In physics, an operator is almost always either a square matrix or a linear mapping from one space of functions (often on $\mathbb{R}^N$ or $\mathbb{C}^N$) to the same or other like space of functions. Operators serve as *observables* and as *time evolution operators* in Quantum Mechanics. This tag ...

learn more… | top users | synonyms

3
votes
0answers
29 views

Baryonic operators in ${\cal N}=1$ $U(N)$ SQCD in four dimensions

Seiberg's duality is usually considered as a duality for $SU(N_c)$ theories with $N_f$ flavors. In his case, the vacuum for $N_f \geq N_c$ is parameterized by mesons $M$ and baryons ${\bar B}$ and $B$....
1
vote
1answer
46 views

Why is the angular momentum added for two independent electron system? (no problem)

There is no problem now. But somebody may be confused by the same analysis when studying QM or Group theory. (actually my motivation for asking this question comes from the SU(5) Grand Unification ...
1
vote
2answers
90 views

Negative powers of operators

This may sound like a strange question, but just to be sure: Suppose I have a general Hermitian operator in Hilbert space whose action on an eigenvector is given by $R|r\rangle = r|r\rangle$. Then, I ...
4
votes
2answers
132 views

Radial quantum number for infinite circular well

For completeness, I will sketch the solution of a particle in an infinite circular well first and then get to my question. I apologize in advance since the introduction is standard undergraduate ...
-2
votes
1answer
61 views

Limit of the position and momentum commutator [closed]

The commutator of position and momentum operator, $\hat{p}$ and $\hat{x}$, respectively is derived as $[\hat{x},\hat{p}]=i\hbar$. Let $\lim_{x\rightarrow x_{o}} [\hat{x},\hat{p}]=\lim_{x\rightarrow x_{...
0
votes
0answers
49 views

Spin-1/2 rotation operator: rotation about an angle of $\pi$

The spin-1/2 rotation operator: $$ R_{n}(\alpha) = \begin{pmatrix} cos(\frac{\alpha}{2})-in_{z}sin(\frac{\alpha}{2}) & (-in_{x}-n_{y})sin(\frac{\alpha}{2}) \\ (-in_{x}+n_{y})sin(\frac{\...
3
votes
1answer
137 views

Is the energy always discrete?

In the von Neumann axioms for quantum mechanics, the first postulate states that a quantum state is a vector in a separable Hilbert space. It means it is assumed the Hilbert space has a basis with at ...
0
votes
1answer
58 views

Spectrum of Laplacian on one hemisphere

as is well-known, the spectrum of the Laplace operator on $S^2$, computed via $-\Delta f=\lambda f$, is positive and discrete. What happens to the spectrum if we just take one hemisphere into account?...
0
votes
0answers
37 views

Lorentz invariance of matrix element of Heisenberg operator

The following text is taken by Weinberg book of QFT Volume 1, pg.437 Let's consider $O_l(x)$ an Heinsenberg-picture operator with the Lorentz transformation properties of some sort of free field $\...
0
votes
0answers
32 views

Can we create ladder operators for any potential? [duplicate]

I understand the mechanism for the harmonic oscillator, so can we generalize the ladder operator in order to work for any 1-D symmetric potential such as Gaussian potential well?
1
vote
2answers
141 views

Are the eigenstates of an operator time independent?

In the Schrodinger picture, are the eigenstates of an operator time independent? Is it their expectation values that evolve in time rather than the actual eigenstates? For example, say I have an ...
1
vote
3answers
137 views

Prove time-dependent hamiltonian is hermitian from unitarity of time-evolution operator

When we solve the Schrodinger equation for the time-evolution operator: \begin{equation} i\hbar\frac{\partial}{\partial t}U(t,t_{0})=HU(t,t_{0}), \end{equation} We have three cases to be treated ...
0
votes
0answers
48 views

The commutator between observable and unit radius vector

As I encounter the commutator relating to unit radius vector, I am quite confused. I have just started the learning of quantum mechanics and all I know about the commutator is based on two identities: ...
3
votes
0answers
53 views

commutation relations in terms of eigenstates scalar product

This question has caught my attention because I was unaware of the fact that the position-momentum canonical commutation relations could be derived out of the only assumption for $\langle x | p\rangle$...
3
votes
0answers
117 views

Why Use the Non-Relativistic Momentum Operator in Relativistic Quantum Mechanics?

In deriving the Klein Gordon equation one starts out with the relativistic energy relation E^2 = p^2 + m^2 and substitutes the quantum momentum operator that corresponds to non-relativistic QM, p = -i ...
0
votes
1answer
91 views

Confusion with time ordering

I am thinking about Proof of correlation function formula in quantum field theory and have realized there is a deeper confusion underpinning that. Consider: $$T\{U_I(T, t_2)\Phi_I(x_1)\}$$ where $...
1
vote
1answer
85 views

Clarification in deriving the radial momentum operator $p_r$

In deriving an expression for $p_r$, a particle's radial momentum, I am unsure what is happening at a certain step. The derivation given in The Physics of Quantum Mechanics by Binney and Skinner is as ...
2
votes
1answer
177 views

Recovering QM from QFT

Reading through David Tong lecture notes on QFT. On pages 43-44, he recovers QM from QFT. See below link: QFT notes by Tong First the momentum and position operators are defined in terms of "...
1
vote
1answer
133 views

Time-ordered product vs path integral

Suppose we have the Green function $$ G(k) \equiv \tag 1\int d^4x e^{ikx}\langle 0| T\left(\partial^{x}_{\mu}A^{\mu}(x)B(0)\right)|0\rangle , $$ which in path integral approach is equal to $$ \tag 2 ...
4
votes
2answers
217 views

Derivation of canonical position-momentum commutator relation

We know that the position-momentum commutator is fundamental in quantum mechanics, but would it be possible to derive it starting from a different set of first principles, more specifically starting (...
-1
votes
2answers
91 views

Fermion Lagrangian with linear momentum versus quadratic momentum

$$ L = \bar{\psi} (\gamma^\mu (p_\mu -A_\mu)- m)\psi \tag{1} $$ $$ L = \bar{\psi} ((\gamma^\mu( p_\mu-A_\mu))^2 - m^2)\psi \tag{2} $$ Is there a difference between the two Lagragians in equations 1 ...
0
votes
1answer
65 views

Definition of leading log terms in one loop corrections for gravity

One loop corrections for gravity usually includes non-local terms in the action such as $R\log(\frac{-\Box}{\mu^2})R$, where $\Box=g^{\mu\nu}\nabla_\mu\nabla_\nu$ is the D'Alembert operator, $R$ is ...
4
votes
2answers
131 views

Interpretation of $\langle \phi | A | \psi \rangle$ [duplicate]

If the current state of some quantum system is $| \psi \rangle$, what is the physical interpretation of $$ \langle \phi | A | \psi \rangle $$ where $|\phi\rangle$ is some other -maybe the same- ...
1
vote
1answer
49 views

Second Quantization: The Identity Operator does not Commute?

Let me take the simplest possible example. Consider the fermonic Fock-space $\Lambda^*(\mathbb{C}^n)$ built out of a finite-dimensional, oriented single-particle Hilbert space $\mathbb{C}^n$ with ...
0
votes
1answer
95 views

Operator algebra for momentum and magnetic vector potential

Let $\vec{A}$ be the magnetic vector potential and $\vec{p}$ be momentum. $$ \vec{p} \cdot \vec{A} \psi = (\vec{p} \cdot \vec{A}) \psi + \vec{A} \cdot (\vec{p} \psi) $$ $$ \vec{A} \cdot \vec{p} \psi =...
2
votes
1answer
122 views

In the algebraic formulation of Quantum Mechanics, how do probability amplitudes naturally arise?

In the algebraic formulation of quantum mechanics, consider $\mathcal{B}(\mathcal{H})$ as the set of all bounded operators on $\mathcal{H}$ (with involution, norm, etc.), which form a C*-algebra $C$. ...
1
vote
1answer
46 views

Expectation to uncertainty

We know that in the case of $O$ being an operator, $\langle O^2\rangle-\langle O\rangle^2$ equals to uncertainty as long as $\langle\rangle$ means the mean value (expectation value). if we have $A$ ...
13
votes
4answers
927 views

Can momentum have a complex expectation value?

I'm making examples of wave functions to incorporate in a QM exam. I came up with the following wave function, which gives me some troubles: $$\psi(x,0) = \begin{cases} A(a-x), & -a \leq x \leq a\...
5
votes
2answers
534 views

Mutual or same set of eigenfunctions if two operators commute

If two operators commute, do they have "a mutual set of eigenfunctions", or "the same set of eigenfunctions"? My quantum chemistry book uses these as if they are interchangeable, but they do not seem ...
1
vote
0answers
33 views

Spin operators in quantum mechanics

I'm reading for my exam on Monday. In my notes I have written that the teacher has told us that: $S_1 \cdot S_2 = [(S_1 + S_2)^2 - S_1^2 - S_2^2]/2 = S_{tot}^2 - \frac{3 \hbar^2}{4}$ Where $S_i = (...
1
vote
1answer
131 views

Expectation value of an operator and commutator relation

I have a quantum operator $A.$ It's expectation value is constant respect to time. I mean $$\langle \psi(t)|A|\psi(t)\rangle$$ is a constant values. If I know $|\psi(t)\rangle$ is not an eigenstate ...
6
votes
2answers
241 views

Is there a time operator in quantum mechanics?

The question in the title has been asked many times on this site before, of course. Here's what I found: Time as a Hermitian operator in QM? in 2011. Answer states time is a parameter. Is there an ...
2
votes
1answer
103 views

Inserting the resolution of identity correctly

In a text on path integrals, I find the following: \begin{equation} \langle q_{j+1}|e^{-i(\hat{p}/2m)\delta t}|q_j\rangle = \int\frac{dp}{2\pi}\langle q_{j+1}|e^{-i(\hat{p}/2m)\delta t}|p\rangle\...
0
votes
0answers
160 views

Perturbation Theory Question

How can you work out the average perturbation, from a normal hamiltonian, of all states that rely on the quantum numbers of s = __ and l = __, with the perturbation being proportional to the product ...
1
vote
1answer
89 views

Momentum in quantum harmonic oscillator with step up and step down operators [closed]

I'm hitting a wall in my understanding of the momentum operator in a quantum harmonic oscillator. I've showed that $p = (a^\dagger - a)\sqrt{\frac{m w \hbar}{2}}i$ where $a^\dagger$ and $a$ are the ...
0
votes
3answers
170 views

Derivation of momentum operator

From a video lecture on quantum mechanics at MIT OCW I found that you didn't need to know Schrödinger's equation to know the momentum operator which is $\frac{\hbar}{i}\frac{\partial}{\partial x}$. ...
2
votes
1answer
78 views

How do we take the limit of this quantum operation?

I am wondering how to take the following limit: \begin{align} L= \lim_{\tau \to \infty} \frac{1}{\tau} \int_{-\tau/2}^{\tau/2} dy \, \left(1 - \frac{1}{\sqrt{ \pi} \sigma } \int_{-\tau/2}^{\tau/2}...
1
vote
0answers
53 views

Hamiltonian in commutator contradiction [duplicate]

Consider the following: $$[ \hat H, \hat x]=\left[-\frac{\hbar^2 \hat p^2}{2m}+V,\hat x\right]\ne0 \text{ in general}$$ But $$[ \hat H, \hat x]=\left[i\hbar \frac{\partial }{\partial t},\hat x\right]...
5
votes
1answer
89 views

Nonabelian global symmetries, $SO(N)$ charges in terms of creation and annihilation operators

Consider an $SO(N)$ symmetric theory of $N$ real scalar fields,$$\mathcal{L} = {1\over2} \partial_\mu \Phi^a \partial^\mu \Phi^a - {1\over2} m^2 \Phi^a \Phi^a - {1\over4} \lambda (\Phi^a \Phi^a)^2.$$...
1
vote
2answers
79 views

Operators for a Perturbed Hamiltonian: Heisenberg Picture ($\hat{x}$, $\hat{p}$)

Problem I am trying to calculate the Equations of Motion in the Heisenberg picture for $\hat{x}$ and $\hat{p}$ in a perturbed Hamiltonian, $$ \tag{1} \hat{H} = \hat{H}_0 + \hat{H}' $$ Assume ...
0
votes
0answers
73 views

How to Fourier transform creation/annihilation operators?

Zee's QFT in a Nutshell pages 65-66. For a complex scalar QFT $$ \varphi(\vec{x},t) = \int\frac{d^Dk}{\sqrt{(2\pi)^D2\omega_k}}\left[a(\vec{k})\mathrm{e}^{-i(\omega_kt-\vec{k}\cdot\vec{x})} + b^\...
0
votes
1answer
69 views

Momentum operator derivation in QFT from QM

In David Tong`s QFT notes there is a chapter about the derivation of the momentum operator from quantum mechanics (page 44) where he is showing that the momentum operator can be expressed by the ...
0
votes
1answer
90 views

Do different creation/annihilation operators always commute?

In a complex (non-hermitian) scalar QFT, is it correct that the creation/annihilation operators $a,a^\dagger$ (particle) and $b,b^\dagger$ (anti-particle) commute, i.e. $[a,b] = [a,b^\dagger] = [a^\...
0
votes
1answer
21 views

State operator map, and scalar fields in $R\times S^{D-1}$ to $R^D$

This question is generalized version of my previous questions State operator map in $R \times S^{D-1}$ to $R^D$ , State-operator map, and scalar fields and State operator correponding $i.e$ $S^1\times ...
-1
votes
1answer
43 views

Parity operator eigenstates [closed]

I have a problem I cannot solve on my own. I have given two states $\psi_1$ and $\psi_2$ and an Operator $O$ such that $P \psi_1 = \epsilon_1 \psi_2$, $P \psi_2 = \epsilon_2 \psi_2$ and $POP^{-1} = ...
0
votes
1answer
54 views

Is the scalar field operator self-adjoint?

In A. Zee's QFT in a Nutshell, he defines the field for the Klein-Gordon equation as $$ \tag{1}\varphi(\vec x,t) = \int\frac{d^Dk}{\sqrt{(2\pi)^D2\omega_k}}[a(\vec k)e^{-i(\omega_kt-\vec k\cdot\vec x)...
0
votes
1answer
41 views

State operator map in $R \times S^{D-1}$ to $R^D$

This question is relevant in my former question State-operator map, and scalar fields and State operator corrponding $i.e$ $S\times S$ to $R^2$. (which was wrong, corrected one was states in $R \times ...
-1
votes
1answer
72 views

Parity operator with other operators

I need to show the following: $$P x P^{-1} = -x, \ P p P^{-1} = -p, \ P L P^{-1} = L$$ where $P$ is the parity operator and $x$, $p$ and $L$ are the position, momentum and angular momentum ...
1
vote
1answer
38 views

State-operator map, and scalar fields

Up so far, i have been studied state-operator correspondence, $i.e$, i have been questioned State operator corrponding $i.e$ $S\times S$ to $R^2$ which was wrong question. By studing Ginsparg's ...
0
votes
1answer
45 views

quantum mechanics probability of +1 spin between arbitrary directions

So there are two unit vectors $\hat{m}$ and $\hat{n}$ with arbitrary directions in 3D space. There is a spin operator along a particular direction in space, say that of $\hat{r}$, is: $\sigma_r= \...