All Questions
4 questions
3
votes
2
answers
833
views
Least action principle in imaginary time
In quantum mechanics, the amplitude of wave function propagation can be found using the Feynman's path integral
$$
\langle z'|e^{-itH/\hbar}|z\rangle=\int\limits_{x(0)=z\\x(t)=z'} Dx(t')\:
\exp\left\{\...
- 2,393
2
votes
0
answers
117
views
How to rigorously take the classical limit of a thermal correlation function
I am interested in how one formally takes the classical limit of:
$$ \langle A(x_0)B(x_t)\rangle = \mathrm{Tr}[e^{-\beta \hat{H}}A(\hat{x})e^{i \hat{H}t/\hbar}B(\hat{x})e^{-i \hat{H}t/\hbar}]$$
i.e. ...
- 418
11
votes
1
answer
605
views
Statistical path integral normalization
So I am looking at a statistical path integral, meaning that I work with an Euclidean action. The propagator of my (Wiener) path integral is given by:
$$
K(x_T,T|x_0,0)=\int\limits_{x(0)=0}^{x(T)=x_T}\...
- 3,132
1
vote
2
answers
1k
views
Phase space derivation of quantum harmonic oscillator partition function
I would like to derive the partition function for the quantum Harmonic oscillator from scratch:
$$\tag{1} Z = \int dp \, dx\, e^{-\beta H}.$$
The free particle appears in many textbooks. $H = p^2$...
- 788