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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28 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:
$$\frac{da_l}{dt} = -...
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0answers
14 views
Conjugate of total spin operator
I got a lattice, and the total spin operator for x and for y, for that lattice. I know that the x component conmutes with an operator called staggered spin operator in y. I also know that the ...
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0answers
12 views
Magnitude of mass polarization
When solving many-body atomic Hamiltonians, (such as Helium), there exists a term in the Hamiltonian which looks something like:
$$
\begin{equation}
\frac{1}{M}\vec{\nabla}_i\cdot\vec{\nabla}_i
\end{...
2
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0answers
18 views
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
$$\int_0^1 dx x^{n-1}f_f^+(x,...
4
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2answers
101 views
Significance when expectation of a commutator is zero
It is clear to me what it means when the commutator of two operator $[A, B]$ is zero and what it implies. However, is there any significance when the expectation of the commutator $\langle[A, B]\...
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1answer
30 views
Action of rotation operator on spin 1/2 system
In Sakurai book on QM in chapter 3, he states the following relation $$e^{\frac{iS_z\phi}{\hbar}}[(\rvert+\rangle\langle-\rvert)+(\rvert-\rangle\langle+\rvert)]e^{\frac{-iS_z\phi}{\hbar}}$$
$$=e^{\...
2
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1answer
69 views
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
$$
\begin{aligned} 1 + (-i)\int_{t_0}^tdt_1 \, V(t_1) +{} &...
1
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1answer
67 views
Goldstone theorem in Weinberg vol 2
I was reading the proof of Goldstone's theorem (the operator proof starting on page 170) in Weinberg's book on QFT (Volume II) and got confused. I am able to follow each line of the proof, but as a ...
7
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1answer
152 views
Proof $\exp(-\beta H)$ trace-class operator
Let $H=\frac{p^2}{2}+\frac{x^2}{2}\, : D(H) \to L^2(\mathbb{R})$ be the Hamiltonian of the harmonic oscillator with $m=\hbar=\omega=1$. Prove that $\exp(-\beta H)$ is a trace-class operator if $\beta&...
2
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1answer
66 views
Trace over configuration basis
Let us take a many-body quantum system, whose phases in the configuration basis are labeled by $\mathbf {\hat q}=(q_1,\cdots, q_N)$ and momenta $\mathbf {\hat p}=\left(-i\frac{\partial}{\partial \hat ...
2
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3answers
110 views
Operators commutation and relation between eigenvalues
If $H$ and $L_i$ are commuting ( $[H, L_i] = 0$ ) could we deduce that the eigenvalues of $H$ depend/ do not depend on $m$ and $\ell$ ( eigenvalue of $L_z, L^2$ )? I don't think so since it does not ...
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1answer
68 views
What is 'definite' variable in QM?
I have gone through a few of the questions on the website regarding this particular query, but I have not understood what they meant.
When a question says that a particle has definite momentum, are ...
4
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1answer
135 views
If $L_z$ has a $0$ eigenfunction, since $[L_x, L_y] = i\hbar L_z,$, then can $L_x, L_y$ have a simultaneous eigenfunctions?
In the lecture Quantum Mechanics by Dr. Adams in ocw.mit.edu, in the 16th lecture at 7:11, it is stated that since
$$[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 ...
0
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1answer
116 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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4answers
85 views
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
$$\begin{align}
\...
0
votes
0answers
25 views
Is there any ambiguity when forming operators for classical observables from other operators? [duplicate]
There are some classical quantities, for which we know corresponding operators: e.g. position, momentum. For some others it's straightforward to compose these operators using expressions similar to ...
2
votes
1answer
58 views
How to define the Hamiltonian properly in quantum field theory
In a rigorous fashion, how does one define the Hamiltonian of QFT as
$$\hat{H}(t) = \int d^3x \hat{\mathcal{H}}(x, t)$$
For now I'm ignoring the fact that $\hat{\mathcal{H}}$ itself may be ill-...
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6answers
587 views
What does vector operator for angular momentum measure?
Consider the vector operator for angular momentum $\hat L=\hat L_x \vec i +\hat L_y \vec j + \hat L_z \vec k$.
Does this mean that if we want to measure the angular momentum of a particle in state $\...
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1answer
41 views
Systems with a finite number of linearly independent states
I am studying systems that admit only a finite number of linearly independent states. In such a case, $|S(t)>$ lives in a N-dimensional vector space and can be represented by a column of N ...
2
votes
2answers
71 views
Relation between destruction/creation operators of harmonic oscillator in QM and Second Quantization
It is well known that in elementary QM the so-called destruction/creation operators
$$ a_i = \frac{Q_j + i P_j }{\sqrt{2}}, \quad a_i^* = \frac{Q_j - i P_j }{\sqrt{2}},$$
are introduced when ...
0
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1answer
44 views
Phase in time evolution operator for time-dependent Hamiltonian [duplicate]
In Quantum Mechanics, a state vector $|\psi\rangle$ will evolve in time according to
$$|\psi(t)\rangle=e^{-\frac{i}{\hbar}\hat H t}|\psi(0)\rangle$$
Imagine we have a system such that, for a short ...
4
votes
0answers
52 views
The kinematic region for the operator product expansion
In Ch.18 of the textbook An Introduction to Quantum Field Theory by Peskin and Schroeder, on P.613 the operator product expansion (OPE) is introduced
$$\mathcal{O}_1(x)\mathcal{O}_2(0)\to \sum_n C_{...
0
votes
1answer
41 views
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
$$ ( 0 | 0 | 0 ) ~?$$
2
votes
0answers
32 views
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 ...
0
votes
1answer
40 views
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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0answers
56 views
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 ...
0
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1answer
58 views
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 $\rho$ ...
6
votes
3answers
116 views
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 $U(t)$, ...
1
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1answer
74 views
Expectation value of a path-ordered exponential
Let us define our path-ordered operator $\overrightarrow{U}\left(t_1,t_2\right)$:
$$
\overrightarrow{U}\left(t_1,t_2\right)=\overrightarrow{\mathcal{P}}\exp\int_{t_1}^{t_2}dt\,\mathcal{O}\left(t\...
1
vote
1answer
78 views
Expectation values of states in QFT via the path integral
The path integral in QFT is usually computed only for the vacuum state,
$$\langle 0 | T\{ A \} | 0 \rangle = \int \mathcal{D}\phi(x) A e^{iS[\phi]}$$
Doing it for different states is a bit trickier,...
4
votes
1answer
71 views
Physical significance of no self-adjoint momentum operator on half line?
I am watching a quantum mechanics lecture by professor Schuller. He mentioned that there does not exist any self-adjoint momentum operator defined on the half line. What is the physical significance ...
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votes
2answers
49 views
Proving the raising and lowering of the raising and lowering operator
I am given a written proof of $\hat A^{\dagger}[u_n] = \sqrt{n+1} \ u_{n+1}$, and from it, and told to similarly prove $\hat A[u_n] = \sqrt{n} \ u_{n-1}$.
However, in the written proof for $\hat A^{\...
1
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2answers
75 views
Should Hamiltonians in quantum field theory be linear operators?
The usual structure of quantum mechanics imposes that Hamiltonians are linear operators. I am not sure if this really holds in quantum field theory. Do non-linear Hamiltonian operators really make ...
1
vote
1answer
94 views
What does Ehrenfest's theorem actually mean?
I am told that Ehrenfest's theorem, applied to a physical observable $\hat A$, is:
$$\frac{d\langle\hat A\rangle}{dt}= \frac{i}{\bar h}\langle[\hat H,\hat A]\rangle$$
I don't understand how to use ...
0
votes
1answer
26 views
Time dependence of expectation value $\hat O$ if $\frac{\partial \hat O}{\partial t} = 0$
I am given the following derivation in my lectures:
$$\frac{\partial}{\partial t} \langle \hat O \rangle = \frac{\partial}{\partial t}\int_{-\infty}^{\infty} \psi^* \hat O \ \psi \ dx$$
$$\implies ...
1
vote
1answer
77 views
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 ...
1
vote
1answer
43 views
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 ...
2
votes
1answer
66 views
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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1answer
20 views
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:
$$ \nabla\cdot\mathbf{u}=0 \...
1
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1answer
58 views
Is a quantum channel essentially either a unitary evolution or a measurement?
I'd like to understand exactly what people mean when they speak of quantum channels. As I understand it, we can represent a channel by a set of Kraus operators, $M_i$, which satisfy $\sum_{i}M^{\...
0
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1answer
48 views
The hermitian conjugate of anti-linear operator
Some quantum mechanics books tell us that the definitions of hermitian are
If $\langle\psi|A\phi\rangle=\langle B\psi|\phi\rangle$ for linear operators, then $B=A^\dagger$
If $\langle\psi|C\...
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0answers
87 views
Weinberg operator question
Similar posts have been made on this site, but none of them seem to answer my specific questions. I am quite new to QFT (especially standard model) so these may seem trivial to you.
We have the ...
6
votes
2answers
420 views
Logarithm of Operators in Quantum Mechanics
In an operators algebra $\mathcal{A}$ one can consider a self-adjoint (i.e. real) operator $H$ and note that $$U=e^{iH}$$ exists and is unitary. A mathematical question will be whether any unitary ...
1
vote
1answer
60 views
Commutive property of the Bra-ket notation
I'm struggling when it comes to understanding the commutive properties of the Bra-ket notation in quantum mechanics. I understand how to work with constants, bra and kets. However, the second I start ...
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2answers
64 views
Physical Interpretation of d'Alembert Operator
$$\mathop{{}\Box}\nolimits=\frac{1}{c^2}\frac{\partial^2}{\partial t^2}-\mathop{{}\bigtriangleup}\nolimits$$ is the d'Alembert-operator. It seems to consist of an oscillation and a diffusion. Is there ...
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0answers
43 views
Operators, gamma matrices and Lorentz invariance
In class, we have define the following operator:
$$\Pi_{\pm} = \frac{1 \pm \gamma^0}{2} \tag1$$
Where, $\gamma^0$ is the usual first gamma matrix in Weyl representation.
Applying it to a 4-...
2
votes
1answer
62 views
Joint Spectral Measure theorem
I want to gain an intuition to understand the joint spectral measure theorem. In the case that operators involved in this theorem have purely discrete spectrum, the theorem should be reduced to the ...
1
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0answers
34 views
QM question of the $z$ Matrix element in Angular Momentum Basis
I found a quite challenge quantum mechanics problem in a preparation sample test for a midterm.
Consider an electron moving in a central potential. Suppose that we know the matrix element of the $...
