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 $\times$, $+$!
2,445
questions
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1answer
36 views
Proof with complete set of eigenvectors
I have to prove, that if $\hat{\textbf{A}}$ and $\hat{\textbf{B}}$ are self-adjoint operators, and each one of them has its complete set of eigenvectors, and if $\hat{\textbf{A}}\hat{\textbf{B}} = \...
1
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1answer
29 views
How does this simplification in expectation value algebra work?
We have a Hamiltonian of form:
$$\hat{H} = \hat{H}_0 + \hat{H}_1$$
Where $\hat{H}_1$ is a time dependent perturbation which can be written as:
$$\hat{H}_1(t) = - \hat{A}F(t)$$
Now $B$ is another ...
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0answers
34 views
What does the probability of finding the system in a random state mean?
In my quantum mechanics course there are lots of problems about finding the probability of a system in some random state given the initial one which involves using the inner product of those two ...
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4answers
63 views
Symmetry transformations: a doubt about the relations that we assume true
When we deal with symmetry transformations in quantum mechanics we assume true that,
If before the symmetry transformation we have this
$ \hat A | \phi_n \rangle = a_n|\phi_n \rangle,$
and after ...
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0answers
17 views
Dimension of space spanned by angular momentum operator eigenvectors
I read that the dimension of the space spanned by
$$ J^2|j,m\rangle=\hbar^2 j(j+1)|j,m\rangle, \qquad J_z |j,m\rangle=\hbar m|j,m\rangle ,\tag{1}$$
the eigenvectors $|j,m\rangle$ is $2j+1$, since $m$...
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0answers
28 views
What the correct way to write the discrete kinetic energy operator?
For a bit of context, I am making simulations of a quantum algorithm that is meant to variationally find the ground state of a quantum harmonic oscillator potential. In one dimension, we know that $\...
1
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0answers
20 views
Convert a Lindbladian time evolution operator to the Kraus operator sum representation
I try to understand how I can convert a Lindbladian time evolution operator to the corresponding Kraus operator sum. Let's assume we have a time independent Hamiltonian $H$ and a set of time ...
1
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1answer
45 views
Vector operators in spherical basis
This question is about the components of a vector operator in the spherical basis. In 3D real Euclidean space, a vector $\mathbf{v}$ can be expanded in the standard Cartesian components as
$$
\mathbf{...
1
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1answer
44 views
Expected value of operator or expected value of observable?
A question about terminology. I have seen both $\langle p\rangle$ and $\langle\hat{p}\rangle$ to calculate the expected value of momentum (same thing with position, energy etc.). The first one would ...
3
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0answers
28 views
Good quantum numbers in solids
We know that good quantum numbers are associated with operators that commute with the Hamiltonian of the system.
For example consider an Hydrogen atom without spin, we know that the good quantum ...
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1answer
37 views
Schrödinger equation for charged particle in potential
This might be a silly question, but I don't think it is trivial.
I am trying to solve an example for my class. In it the Schrödinger equation for a charged particle in a vector potential is given:
$$i\...
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1answer
51 views
How can you subtract a value from an operator/matrix?
I'm currently following Quantum Computation and Quantum Information by Nielsen & Chuang. I'm struggling to understand the derivation of The Heisenberg Uncertainty Principle in Box 2.4 page 89. I ...
1
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1answer
31 views
Rotation operator in the matrix representation in the standard basis
I recently been introduced to rotation operators in Quantum-Mechanics.
Can someone please provide an explanation on what is means to represent a operator for the rotation of a quantum-mechanical ...
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2answers
40 views
How to find matrix representation of an operator in new basis
I have recently begun to learn QM and I cannot solve this task:
Let's say I have operator $\hat H = \begin{bmatrix}\epsilon & \upsilon \\ \upsilon & \epsilon \end{bmatrix} ,(\upsilon \in \...
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1answer
40 views
Commutator of the number and momentum operator [closed]
Does the number and momentum operator commute?
$[\hat p,\hat N]=0$?
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1answer
57 views
Operator acting on bras
I need some help. Suppose, $\hat{\textbf{A}}$ and $\hat{\textbf{B}}$ are operators and $|\psi\rangle$ is any state, so that
$$ \hat{\textbf{A}}|\psi\rangle=a|\psi\rangle. $$
And I wonder if this ...
4
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2answers
156 views
Physical meaning of the operator $\exp(-a {\hat{p}}^2)$
I am curious about the physical meaning of the operator $\exp(-a {\hat{p}}^2)$ with $a$ being a positive constant. With respect to the coordinate basis, I find that $\langle x |\exp(-a {\hat{p}}^2)|x' ...
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0answers
46 views
Rigorous definition of position operator in 3D
In 1D, let $$\mathcal{D}:=\{f\in L^2(\mathbb{R};\mathbb{C})\bigg|\int_{\mathbb{R}}|x^2f(x)^2|\text{ d}x)<+\infty \}.$$
Then, $\mathcal{D}$ is a dense subspace of $L^2(\mathbb{R},\mathbb{C})$ and ...
4
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4answers
150 views
Time derivative in Schrödinger equation
In quantum mechanics, a system is descibed by an element $|\psi\rangle\in\mathcal{H}$, where $\mathcal{H}$ is a Hilbert space.
Then on $\mathcal{H}$ (or on a dense subspace of $\mathcal{H}$), we can ...
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0answers
13 views
Calculate $\langle 2s |V| 2p,m= 0\rangle =3ea_0 |E|$ in linear Stark effect with operator properity
From Sakurai Eq 5.1.65 and Eq 5.2.19, one would be able to obtain the matrix element for degenerate perturbation theory of linear Stark effect under the potential $V=-ez|E|$.
However, I'm wondering ...
2
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2answers
49 views
Commutator between Angular Momentum J and Cross Product
I'm trying to show that a cross product $C$ of two vectorial operators (let's say $A$ and $B$) it's a vector by it's own, which means, I want to show
$$\left [J_i,C_j \right ] = i\hbar \epsilon_{...
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0answers
34 views
Does no-level-crossing theorem (aka avoided crossing) always hold in perturbation theory?
In perturbation, J.J. Sakurai Modern Quantum Mechanics Second Edition page 310 stated a no-level-crossing theorem stated that
"a pair of energy levels connected by perturbation do not cross as ...
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2answers
68 views
Annihilation operator in the Heisenberg picture
I study the monograph An Introduction to the Standard Model of Particle Physics written by W. N. Cottingham and D. A. Greenwood. I can't understand the equation about the annihilation operator in the ...
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1answer
46 views
Division of two operators (or polonomials of operators) in quantum mechanics
Consider a function of an operator $\hat{A}$, which is like follows
$$ f\left(\hat{A}\right) = \frac{a + i b\hat{A}-c\hat{A}^2}{3-\hat{A}} $$
where $a$, $b$ and $c$ are complex numbers. My question ...
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0answers
68 views
How do you write commutators in polar coordinates?
In quantum field theory we have the commutators of fields must be zero outside the light cone.
$[\phi(x),\phi(y)]=0$ if $|x-y|^2<0$
How can one write this in polar coordinates or a general ...
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0answers
35 views
Stone's theorem on one-parameter unitary groups and new self-adjoint operators
I have been following the proof of the Stone's theorem on one-parameter unitary groups.
The question is if the current list of self-adjoint operators used in quantum mechanics, including position, ...
1
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1answer
42 views
How to calculate $\left[ \vec{L}^2, x_i \right]$
I've been asked to prove $\left[ \vec{L}^2, x_i \right] = -2i\hbar \varepsilon_{ijk}L_j x_k -2\hbar^2 x_i $ and I don't seem to get it correctly.
I propose $\left[ \vec{L}^2, x_i \right] = \left[ L_l ...
0
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1answer
43 views
Radial Commutativity in AdS/CFT
I am reading Daniel Harlow's Tasi lecture notes on the emergence of bulk physics in AdS/CFT, and the question that I have is regarding the radial commutativity puzzle given in these notes.
First, I ...
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0answers
30 views
Minimal coupling to electric dipole form - II
In addition to link, may I ask you for details of the Hamiltonian transformation.
Knowing that:
$[\textbf{x},\textbf{p}]=i\hbar$, $[\textbf{x},\textbf{p}^2]=2i\hbar\textbf{p}$ and $e^{-i\alpha A}Be^{...
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2answers
83 views
Dirac expression derivation
In Quantum Mechanics, 2nd Edition by Davies & Betts on page 78 it states that there is a symmetry implied by the following Hermitian operator equation:
$${\displaystyle \int \phi^{*}(A \psi)d \,\...
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107 views
How to operationally realize the following type of equations of motion?
It is well known that for a free particle, described by $H=\hat{p}^2/2m$,
$\hat{p}_{x}(t)=$ constant (similarly for other components of momentum). Meanwhile, $\hat{x}(t)$ is not a constant, being ...
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0answers
48 views
Generalized momenta and quantum mechanics
In my introductory quantum mechanics book it was stated that the operator $-i\hbar\vec{\nabla}$ represents the momentum $\vec{p}=m\vec{v}$ of a particle. In the book "Physics of atoms and molecules" ...
4
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1answer
62 views
Representations of Conformal Group
I want to work out the Representations of the Conformal Group. I work with Francesco's Conformal Field Theory.
He stats in equation 4.30 that
$$e^{i x^\rho P_\rho}K_\mu e^{-i x^\rho P_\rho}= K_\mu +...
2
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0answers
85 views
What does the operator's explicit dependence or independence on time actually mean in Quantum mechanics?
Consider the equation of motion for the expectation value of an operator $A$
$$\frac{d\langle A\rangle}{dt} = \frac{1}{i\hbar}\langle [A,H]\rangle + \left \langle \frac{\partial A}{\partial t} \right \...
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2answers
56 views
Correlation function of single annihilation/creation operator vanishes
I could not find anything on that on google, or here on physics stack exchange, which surprises me. My problem is, that I do not see, why exactly
$<a> = <a^{\dagger}> = 0$
where <...> ...
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1answer
99 views
Trick with “functional” derivative to evaluate commutators between diagonal hamiltonian and creation fermionic operator
I found a theorem that states that if $A$ and $B$ are 2 endomorphism that satisfies $[A,[A,B]]=[B,[A,B]]=0$ then $[A,F(B)]=[A,B]F'(B)=[A,B]\frac{\partial F(B)}{\partial B}$.
Now i'm trying to apply ...
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0answers
30 views
Norm of a specific operator
I am working on a problem for my Quantum Mechanics class. Given the operator
$\operatorname{M}: \, L^2(0,1) \rightarrow L^2(0,1): \, f(x) \mapsto (\operatorname{M}f)(x) = m(x)f(x)$
I shall prove, ...
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0answers
22 views
Is there any operator for the scattering problem of an elastic wave in Hilbert space?
From an algebraic perspective, there is a common shape for wave propagation that includes operators in Hilbert space. I want to know is there any operator in Hilbert space that describes the elastic ...
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1answer
53 views
Quantum measurement:can we recover evolution operator from measurement?
Suppose there is a quantum system in state $|a\rangle$$\in$ H,an instrument designed to measure some physical quantity, which itself is in state $|b\rangle$ $\in$ H'. If we consider the bigger system "...
1
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1answer
56 views
Show that a wavefunction is an eigenfunction of the lowering operator
Given $\psi_{\lambda}=e^{\lambda a_{+}}\psi_{0}$, show that $a_{-}\psi_{\lambda}$ is an eigenfunction of $\psi_{\lambda}$ with eigenvalue $\lambda$.
In this case $a_{\pm}=\frac{1}{\sqrt{2\hbar m \...
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62 views
Simplifying Exponentials of Matrix Operators [closed]
I have been given the following question and would really appreciate any insight. I apologise for some of the symbols as I can't find a way to compile them properly.
Assume the environment is given ...
4
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3answers
121 views
Is there an identity for anti-commutator $\{ A B, \, C D \}$ in terms of commutators $[\, , \,]$ only?
I'm looking for an identity that could express the anti-commutator
$$\tag{1}
\{ A B , \, C D \} \equiv A B C D + C D A B
$$
expressed as a combination of commutators only: $[A,\, C]$, $[A, \, D]$, etc....
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1answer
44 views
Anti-commutator for annihilation and creation operators: ordering of indices
I'm trying to prove that $\{\tilde a_i,\tilde a_j^{\dagger} \}=\delta_{ij}$, by defining
$\tilde a_i=\sum_j \bar U_{ji}a_j$. U is an unitary matrix and $a_i$ refers to an element of the operator $a$. ...
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0answers
43 views
Dirac matrix algebra from bosonic creation/anihilation operators?
Using some generic fermionic creation/anihilation operators $a_i$ and $a_i^{\dagger}$ ($i, j = 1, 2, 3, \dots, N$) such that
\begin{align}
\{ a_i, \, a_j^{\dagger} \} &= \delta_{ij}, \tag{1} \\[...
8
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0answers
148 views
QFT in Curved Spacetime - Commutation Relations for Annihilation Operators
I am currently learning about QFT in curved spacetime using these notes.
Given a subspace $S_p$ of positive frequency solutions to the Klein-Gordon equation, we defined the corresponding ...
3
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1answer
60 views
Derivation of Single Particle Operator in second quantization?
See, I have looked through a bunch of scripts about second quantization on the internet, but everywhere at some point something weird is happening so I get stuck over and over and over again, which is ...
1
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4answers
194 views
The exponential of an operator
I have a problem to have an intuitive idea about these operators:
$\hat{D_x}(x)=e^{-i\frac{x}{\hbar}\hat{p_{x}}}$: the spatial displacement operator, moves the wave function $\psi$ along the x ...
2
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2answers
88 views
Trace of the Operator
I want to ask a question about the fundamental knowledge of trace of the an operator. The operator $A$ is $$A = v (G_r-G_a)$$
where v is the velocity operator of the Hamiltonian ($v=dH/dk$); $G_r$ and ...
2
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1answer
75 views
Fermion operators: why $\rm SU(2)$ symmetry and not $\rm U(2)$ symmetry?
Let us consider operators $c_{\uparrow}$ and $c_{\downarrow}$ which destroy a fermion with spin up and a fermion with spin down, respectively. These operators can be found, for example, in the Hubbard ...
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1answer
52 views
Understanding bra-ket outer product infinite sum
I have an elementary question on clarifying the following expression:
$ \sum_{n=0}^{\infty} |n+2\rangle \langle n| $
Can the term $|n+2\rangle \langle n|$ be observed as the outer product of the ...
