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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 $\times$, $+$!

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Slater determinant in second quantization using the creation operators help

$$ \left\langle 0\left|\hat{\Psi}\left(x_{1}\right) c_{\alpha_{1}}^{\dagger}\right| 0\right\rangle=\left\langle 0\left|\varphi_{\alpha_{1}}\left(x_{1}\right)-c_{\alpha_{1}}^{\dagger} \hat{\Psi}\left(...
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3answers
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Is there any intuitive reason behind why should the eigenfunctions of observables form a basis for our Hilbert space?

Is there any intuitive reason behind why should the eigenfunctions of observables form a basis for our Hilbert space ? For example, in the case of Stern-Gerlach experiment, sending the beam that has ...
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0answers
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Double-trace operators in CFT?

What is the conceptual difference between so called "single-trace" and "double-trace" (or "multi-trace") operators e.g., in a Conformal Field Theory?
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1answer
42 views

Anti-commutation relations in annihilation operators

It is claimed that $$\{c_\alpha,c_\beta \} = c_\alpha c_\beta + c_\beta c_\alpha = 0$$ where $c_\alpha$ and $c_\beta$ are the fermionic annihilation operators in second quantization. Why is that ...
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How to write an operator in matrix form?

Say I have the following operator: $$\hat { L } =\hbar { \sum_{ \sigma ,l,p } { l } \int_{ 0 }^{ \infty }\!{ \mathrm{d}{ k }_{ 0 }\,\hat { { { a }}}_{ \sigma ,l,p }^{ \dagger } } } \left({ k }_{...
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1answer
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Negative unity matrix not hermitian? (stabilizer formalism)

I read the section in the attached picture about the stabilizer formalism and was wondering about the last sentence in the pic. It says that all operators of the stabilizer group are hermitian, ...
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34 views

Operators evolution

I have a little question about equation for creation/annihilation operators. Usually we obtain time evolution equation for these operators from Heisenberg equation. for example: $$\frac{da_l}{dt} = -...
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15 views

Conjugate of total spin operator

I got a lattice, and the total spin operator for x and for y, for that lattice. I know that the x component conmutes with an operator called staggered spin operator in y. I also know that the ...
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Magnitude of mass polarization

When solving many-body atomic Hamiltonians, (such as Helium), there exists a term in the Hamiltonian which looks something like: $$ \begin{equation} \frac{1}{M}\vec{\nabla}_i\cdot\vec{\nabla}_i \end{...
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Contribution from $u$-channel and $t$-channel processes in OPE analysis for deep inelastic scattering

In Ch.18 of the textbook An Introduction to Quantum Field Theory by Peskin and Schroeder, on P.633 the moment sum rules for the deep inelastic form factors are discussed $$\int_0^1 dx x^{n-1}f_f^+(x,...
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2answers
102 views

Significance when expectation of a commutator is zero

It is clear to me what it means when the commutator of two operator $[A, B]$ is zero and what it implies. However, is there any significance when the expectation of the commutator $\langle[A, B]\...
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1answer
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Action of rotation operator on spin 1/2 system

In Sakurai book on QM in chapter 3, he states the following relation $$e^{\frac{iS_z\phi}{\hbar}}[(\rvert+\rangle\langle-\rvert)+(\rvert-\rangle\langle+\rvert)]e^{\frac{-iS_z\phi}{\hbar}}$$ $$=e^{\...
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1answer
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Utility of the time-ordered exponential

Is the time-ordered exponential $$ \mathcal{T}\exp\left\{-i\int_{t_0}^tdt'V(t')\right\}\tag{1} $$ just a mnemonic device for the series $$ \begin{aligned} 1 + (-i)\int_{t_0}^tdt_1 \, V(t_1) +{} &...
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1answer
67 views

Goldstone theorem in Weinberg vol 2

I was reading the proof of Goldstone's theorem (the operator proof starting on page 170) in Weinberg's book on QFT (Volume II) and got confused. I am able to follow each line of the proof, but as a ...
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Proof $\exp(-\beta H)$ trace-class operator

Let $H=\frac{p^2}{2}+\frac{x^2}{2}\, : D(H) \to L^2(\mathbb{R})$ be the Hamiltonian of the harmonic oscillator with $m=\hbar=\omega=1$. Prove that $\exp(-\beta H)$ is a trace-class operator if $\beta&...
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1answer
66 views

Trace over configuration basis

Let us take a many-body quantum system, whose phases in the configuration basis are labeled by $\mathbf {\hat q}=(q_1,\cdots, q_N)$ and momenta $\mathbf {\hat p}=\left(-i\frac{\partial}{\partial \hat ...
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3answers
112 views

Operators commutation and relation between eigenvalues

If $H$ and $L_i$ are commuting ( $[H, L_i] = 0$ ) could we deduce that the eigenvalues of $H$ depend/ do not depend on $m$ and $\ell$ ( eigenvalue of $L_z, L^2$ )? I don't think so since it does not ...
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1answer
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What is 'definite' variable in QM?

I have gone through a few of the questions on the website regarding this particular query, but I have not understood what they meant. When a question says that a particle has definite momentum, are ...
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1answer
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If $L_z$ has a $0$ eigenfunction, since $[L_x, L_y] = i\hbar L_z,$, then can $L_x, L_y$ have a simultaneous eigenfunctions?

In the lecture Quantum Mechanics by Dr. Adams in ocw.mit.edu, in the 16th lecture at 7:11, it is stated that since $$[L_x, L_y] = i\hbar L_z,$$ there is no state s.t it is eigenfunction of both $L_x, ...
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1answer
60 views

A delicate measurement?

The identity operation means that we are not disturbing the system at all. However, if we consider the following operation $$ \begin{pmatrix} 1 + \epsilon & 0\\ 0 & 1 + \...
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1answer
55 views

Where did mess up while calculating the expected value of the momentum squared?

I have the correct answer except with a negative sign. The wave function is given as, $$\Phi=A\exp\left[-a\left(\frac{mx^2}{\hbar} + it\right)\right]$$ By squaring the momentum quantity, I found ...
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1answer
121 views

What is the time derivative $\frac{d}{dt}(\exp(\hat A))$ of operator exponential $\exp(\hat{A})$? [duplicate]

Assuming that the operator $\hat{A}$ does not necessarily commute with $d \hat A/dt$, what is $$\frac{d}{dt}(\exp(\hat A))~?\tag{1}$$ Is it $\exp(\hat A(t)) \frac{d \hat A}{dt}$ or $\frac {d\hat A}{dt}...
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Action of momentum operator on wavefunction in momentum space

In a previous question How to get the position operator in the momentum representation from knowing the momentum operator in the position representation? it was mentioned that $$\begin{align} \...
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0answers
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Is there any ambiguity when forming operators for classical observables from other operators? [duplicate]

There are some classical quantities, for which we know corresponding operators: e.g. position, momentum. For some others it's straightforward to compose these operators using expressions similar to ...
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1answer
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How to define the Hamiltonian properly in quantum field theory

In a rigorous fashion, how does one define the Hamiltonian of QFT as $$\hat{H}(t) = \int d^3x \hat{\mathcal{H}}(x, t)$$ For now I'm ignoring the fact that $\hat{\mathcal{H}}$ itself may be ill-...
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6answers
591 views

What does vector operator for angular momentum measure?

Consider the vector operator for angular momentum $\hat L=\hat L_x \vec i +\hat L_y \vec j + \hat L_z \vec k$. Does this mean that if we want to measure the angular momentum of a particle in state $\...
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1answer
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Systems with a finite number of linearly independent states

I am studying systems that admit only a finite number of linearly independent states. In such a case, $|S(t)>$ lives in a N-dimensional vector space and can be represented by a column of N ...
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2answers
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Relation between destruction/creation operators of harmonic oscillator in QM and Second Quantization

It is well known that in elementary QM the so-called destruction/creation operators $$ a_i = \frac{Q_j + i P_j }{\sqrt{2}}, \quad a_i^* = \frac{Q_j - i P_j }{\sqrt{2}},$$ are introduced when ...
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1answer
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Phase in time evolution operator for time-dependent Hamiltonian [duplicate]

In Quantum Mechanics, a state vector $|\psi\rangle$ will evolve in time according to $$|\psi(t)\rangle=e^{-\frac{i}{\hbar}\hat H t}|\psi(0)\rangle$$ Imagine we have a system such that, for a short ...
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The kinematic region for the operator product expansion

In Ch.18 of the textbook An Introduction to Quantum Field Theory by Peskin and Schroeder, on P.613 the operator product expansion (OPE) is introduced $$\mathcal{O}_1(x)\mathcal{O}_2(0)\to \sum_n C_{...
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1answer
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What do I get by multiplying a 0 operator on a 0 eigenvector?

I don't know how to write the equation form. Assuming my notation as Dirac notation, what do I get from $$ ( 0 | 0 | 0 ) ~?$$
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0answers
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How to define an Operator Product Expansion (OPE) on arbitrary Riemann surface for a CFT?

Whenever we define the OPE of a 2D CFT, we do so (at least in the literature that I have come across) on the complex plane. Similarly, the commutation relations between conformal generators $L_n$ and ...
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1answer
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Commutator of $\hat {L}_x$ and $\hat{V}(\hat{r})$ [duplicate]

Consider the angular momentum operator $\hat{L_x}=\hat y\hat{p}_z-\hat{z}\hat{p}_y$ and the potential operator $\hat{V}$ where the potential $\hat{V}=\hat{V}(\hat{r})$ is spherically symmetric. It ...
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What is radial ordering?

In my String theory lecture radial ordering was introduced and I don't understand what it is. My first guess was $$R(A(z)B(w)) = A(z)B(w)\Theta(|z|-|w|) + B(w)A(z)\Theta(|w|-|z|).$$ But then we have ...
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1answer
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Some basics about Bracket Notation

I'm trying to prove something. Sorry this post is so long but I wanted to keep things as basic as possible so people have an easier time understanding. Let's assume we have a quantum system $\rho$ ...
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3answers
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Domains of $H$ and $U(t) = \exp(-iH t )$

I am not so familiar with functional analysis. But in my impression, the Hamiltonian $H$ is often not defined everywhere on the Hilbert space. On the other hand, the time evolution operator $U(t)$, ...
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1answer
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Expectation value of a path-ordered exponential

Let us define our path-ordered operator $\overrightarrow{U}\left(t_1,t_2\right)$: $$ \overrightarrow{U}\left(t_1,t_2\right)=\overrightarrow{\mathcal{P}}\exp\int_{t_1}^{t_2}dt\,\mathcal{O}\left(t\...
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Expectation values of states in QFT via the path integral

The path integral in QFT is usually computed only for the vacuum state, $$\langle 0 | T\{ A \} | 0 \rangle = \int \mathcal{D}\phi(x) A e^{iS[\phi]}$$ Doing it for different states is a bit trickier,...
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1answer
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Physical significance of no self-adjoint momentum operator on half line?

I am watching a quantum mechanics lecture by professor Schuller. He mentioned that there does not exist any self-adjoint momentum operator defined on the half line. What is the physical significance ...
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2answers
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Proving the raising and lowering of the raising and lowering operator

I am given a written proof of $\hat A^{\dagger}[u_n] = \sqrt{n+1} \ u_{n+1}$, and from it, and told to similarly prove $\hat A[u_n] = \sqrt{n} \ u_{n-1}$. However, in the written proof for $\hat A^{\...
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2answers
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Should Hamiltonians in quantum field theory be linear operators?

The usual structure of quantum mechanics imposes that Hamiltonians are linear operators. I am not sure if this really holds in quantum field theory. Do non-linear Hamiltonian operators really make ...
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1answer
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What does Ehrenfest's theorem actually mean?

I am told that Ehrenfest's theorem, applied to a physical observable $\hat A$, is: $$\frac{d\langle\hat A\rangle}{dt}= \frac{i}{\bar h}\langle[\hat H,\hat A]\rangle$$ I don't understand how to use ...
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1answer
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Time dependence of expectation value $\hat O$ if $\frac{\partial \hat O}{\partial t} = 0$

I am given the following derivation in my lectures: $$\frac{\partial}{\partial t} \langle \hat O \rangle = \frac{\partial}{\partial t}\int_{-\infty}^{\infty} \psi^* \hat O \ \psi \ dx$$ $$\implies ...
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1answer
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Is this a “good enough” statement of Wigner's theorem from Quantum Mechanics?

I posted this on math StackExchange and got no replies, so I'm trying my luck here! I'm a fourth year physics and math student who is writing up a report on some quantum mechanical symmetries and ...
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1answer
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What is physical meaning of state getting by acting of Hermitian operator to non-eigenstate?

The definition of an operator is $$ \hat{Q}\left|\Psi\right> = \left|\Phi\right>, $$ the thing that convert one state vector $\left|\Psi\right>$ to another $\left|\Phi\right>$. For ...
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1answer
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Domain of symmetric momentum operator vs self-adjoint momentum operator

Is there an example of a function that is not in the domain of the 'naive' symmetric (but not self-adjoint) momentum operator $p:=-i\frac{d}{dx}$ but is in the 'true' self-adjoint momentum operator $p:...
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Question about Divergence of Disturbance Flow

If we break down a flow past a particle into both the incident and disturbance flow like:$$\mathbf{u}^\infty+\mathbf{u}^D=\mathbf{u}^{total} $$ Can we show that: $$ \nabla\cdot\mathbf{u}=0 \...
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1answer
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Is a quantum channel essentially either a unitary evolution or a measurement?

I'd like to understand exactly what people mean when they speak of quantum channels. As I understand it, we can represent a channel by a set of Kraus operators, $M_i$, which satisfy $\sum_{i}M^{\...
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1answer
48 views

The hermitian conjugate of anti-linear operator

Some quantum mechanics books tell us that the definitions of hermitian are If $\langle\psi|A\phi\rangle=\langle B\psi|\phi\rangle$ for linear operators, then $B=A^\dagger$ If $\langle\psi|C\...
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2answers
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Logarithm of Operators in Quantum Mechanics

In an operators algebra $\mathcal{A}$ one can consider a self-adjoint (i.e. real) operator $H$ and note that $$U=e^{iH}$$ exists and is unitary. A mathematical question will be whether any unitary ...