Linked Questions

10
votes
1answer
5k views

Why/How is this Wick's theorem?

Let $\phi$ be a scalar field and then I see the following expression (1) for the square of the normal ordered version of $\phi^2(x)$. \begin{align} T(:\phi^2(x)::\phi^2(0):) &= 2 \langle 0|T(\...
17
votes
5answers
566 views

Can I swap quantum mechanical ground state for some classical trajectory distribution and have it sit still after the swap?

Suppose that I have a single massive quantum mechanical particle in $d$ dimensions ($1\leq d\leq3$), under the action of a well-behaved potential $V(\mathbf r)$, and that I let it settle on the ground ...
8
votes
2answers
1k views

How to promote algebraic expressions to operators in quantum mechanics?

Okay, I know that in quantum mechanics the quantum observable is obtained from the classical observable by the prescription $$ X \rightarrow x,\quad P \rightarrow -i\hbar\frac{\partial}{\partial x} $...
5
votes
2answers
1k views

Normal Order of Normal Order

in first volume of Polchinski page 39 we can read a compact formula to perform normal-order for bosonic fields $$ :\cal F:=\underbrace{\exp\left\{\frac{α'}{4}∫\mathrm{d}^2z\mathrm{d}^2w\log|z-w|^2\...
10
votes
2answers
517 views

Motivation for Wigner phase space distribution

Most sources say that Wigner distribution acts like a joint phase-space distribution in quantum mechanics and this is justified by the formula $$\int_{\mathbb{R}^6}w(x,p)a(x,p)dxdp= \langle \psi|\...
10
votes
1answer
807 views

In quantum mechanics, how exactly do we associate Hermitian operators to classical observables? [duplicate]

In a first course on quantum mechanics, everybody learns some version of the following statement: Postulate: To every classical observable $A$ of a physical system, there corresponds a Hermitian ...
3
votes
2answers
999 views

Examples of Weyl transforms of nontrivial operators

I've been able to find examples of Weyl transforms of operators like $\hat{x}$,$\hat{p}$, and $\hat{1}$, but not anything more complicated. Are there derivations of the Weyl transforms of more ...
2
votes
1answer
538 views

Can I Weyl-order the following Hamiltonian?

I am trying to perform a path integral but I am having trouble with the Weyl ordering of my Hamiltonian. The Lagrangian of the system in question is $$L~=~\frac{1}{2}f(q)\dot{q}^2,$$ where $f(q)$ ...
5
votes
2answers
189 views

Why any operator is specified by this characteristic function?

On the paper "Tutorial Notes on One-Party and Two-Party Gaussian States", arXiv:quant-ph/0307196, the author states on section 2: Any operator referring to a harmonic oscillator — position operator ...
1
vote
1answer
321 views

Weyl and Normal ordering in QFT

In my QFT course, if I understood well, we define the normal ordering as the way to quantize a system where we put the creation operators at the left and the annihilation ones at the right. For ...
1
vote
1answer
88 views

How does the following commutator for measured observables and this operator relation imply the following relation?

$$ \hat{\Omega}_j{(\tilde{q}_j)}=\Omega_j(\tilde{q}_j-\hat{q}_j) $$ $$ [\hat{q}_j,\hat{q}_l]=ik_{jl} $$ Implies $$ [\hat{q}_j,\hat{\Omega}_l]= \frac{\partial\Omega_l(\tilde{q}_l-\hat{q}_l)}{\...