# Questions tagged [operators]

In physics, an operator is almost always either a square matrix or a linear mapping between two function spaces (defined on, say, $\mathbb R^n$). Operators serve as *observables* and as *time evolution operators* in Quantum Mechanics. This tag will most often find valid use in quantum mechanics; don't use this tag just because your equations contain "everyday operations" like $\times$, $+$!

2,066 questions
1answer
62 views

### What does the notation $\langle a|b|c\rangle$ mean? [on hold]

What does the notation $\langle a|b|c\rangle$ mean? I saw this in a Quantum Mechanics book and couldn’t understand it
1answer
120 views

0answers
27 views

1answer
67 views

### Expectation value of derivative of operator

I was given the following question: Let $A(\lambda)$ be a Hermitian operator, which is dependent on some real parameter $\lambda$. Let us denote the eigenvalues and corresponding eigenstates of $A$ ...
1answer
62 views

### Question about field quantization

I'm reading a paper by Rubin, Klyshko, Shih and Sergienko titled "Theory of two-photon entanglement in type_II optical parametric down-conversion", 1994 (link to the paper). I'm stumped by equation ...
3answers
41 views

### Does the set of the degenerated eigenfunctions of hamiltonian forms a subspace?

I have read in a book that the set $\{ \Psi_{n}^{(\nu)} \in \mathcal{H} | \ \ \hat{H}\Psi_{n}^{(\nu)} = E_{n}\Psi_{n}^{(\nu)} \}$ (that is, the set of all eigenfunctions of the hamiltonian with the ...
1answer
51 views

### Problem with expansion of normal ordering

I am reading normal ordering..and far now I'm able to understand. I am stuck in third line from second expression in the book Lectures On Quantum Field Theory By Ashok Das in page no. 237. It is ...
1answer
21 views

### Questions about finding “top rung” and “bottom rung” of angular momentum operator (Proof in Griffiths)

The problem is like this: Let $$L_x = yp_z - zp_y, L_y = zp_x - xp_z, L_z = xp_y- yp_x, \\ L^2 = {L_x}^2 + {L_y}^2 + {L_z}^2 \\ L_\pm \equiv L_x \pm iL_y$$ We wish to find a "top rung" $f_t$ and ...
1answer
59 views

### Rising/lowering operators and trigonometric functions

I've just started learning about angular momentum and spin theory, and when I came across the definitions of the rising and lowering operators, I noticed the inverse form looks suspiciously like the ...
1answer
88 views

### What is Wick's theorem and what this is use for? [closed]

I am reading Wick's theorem but although I look for it to clearly understand in some textbooks and youtube videos but still it is unclear to me. I cannot get my head over what is normal ordering ...
3answers
100 views

### What are the eigenvector's of the $\hat a^2$ operator?

Since $\hat a^2$ and $\hat a$ commute, then one of the eigenvectors of $\hat a^2$ will be, the coherent state $|\alpha\rangle$. Are there others states as well?
2answers
126 views

### Rigorously why there should be an operator product expansion in conformal field theory?

This is probably something quite trivial I'm not getting. I'm studying CFT (conformal field theory) through David Tong's lecture notes and on page 9 he says: We now define the operator product ...
0answers
37 views

### Which orderings are commonly used in modern physics?

I am curious as to which mathematical orderings are used in contemporary theoretical physics, and in what context/situations. So far, I have encountered the following: Time ordering: Commonly used ...
2answers
116 views

### Transformation connecting two representations - Quantum mechanics [duplicate]

I am working on Dirac's paper The Lagrangian in Quantum Mechanics. He looks for the analogy between a classical transformation between two sets of coordinates and momenta $p_r$, $q_r$ and $P_r$, $Q_r$ ...
2answers
85 views

### Why does the number operator gives the number of excitations?

Could somebody explain why the number operator (for a simple harmonic oscillator) gives the number of excitations? I understand its definition and its relation to the Hamiltonian, but I just can't ...
1answer
49 views

### Derivating operator acting on ket

I'm deducing a formula, and I used the "product rule" $\frac{\partial}{\partial t}(A|\phi>)=(\frac{\partial A}{\partial t})|\phi>+A\frac{\partial}{\partial t}|\phi>$. I'm actually getting the ...
1answer
39 views

### Quantum mechanics angular momentum spherical tensor components

In Sakurai Quantum Mechanics, problem 3.25b we imagine $J_z^2$ as the component of a tensor with components $T_{ij} = J_iJ_j$. $J_z^2 = \frac{1}{3}\pmb{J}^2 + (J_z^2 - \frac{1}{3}\pmb{J}^2)$ The ...
2answers
129 views

### $T \bar{T}$ OPE

In page 157 of Di Francesco (Conformal Field Theory) it is said that the holomorphic and antiholomorphic components of the energy-momentum tensor have the trivial OPE $T(z) \bar{T}(\bar{w}) \sim 0$....
1answer
57 views

1answer
48 views

### How to derive the analytical expression for the retarded Green's function with quadratic Hamiltonian?

For two operators, $A(t)$ and $B(t)$ the retarded Green’s function is defined as \begin{equation} G^R(t,t') \equiv \langle \langle A(t)|B(t) \rangle \rangle^R = -i\theta(t-t')\langle \{A(t),B(t')\} \...
1answer
62 views

### Proof for $\langle i[A,B]\rangle$ [closed]

I have to prove the following equation: $$\langle i[A,B]\rangle = 2\mathfrak{Im}\left[\int dV(\overline{B\psi)}(A\psi)\right]\,,$$ where A,B are hermitian operators. Here is my calculation, but I don'...
0answers
37 views

### How to understand notation in “Introduction to Quantum Mechanics (3rd Edition)” by David Griffiths, Chapter 3.6.2?

In the 3rd edition, on page 118, the projection operator is introduced as $$\hat{P}=|\alpha\rangle\langle\alpha|.$$ Then Griffiths says that when $\hat{P}$ acts on another vector, it looks like ...
2answers
81 views

### Finding the Green function of an operator in QFT

I'm working on some quantum field theory and have to operate on a field with the following operator: $$(x^\mu \partial_\mu + 1)^{-1}$$ I've been trying to find an explicit form of this operator, ...
1answer
62 views

1answer
173 views

### What Lie group structure is used for infinite-dimensional Unitary Groups in Quantum Mechanics?

Given an infinite-dimensional Hilbert space $H$, the set $U(H)$ of all unitary operators on $H$ forms a group, known as the unitary group. Now several subgroups of this group play an important role ...