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 $\times$, $+$!
-1
votes
1answer
50 views
Slater determinant in second quantization using the creation operators help
$$
\left\langle 0\left|\hat{\Psi}\left(x_{1}\right) c_{\alpha_{1}}^{\dagger}\right| 0\right\rangle=\left\langle 0\left|\varphi_{\alpha_{1}}\left(x_{1}\right)-c_{\alpha_{1}}^{\dagger} \hat{\Psi}\left(...
1
vote
3answers
106 views
Is there any intuitive reason behind why should the eigenfunctions of observables form a basis for our Hilbert space?
Is there any intuitive reason behind why should the eigenfunctions of observables form a basis for our Hilbert space ?
For example, in the case of Stern-Gerlach experiment, sending the beam that has ...
1
vote
0answers
17 views
Double-trace operators in CFT?
What is the conceptual difference between so called "single-trace" and "double-trace" (or "multi-trace") operators e.g., in a Conformal Field Theory?
0
votes
1answer
42 views
Anti-commutation relations in annihilation operators
It is claimed that
$$\{c_\alpha,c_\beta \} = c_\alpha c_\beta + c_\beta c_\alpha = 0$$
where $c_\alpha$ and $c_\beta$ are the fermionic annihilation operators in second quantization. Why is that ...
0
votes
3answers
80 views
How to write an operator in matrix form?
Say I have the following operator:
$$\hat { L } =\hbar { \sum_{ \sigma ,l,p } { l } \int_{ 0 }^{ \infty }\!{ \mathrm{d}{ k }_{ 0 }\,\hat { { { a }}}_{ \sigma ,l,p }^{ \dagger } } } \left({ k }_{...
-1
votes
1answer
38 views
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, ...
0
votes
0answers
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:
$$\frac{da_l}{dt} = -...
0
votes
0answers
15 views
Conjugate of total spin operator
I got a lattice, and the total spin operator for x and for y, for that lattice. I know that the x component conmutes with an operator called staggered spin operator in y. I also know that the ...
0
votes
0answers
12 views
Magnitude of mass polarization
When solving many-body atomic Hamiltonians, (such as Helium), there exists a term in the Hamiltonian which looks something like:
$$
\begin{equation}
\frac{1}{M}\vec{\nabla}_i\cdot\vec{\nabla}_i
\end{...
2
votes
0answers
22 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
$$\int_0^1 dx x^{n-1}f_f^+(x,...
4
votes
2answers
102 views
Significance when expectation of a commutator is zero
It is clear to me what it means when the commutator of two operator $[A, B]$ is zero and what it implies. However, is there any significance when the expectation of the commutator $\langle[A, B]\...
1
vote
1answer
31 views
Action of rotation operator on spin 1/2 system
In Sakurai book on QM in chapter 3, he states the following relation $$e^{\frac{iS_z\phi}{\hbar}}[(\rvert+\rangle\langle-\rvert)+(\rvert-\rangle\langle+\rvert)]e^{\frac{-iS_z\phi}{\hbar}}$$
$$=e^{\...
2
votes
1answer
70 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
$$
\begin{aligned} 1 + (-i)\int_{t_0}^tdt_1 \, V(t_1) +{} &...
1
vote
1answer
67 views
Goldstone theorem in Weinberg vol 2
I was reading the proof of Goldstone's theorem (the operator proof starting on page 170) in Weinberg's book on QFT (Volume II) and got confused. I am able to follow each line of the proof, but as a ...
7
votes
1answer
155 views
Proof $\exp(-\beta H)$ trace-class operator
Let $H=\frac{p^2}{2}+\frac{x^2}{2}\, : D(H) \to L^2(\mathbb{R})$ be the Hamiltonian of the harmonic oscillator with $m=\hbar=\omega=1$. Prove that $\exp(-\beta H)$ is a trace-class operator if $\beta&...
2
votes
1answer
66 views
Trace over configuration basis
Let us take a many-body quantum system, whose phases in the configuration basis are labeled by $\mathbf {\hat q}=(q_1,\cdots, q_N)$ and momenta $\mathbf {\hat p}=\left(-i\frac{\partial}{\partial \hat ...
2
votes
3answers
112 views
Operators commutation and relation between eigenvalues
If $H$ and $L_i$ are commuting ( $[H, L_i] = 0$ ) could we deduce that the eigenvalues of $H$ depend/ do not depend on $m$ and $\ell$ ( eigenvalue of $L_z, L^2$ )? I don't think so since it does not ...
1
vote
1answer
68 views
What is 'definite' variable in QM?
I have gone through a few of the questions on the website regarding this particular query, but I have not understood what they meant.
When a question says that a particle has definite momentum, are ...
4
votes
1answer
136 views
If $L_z$ has a $0$ eigenfunction, since $[L_x, L_y] = i\hbar L_z,$, then can $L_x, L_y$ have a simultaneous eigenfunctions?
In the lecture Quantum Mechanics by Dr. Adams in ocw.mit.edu, in the 16th lecture at 7:11, it is stated that since
$$[L_x, L_y] = i\hbar L_z,$$
there is no state s.t it is eigenfunction of both $L_x, ...
1
vote
1answer
60 views
A delicate measurement?
The identity operation means that we are not disturbing the system at all. However, if we consider the following operation
$$
\begin{pmatrix}
1 + \epsilon & 0\\
0 & 1 + \...
0
votes
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
votes
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}...
0
votes
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
$$\begin{align}
\...
0
votes
0answers
25 views
Is there any ambiguity when forming operators for classical observables from other operators? [duplicate]
There are some classical quantities, for which we know corresponding operators: e.g. position, momentum. For some others it's straightforward to compose these operators using expressions similar to ...
2
votes
1answer
59 views
How to define the Hamiltonian properly in quantum field theory
In a rigorous fashion, how does one define the Hamiltonian of QFT as
$$\hat{H}(t) = \int d^3x \hat{\mathcal{H}}(x, t)$$
For now I'm ignoring the fact that $\hat{\mathcal{H}}$ itself may be ill-...
10
votes
6answers
591 views
What does vector operator for angular momentum measure?
Consider the vector operator for angular momentum $\hat L=\hat L_x \vec i +\hat L_y \vec j + \hat L_z \vec k$.
Does this mean that if we want to measure the angular momentum of a particle in state $\...
0
votes
1answer
41 views
Systems with a finite number of linearly independent states
I am studying systems that admit only a finite number of linearly independent states. In such a case, $|S(t)>$ lives in a N-dimensional vector space and can be represented by a column of N ...
2
votes
2answers
71 views
Relation between destruction/creation operators of harmonic oscillator in QM and Second Quantization
It is well known that in elementary QM the so-called destruction/creation operators
$$ a_i = \frac{Q_j + i P_j }{\sqrt{2}}, \quad a_i^* = \frac{Q_j - i P_j }{\sqrt{2}},$$
are introduced when ...
0
votes
1answer
44 views
Phase in time evolution operator for time-dependent Hamiltonian [duplicate]
In Quantum Mechanics, a state vector $|\psi\rangle$ will evolve in time according to
$$|\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 ...
1
vote
0answers
57 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
votes
1answer
60 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
123 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
vote
1answer
74 views
Expectation value of a path-ordered exponential
Let us define our path-ordered operator $\overrightarrow{U}\left(t_1,t_2\right)$:
$$
\overrightarrow{U}\left(t_1,t_2\right)=\overrightarrow{\mathcal{P}}\exp\int_{t_1}^{t_2}dt\,\mathcal{O}\left(t\...
1
vote
0answers
79 views
Expectation values of states in QFT via the path integral
The path integral in QFT is usually computed only for the vacuum state,
$$\langle 0 | T\{ A \} | 0 \rangle = \int \mathcal{D}\phi(x) A e^{iS[\phi]}$$
Doing it for different states is a bit trickier,...
4
votes
1answer
73 views
Physical significance of no self-adjoint momentum operator on half line?
I am watching a quantum mechanics lecture by professor Schuller. He mentioned that there does not exist any self-adjoint momentum operator defined on the half line. What is the physical significance ...
0
votes
2answers
51 views
Proving the raising and lowering of the raising and lowering operator
I am given a written proof of $\hat A^{\dagger}[u_n] = \sqrt{n+1} \ u_{n+1}$, and from it, and told to similarly prove $\hat A[u_n] = \sqrt{n} \ u_{n-1}$.
However, in the written proof for $\hat A^{\...
1
vote
2answers
75 views
Should Hamiltonians in quantum field theory be linear operators?
The usual structure of quantum mechanics imposes that Hamiltonians are linear operators. I am not sure if this really holds in quantum field theory. Do non-linear Hamiltonian operators really make ...
1
vote
1answer
95 views
What does Ehrenfest's theorem actually mean?
I am told that Ehrenfest's theorem, applied to a physical observable $\hat A$, is:
$$\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
78 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
68 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:...
0
votes
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:
$$ \nabla\cdot\mathbf{u}=0 \...
1
vote
1answer
58 views
Is a quantum channel essentially either a unitary evolution or a measurement?
I'd like to understand exactly what people mean when they speak of quantum channels. As I understand it, we can represent a channel by a set of Kraus operators, $M_i$, which satisfy $\sum_{i}M^{\...
0
votes
1answer
48 views
The hermitian conjugate of anti-linear operator
Some quantum mechanics books tell us that the definitions of hermitian are
If $\langle\psi|A\phi\rangle=\langle B\psi|\phi\rangle$ for linear operators, then $B=A^\dagger$
If $\langle\psi|C\...
6
votes
2answers
421 views
Logarithm of Operators in Quantum Mechanics
In an operators algebra $\mathcal{A}$ one can consider a self-adjoint (i.e. real) operator $H$ and note that $$U=e^{iH}$$ exists and is unitary. A mathematical question will be whether any unitary ...
