1 Classical limit

The classical limit of the path integral is when we look at the path that dominates--where the phase of the path integral changes as little as possible with a change of path, i.e.

This is the path described by the classical equations of motion. We can convince ourselves that if

, the action for the classical path, is much larger than

, this indeed dominates. If the action changes rapidly for neighbouring paths, then a group of nearby paths contributes each with an essentially random phase, and this averages out to zero. Near the classical path, all paths have almost the same phase, and these phases add coherently.

5.7.2 Asymptotic analysis of integrals

The argument can be made more precise by considering an integral of the form

in the limit

. Suppose that the integral has an extremum at

. we can then approximate the exponential by

Since

tends to zero, the Gaussian is incredibly sharply peaked, and we can move the boundaries on the integral without making an error,

Since in a Gaussian integral the typical scale of

, we can convince ourselves that a cubic term would contribute

On dimensional grounds we know that

suggesting thatwe can order this as an expansion in powers of

, since we alwasy have at least one more power of

than of

5.7.3 Gaussian expansion of path integrals

Now let us tackle the path integral again, and assume that the classical action

, so that we effectively are in the asymptotic regime discussed above [Y] [Y] This is usually (and incorrectly!!!) called the “limit

”. Since

is a physical and dimension-carrying quantity, there is no way we can take that limit in a meaningful way. As ever, in a mathematical analysis only dimensionless numbers play a role.. We expand about the classical path(s)--there can be multiple ones, but we shall assume a single one for the case of this argument--to second order, using the definition

. With the definitions

we can make an expansion to second order and change integration variables,

in complete analogy with the previous section. It is very easy to see from

that

with

We can evaluate the integral, since we have once again obtained the time-dependent harmonic oscillator. The boundary conditions on

mean that we don’t have to consider the two end points (their contribution is zero since

is zero); the classical action for

is also obviously equal to zero. Thus we find that we only need to evaluate the determinant

from its differential equation

and we get

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