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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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The meaning of a representation in one-dimensional quantum mechanics

In many places, one reads about chosing a representation for studying a particular one-dimensional quantum system. Usual representations are the position representation, the momentum representation or ...
user536450's user avatar
7 votes
3 answers
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Negative kinetic energy on a step potential

I'm doing an introductory course on quantum mechanics. I'm having trouble with the explanation of the kinetic energy on the classically forbbiden region on a step potential ($V=0$ for $x<0$, $V=V_0$...
Vito P.'s user avatar
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On which bundle do QFT fields live?

In QFT, there is a vector field of electromagnetism, usually notated by $A$, which transforms as a 1-form under coordinate changes. Since quantum fields are operator-valued, I thought it is a section ...
Sung Kan's user avatar
2 votes
2 answers
79 views

How does inserting an operator in the path integral change the equation of motion?

I am reading this review paper "Introduction to Generalized Global Symmetries in QFT and Particle Physics". In equation (2.43)-(2.47), the paper tried to prove that when $$U_g(\Sigma_2)=\exp\...
gshxd's user avatar
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Mahan's derivation of energy current of a free-particle system

On p.25 of the 3rd Ed. analogously to the polarisation operator $\textbf{P}$ for particle currents $$\textbf{P}=\int\textbf{r}\rho(\textbf{r})d^3r$$ he defines an operator $$\textbf{R}_E=\frac{1}{2}\...
Redcrazyguy's user avatar
1 vote
1 answer
107 views

Can the Parity Operator in polar coordinates be defined as $\hat\Pi\psi(r,\theta,\phi) = \psi(r,\theta+\pi,\phi).$?

I was reading about Symmetries & Conservation Laws from Introduction to Quantum Mechanics, David J. Griffiths when I came across this question about the parity operator in three dimensions: ...
Kapil's user avatar
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Positive Constant in Lindbladian

Consider the following Lindbladian $$ L \left[X \right] = i \left[H, X \right] + \gamma \left( 2QXQ - \left(Q^2X + XQ^2 \right) \right), $$ where X is observable, in other words, Heisenberg picture. ...
ets_ets's user avatar
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Derivation of two-body Coulomb interaction in momentum space

$\newcommand{\vec}{\mathbf}$ In Condensed Matter Field Theory by Altland and Simons, they claim the two-body Coulomb interaction for the nearly-free electron model for a $d$-dimensional cube with side ...
zeroknowledgeprover's user avatar
0 votes
3 answers
104 views

Can I use any linearly independent, orthogonal, eigenkets as starting basis to construct $S_x$, $S_y$ and $S_z$? [closed]

I know how to construct $S_z$ using $|\uparrow\rangle$=$\left(\begin{matrix}1\\0\end{matrix}\right)$ and $|\downarrow\rangle$=$\left(\begin{matrix}0\\1\end{matrix}\right)$ as starting basis. And I can ...
Siddaram's user avatar
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Completeness meaning (complete basis vs complete metric space) [migrated]

Today my professor started talking about the formalism of QM. We talked about that that eigenvectors of a Hermitian operator (over Hilbert space) is a "complete set". He also mentioned ...
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Alternative way to compute expectation value of momentum? [closed]

This might be ridiculously incorrect, but is it possible to find the expectation value of momentum like this? In the position space: $$\langle x | \psi \rangle = \psi(x)$$ $$\langle \hat{A} \rangle_{x\...
Aryan MP's user avatar
2 votes
2 answers
214 views

Why we use trace-class operators and bounded operators in quantum mechanics?

The set of trace-class operators $\mathcal{B_1(H)}$ on the Hilbert space $\mathcal{H}$ is like the Banach space $l^1$, while the set of bounded operators $\mathcal{B_\infty(H)}$ is like the Banach ...
Godfly666's user avatar
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How to go from a vector operator to its components?

(I'm sorry if this question is a duplicate, I couldn't find anything that answered my question.) I'm doing an exercise where I'm supposed to get the matrix elements for the vector operator $D$ (the ...
Hector Freire's user avatar
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Why the Slavnov operator is self-adjoint? [duplicate]

In the context of BRST we can define the Slavnov operator $\Delta_{BRST}$ which generates BRST transformations. My lecture notes claim that $\Delta_{BRST}$ is self-adjoint, but I don't see why.
Alex's user avatar
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Conjugate observables - can the commutation relations be generalised?

Conjugate variables are variables that are Fourier transforms of one another (that is they are Fourier transform duals) and consequently have an uncertainty relation existing between them. In quantum ...
Martin Vaughan's user avatar
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2 answers
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Do different bases of Fock space commute?

$\newcommand\dag\dagger$ Suppose we have a Fock space $\mathcal{F}$ with two different bases of creation and annihilation operators $\{a_\lambda, a^\dag_\lambda\}$ and $\{a_{\tilde \lambda}, a^\dag_{\...
zeroknowledgeprover's user avatar
2 votes
1 answer
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$Ad\circ\exp=\exp\circ ad$ and $e^{i(\theta/2)\hat{n}\cdot\sigma}\sigma e^{-i(\theta/2)\hat{n}\cdot\sigma}=e^{\theta\hat{n}\cdot J}\sigma$

This question is inspired by my recent question How to prove $e^{+i(\theta/2)(\hat{n}\cdot \sigma)}\sigma e^{-i(\theta/2)(\hat{n}\cdot \sigma)} = e^{\theta \hat{n}\cdot J}\sigma$? with answer https://...
Jagerber48's user avatar
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Do operators change variance under unitary transformations?

Assume we have an unspecified source of quantum optical states and a measurement device that allows us to measure a certain property of states produced by this source. Providing we can take a large ...
pcalc's user avatar
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Must all observables (operators that describe measurable physical quantities) be hermitian? [duplicate]

I read a book where they said all operators in QM must be hermitian to get real eigenvalues(result of measurements) as it's counter intuitive to get a complex result in measuring..... Say distance ...
Folly 's user avatar
3 votes
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The commutation relations of photon and gluon?

In QED, the photon field has the following commutation relations: \begin{equation} [A^{\mu}(t,\vec{x}),A^{\nu}(t,\vec{y})]=0, \tag{1} \end{equation} where $A^{\mu}(t,\vec{x})$ is the photon filed. ...
Qin-Tao Song's user avatar
3 votes
1 answer
284 views

Time-evolution operator in QFT

I am self studying QFT on the book "A modern introduction to quantum field theory" by Maggiore and I am reading the chapter about the Dyson series (chapter 5.3). It states the following ...
Andrea's user avatar
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How to get $ H=\int\widetilde{dk} \ \omega a^\dagger(\mathbf{k})a(\mathbf{k})+(\mathcal{E}_0-\Omega_0)V $ in Srednicki 3.30 equation?

We have integration is \begin{align*} H =-\Omega_0V+\frac12\int\widetilde{dk} \ \omega\Big(a^\dagger(\mathbf{k})a(\mathbf{k})+a(\mathbf{k})a^\dagger(\mathbf{k})\Big)\tag{3.26} \end{align*} where \...
liZ's user avatar
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2 votes
1 answer
174 views

Problem Deriving "The General Uncertainty Principle" in Section 5.7 of Susskind's "Quantum Mechanics"

I'm having a problem in section 5.7 of Susskind's "Quantum Mechanics, The Theoretical Minimum". Specifically, I'm trying to derive eq. 5.11, $$ 2\sqrt{ \langle \mathbf{A}^2 \rangle \langle \...
BoCoKeith's user avatar
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2 answers
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I need to find the state of the system at a general time, knowing the Hamiltonian and the state at $t=0$ [closed]

The Hamiltonian for a certain three-level system is represented by the matrix $$H = \begin{pmatrix}a & 0 & b \\ 0 & c & 0 \\ b & 0 & a\end{pmatrix},$$ where $a$, $b$, and $c$ ...
zzzzzzzzz's user avatar
1 vote
0 answers
25 views

OPE limit of four-point function in de Sitter space

I have been trying to read the paper 'Cosmological Collider Physics'. This paper studies several things, of which the most interesting to me was studying the correlation function in de Sitter space by ...
Chandra Prakash's user avatar
3 votes
1 answer
106 views

Conceptual Difference Between OPE and Propagator

I'm specifically working with a 2d free scalar CFT. In this case, the propagator is $$\langle X(\sigma) X(\sigma')\rangle=-\frac{\alpha'}{2}\ln(\sigma-\sigma')^2\tag{p.78}$$ while the OPE between $X(\...
Sam's user avatar
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1 answer
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How is the quantum harmonic oscillator related to Fock states?

The question is basically in the title. From what I understand, in the Fock state there is a certain number of particles in each energy level. The creation/annihilation operators create or destroy a ...
Andris Erglis's user avatar
1 vote
1 answer
129 views

The expectation of $L^2_x$

In some problems, we use $$\langle L^2_x\rangle=\frac{1}{2}(\langle L^2\rangle-\langle L^2_ z\rangle)$$ But in other problems, we use $$\langle L^2_x\rangle=\frac{1}{4}\langle[L^2_+ + L^2_- +2(L^2-L^...
Suhail Sarwar's user avatar
4 votes
1 answer
124 views

Jensen's inequality on (super)operator exponential

Let us define the expectation value $\langle A\rangle_{\rho}$ of a superoperator $A$ over a density matrix $\rho$ as $(\rho, A(\rho))$, where the scalar product between operators reads $(O_1,O_2):= Tr[...
lgotta's user avatar
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4 votes
0 answers
105 views

Canonical commutation relation in QFT

The canonical commutation relation in QFT with say one (non-free) scalar real field $\phi$ is $$[\phi(\vec x,t),\dot \phi(\vec y,t)]=i\hbar\delta^{(3)}(\vec x-\vec y).$$ Is this equation satisfied by ...
MKO's user avatar
  • 2,200
1 vote
2 answers
98 views

Identity for squeezing spin coherent state

Let $S_{\pm} := S_x \pm iS_y$ where $S_i$ are the spin operators satisfying the usual commutation relations $[S_i, S_j] = i\hbar \epsilon_{ijk}S_k$. In Kitagawa & Ueda (1993) it is claimed that $$...
Silly Goose's user avatar
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2 votes
0 answers
63 views

Possible ambiguities of quantization

Quantization means to replace $p$ (the momentum) in the expressions of classical physical quantities with $-i\hbar\nabla$, so we get an operator belonging to each physical quantity. However, an ...
mma's user avatar
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0 answers
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Conmutation of operators that act on many electrons

If I define the operator L as: $$\vec{L}=\sum{\vec{l_i}}$$ where $\vec{l_i}$ is the operator of angular momentum of a single electron. How can I justify formally that $$\left[l_i ^2 , L^2\right]=0 $$ ...
paula garcia's user avatar
3 votes
0 answers
75 views

Application of Callias operator in physics

In his article "Axial Anomalies and Index Theorems on Open Spaces" C.Callias shows how the index of the Callias-type operator on $R^{n}$ can be used to study properties of fermions in the ...
C1998's user avatar
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2 votes
1 answer
61 views

Operator systems in functional analysis & quantum mechanics: intuition

I saw this concept of operator systems in here but I am not sure if I want to get deep into it before knowing roughly what it is used for in, say, quantum information or quantum mechanics. My very ...
Evangeline A. K. McDowell's user avatar
8 votes
4 answers
1k views

Examples of systems with infinite dimensional Hilbert space, whose energy is bounded from above

We often encounter (and love to!) deal with systems whose energy is bounded from below, for good reasons like stability, etc. But what about systems whose energy is bounded from above? In finite ...
Sanjana's user avatar
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1 vote
0 answers
35 views

Vacuum expectation of polynomial of bosonic creation and annihilation operators [duplicate]

Let $\hat{a}^\dagger,\hat{a}$ be creation and annihilation operators with commutator $$ [\hat{a},\hat{a}^\dagger] = 1. $$ Let $|0\rangle$ be vacuum state that $$ \hat{a} |0\rangle=0. $$ Let $\beta$ be ...
Luessiaw's user avatar
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2 votes
1 answer
88 views

Why does the mass term not violate particle number conservation in a free theory?

The Lagrangian of a free real scalar field theory is $$ \mathcal{L} = \frac{1}{2} \partial_{\mu} \phi\; \partial^{\mu} \phi \; - \frac{1}{2} m^2 \phi^2. $$ If we decompose $\phi$ in terms of the ...
ratchet411's user avatar
4 votes
1 answer
77 views

Solving for unitary operation using perturbation theory

Let the time-dependent Hamiltonian be \begin{equation} H(t) = H_0(t) + \lambda H_1(t), \end{equation} where $\lambda$ is a small parameter. In the interaction picture (i.e. rotating frame w.r.t ...
Hailey Han's user avatar
4 votes
1 answer
126 views

Is the factorization method of Hamiltonian related to the theory of Lie groups?

I was learning about algebraic methods to solve the H atom, when I came across the factorization method. It is mentioned in various textbooks, notes and papers, like the one from Infeld and Hull. I am ...
Po1ynomial's user avatar
3 votes
1 answer
51 views

Deriving OPE between vertex operator: Di Francesco Conformal Field Theory equation 6.65

How does one get Di Francesco Conformal Field Theory equation 6.65: $$ V_\alpha(z,\bar{z})V_\beta(w,\bar{w}) \sim |z-w|^{\frac{2\alpha\beta}{4\pi g}} V_{\alpha+\beta}(w,\bar{w})+\ldots~?\tag{6.65}$$ ...
Jens Wagemaker's user avatar
1 vote
1 answer
93 views

Number Operator "Ordering" for Higher Order Bosonic Operators

I'm considering the algebra of a single harmonic oscillator where $[\hat{a},\hat{a}^\dagger]=\hat{\mathbb{I}}$. Typically, one is interested in normal, antinormal or symmetric ordering. I am ...
Lost In Euclids 5th Postulate's user avatar
0 votes
1 answer
44 views

Commutation in the Local Gauge Transformations

Let's say that I have a Unitary Local Gauge Transformation $U$, in which the Lie Generators are $T$: $$ \partial_{\mu} U = \partial_{\mu} e^{-i T^{a} \alpha_{a}(x)} = U \partial_{\mu} \left( -i T^{a} \...
user avatar
2 votes
1 answer
56 views

For any pure state, can I find a pair of non-commuting observables which saturate the uncertainty bound?

Given some pure state $|\psi\rangle$ we have the following bound on the uncertainty for two non-commuting operators $A$ and $B$ \begin{equation} \sigma_A\sigma_B\geq\left|\frac 1{2i}\langle[A,B]\...
Andrew Forbes's user avatar
0 votes
0 answers
26 views

Classical Hamilton’s equations in quantum mechanics [duplicate]

How can one derive what the position operator is in momentum space for a quantum wave function from the classical Hamilton’s equations? Similarly, is a concept of an “angular momentum space” ...
TheorVHP's user avatar
1 vote
1 answer
75 views

"Deriving" Poisson bracket from commutator

This Q/A shows that deriving P.B.s from commutators is subtle. Without going into deep deformation quantization stuff, Yaffe manages to show that $$\lim_{\hbar \to 0}\frac{i}{\hbar}[A,B](p,q)=\{a(p,q),...
Sanjana's user avatar
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0 votes
0 answers
48 views

What are the similarities and differences between the Magnus expansion and the Schrieffer-Wolff transformation?

The Magnus expansion and the Schrieffer-Wolff transformation are both methods used to get certain effective Hamiltonians. I know that at a basic level, the Schrieffer-Wolff transformation eliminates ...
NikNack's user avatar
  • 19
0 votes
2 answers
89 views

Energy and momentum operators using Hamilton's equations

The energy operator is: $${\displaystyle {\hat {E}}=i\hbar {\frac {\partial }{\partial t}}}\tag1$$ and the momentum operator is $${\displaystyle {\hat {p}}=-i\hbar {\frac {\partial }{\partial x}}}.\...
User198's user avatar
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2 votes
0 answers
39 views

Taylor condition on the general formula for momentum commutator [closed]

My quantum homework asked me the following question: Prove that for any $f(x)$ such that $f$ admits a Taylor expansion, the following is true: $$[f(x), \hat{p}] = i\hbar\frac{\mathrm{d}f}{\mathrm{d}x}...
Trips73's user avatar
  • 21
1 vote
1 answer
63 views

Quantised Newtonian potential as an operator in non-relativistic QM [closed]

Suppose we have two slowly moving (effectively static) masses $m_1,m_2$, interacting through gravity, that are not occupying a definite state of position i.e. that matter is being treated quantum ...
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