To be used in future
 Section 5.11: Partition sum Up Chapter 5: Path Integrals Section 5.13: Potential scattering 

5.12 Perturbation theory for the ground state energy

We can use the result for the action to get a very powerful way to derive the time-independent perturbation theory for the ground state. The starting point is the fact that in the low temperature limit the only state contributing to the partition sum is the ground state, and use the path integral representation for . Actually, we only need to look at since as well.
So lets now look at this object. We first shift the integration boundaries so the result looks more symmetric, and for definiteness we shall look at the quartic anharmonic oscillator The path integral we need to evaluate (in the limit is The cute trick is to evaluate a different path integral instead, and the relate the two: If we functionally differentiate w.r.t. we bring down an : We thus find that we can write (formally) that Specifically, we can get the simple expression to first order in that We now will evaluate --that this can be done should be obvious, because it is just the Euclidean version of the harmonic oscillator in an external field , which is still quadratic. We can’t trivially complete the square since depends on . Instead we integrate by parts to find Taking the stationary value of the action, i.e., the exponential in (↓), we get the Euclidean (imaginary time) equations of motion
This can be solved if we can find the Green function of the differential equation, by and we can simplify (↓) to
As the Green function is simple (fill in the blanks), and we have
 Section 5.11: Partition sum Up Chapter 5: Path Integrals Section 5.13: Potential scattering