In physics, an operator is almost always either a square matrix or a linear mapping from one space of functions (often on $\mathbb{R}^N$ or $\mathbb{C}^N$) to the same or other like space of functions. Operators serve as *observables* and as *time evolution operators* in Quantum Mechanics. This tag ...

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170 views

Explaining causal completion axiom in Haag-Kastler axioms?

There are several variants of the Haag-Kastler axioms for algebraic quantum field theory. Usually one associates an algebra $\mathcal{A}(O)$ to each open region $O$ of spacetime. An often-suggested ...
-3
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2answers
87 views

Is the Wave Function a Unitary Operator? [closed]

A unitary operator can be represented as an exponential $$e^{iA}$$ and as we represent the wave function in general as $$e^{i k x}.$$ Does that mean that the wavefunction is unitary as the exponent is ...
4
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2answers
105 views

Is the eigenvalue of Hamiltonian invariant under linear transformation of momentum operator?

It is given The dynamics of a particle moving one-dimensionally in a potential V(x) is governed by the Hamiltonian $H_0 = p^2 /2m + V(x) $, where $p = -i\hbar d/dx$ is the momentum operator. ...
0
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0answers
81 views

Eigenfunctions of translation operator

I had an HW assignment in which we were asked to find the eigenfunctions of the translation operator which is defined as follows: $$\hat{D}(a)=e^{-(i/\hbar)a\hat{P}}$$ where $\hat{P}$ is the momentum ...
11
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1answer
316 views

Shape of the state space under different tensor products

I am currently studying generalized probabilistic theories. Let me roughly recall how such a theory looks like (you can skip this and go to "My question" if you are familiar with this). Recall: In a ...
-1
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1answer
46 views

Deriving eigen values of $\hat{N}$

So let's say we have an operator $\hat{a}$ (ladder operator), where $\left[\hat{a},\hat{a}^\dagger\right] = 1$, and $\hat{a}^2 |\phi\rangle = 0$. How do I show that the eigenvalues of ...
0
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2answers
457 views

Expectation values of commutator and anti-commutator (momentum and position)

What are the expectation values of commutator and anti-commutator for momentum and position operators? In the case of commutator: $$\langle[x,p]\rangle=\langle i\hbar\rangle=~?$$ In the case of ...
1
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0answers
14 views

Relating $C_j(t-t') = \left<\hat{B}_j(t)\hat{B}_j(t')\right>$ to $\left<\hat{B}_j(t)^2\right>$

I'm trying to relate a known quantum correlation function $C_j(t-t') = \left<\hat{B}_j(t)\hat{B}_j(t')\right>$ (which is not real!) of a (time dependent, but this is not super relevant) quantum ...
0
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1answer
33 views

Can a sum of Pauli matrices be a real value?

In the page 2 of Quantum Annealing for Constrained Optimization, the authors introduced a constraint term under the Constrained quantum annealing (CQA) section. The ultimate goal is to work out a ...
3
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1answer
40 views

Generators of a certain symmetry in Quantum Mechanics

In Classical Mechanics to describe symmetries like translations and rotations we use diffeomorphisms on the configuration manifold. In Quantum Mechanics we use unitary operators in state space. We ...
3
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1answer
355 views

Ladder Operator killing state

So in my last question, @joshphysics showed me how to prove $K_\pm$ were ladder operators. Now I need to show that there is a lowest state, i.e $$\langle m_0|K_+=K_-|m_0\rangle=0$$ I am not ...
5
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832 views

Why is the “canonical momentum” for the Dirac equation not defined in terms of the “gauge covariant derivative”?

The canonical momentum is always used to add an EM field to the Schrödinger/Pauli/Dirac equations. Why does one not use the gauge covariant derivative? As far as I can see, the difference is a factor ...
1
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2answers
46 views

Quantization of the Hamiltonian of a particle in a uniform magnetic field

If a particle of mass $m$ and charge $q$ is subject to a uniform magnetic field and if we have a vector potential $\mathbf{A}$ then we know that classically the dynamics of the particle will be ...
8
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2answers
3k views

Weyl Ordering Rule

While studying Path Integrals in Quantum Mechanics I have found that [Srednicki: Eqn. no. 6.6] the quantum Hamiltonian $\hat{H}(\hat{P},\hat{Q})$ can be given in terms of the classical Hamiltonian ...
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0answers
36 views

Lippmann-Schwinger equation and time dependence

Consider the Lippmann-Schwinger equation (LSE) $$ |\psi\rangle = |\phi\rangle + \hat{G}_0(\epsilon) \hat{V} |\psi\rangle \tag{1}$$ where $\hat{G}_0(\epsilon) = \frac{1}{\epsilon - \hat{H}_0 + ...
1
vote
1answer
107 views

Three dimensional isotropic harmonic oscilator Hamiltonian

Let us consider the Hamiltonian for the isotropic three dimensional harmonic oscilator: $$H = \dfrac{\mathbf{P}^2}{2m}+\dfrac{m\omega^2\mathbf{R}^2}{2},$$ where $\mathbf{P}$ and $\mathbf{R}$ are the ...
0
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0answers
20 views

Field Coherent State relationship to annihilation operator

I am trying to show that $|\Psi_{\lambda\bar{n}}> = \sum c_{\lambda\bar{n}m}\exp(-i(n+\frac{1}{2})\omega_\lambda t)|n_\lambda>$, where $c_{\lambda\bar{n}m} = ...
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0answers
22 views

Particle in a $V(\rho)$ potential in cylindrical coordinates

Consider cylindrical coordinates $(\rho,\phi,z)$ and consider a particle with a potential energy $V(\rho)$. If we write the Hamiltonian operator in these coordinates we find that $$H = ...
1
vote
1answer
55 views

Going to the interaction picture in the Jaynes–Cummings model [closed]

In the Jaynes–Cummings model for a two level atom, the Hamiltonian for the atom is defined as (I let $\bar{h}=1$) $$H_a=\omega_a\frac{\sigma_z}{2}$$ and the field Hamiltonian is ...
0
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0answers
27 views

Construct any Hamiltonian that is the linear combination of existing constructable Hamiltonians

In the paper Quantum Computation over Continuous Variables, it states that since $$e^{iAt}e^{iBt}e^{-iAt}e^{-iBt} = e^{-[A,B] t^2} + O(t^3)$$ when $t\rightarrow 0$, if one can apply a set of ...
0
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1answer
66 views

Single particle tunneling Hamiltonian

In reference to Problem 9, Chapter 2 in Modern Quantum Mechanics by JJ Sakurai, For a single particle tunneling in a 1D double well potential, with position eigenkets $\mid R\rangle$, $\mid ...
1
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0answers
71 views

CFT: from States to Operators

I'm having trouble finding the general algorithm for moving from states to operators under the state-operator correspondence in a CFT. Does anyone have any hints as to how one might go about ...
6
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2answers
211 views

What is $\langle \phi | H | \psi \rangle$ in QM?

I know that $\langle \phi | \psi \rangle$ is the probability of going from the $\psi$-state to the $\phi$-state, and that $\langle \phi | H | \phi \rangle$ is the expectation value of the energy for ...
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0answers
47 views

Action of Swap operator

I am trying to understand concept of swap operator, introduced in the article http://arxiv.org/pdf/1001.2335v2.pdf by means of simple example. Swap operator is supposed to act on two identical(?) ...
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16 views

Linear viscoelastic differential operators

I am starting with differential operators: $P = \sum_{i=0}^{N}p_i \cfrac{d^i}{dt^i}$ $Q = \sum_{i=0}^{N}q_i \cfrac{d^i}{dt^i}$ $p_i$ and $q_i$ are functions of time only. $K$ is a constant that ...
1
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1answer
75 views

Fermionic ladder operators [closed]

After reading Dirac's method for finding the eigen energies of a harmonic oscillator by means of ladder operators and commutation relations, I tried making some exercises on them. First I did a ...
2
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2answers
150 views

Complex conjugate of the Schrödinger equation?

This might be a very simple question but I don't understand how to compute the complex conjugate of the Schrödinger equation: $$ i\partial_t \psi = H\psi $$ where $H$ is an hermitian operator. How to ...
3
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4answers
169 views

Help understanding proof in simultaneous diagonalization

The proof is from Principles of Quantum Mechanics by Shankar. The theorem is: If $\Omega$ and $\Lambda$ are two commuting Hermitian operators, there exists (at least) a basis of common eigenvectors ...
1
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3answers
82 views

Show that a linear operator can be written in terms of its spectral decomposition [closed]

Let $\hat Q$ be an operator with a complete set of orthonomal eigenvectors: $$ \hat Q |e_n\rangle=q_n|e_n\rangle\ \ (n=1,2,3,...) $$ Show that $\hat Q$ can be written in terms of its spectral ...
3
votes
1answer
42 views

Lower bound on energy is potential minimum

Suppose we have a particle of mass $m$ that is in an eigenstate $|\psi\rangle$ of the Hamiltonian $\hat{H}=\hat{T}+\hat{V}$, where $\hat{T}$ is the kinetic energy operator and $\hat{V}=V(\hat{r})$ is ...
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0answers
29 views

Statement of vector model of angular momentum in Eisberg and Resnick

While serving as a teaching assistant for a sophomore-level Quantum Physics course, I came across the following paragraph in Eisberg and Resnick regarding orbital angular momentum (section 7-8, page ...
7
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1answer
410 views

Kraus operator rank

All quantum operations $\mathcal{E}$ on a system of Hilbert space dimension $\mathcal{d}$ can be generated by an operator-sum representation containing at most $\mathcal{d^2}$ elements. Extending ...
1
vote
1answer
42 views

Is it necessary to prove the existence of an operator representing symmetry on Hilbert space?

Is there any need to prove the existence of an operator $U$ which represents the action of symmetry transformation on rays in Hilbert space? Or is it enough just to prove that it is unitary and linear ...
1
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1answer
41 views

Correct way to write Pauli matrices

This is purely a question of notation for the Pauli matrices. What is the correct way to write them for use as operators? Would I just write the vector of the matrices as a vector i.e ...
4
votes
1answer
277 views

Virasoro operators commutation relations [closed]

For the commutation relation in quantising the bosonic string $$\left[L_n,L_{m}\right]=(n-m)L_{n+m}+\frac{D}{12}n(n^2-1)\delta_{n+m,0}$$ we can then calculate this for $m=-n$ in between the vacuum ...
3
votes
2answers
116 views

Clarification from Griffiths Introduction to quantum mechanics

A question from Appendix Linear algebra A.3 Matrices on page 441 "If you know what a prticular linear transformation does to a set of basis vectors, you can easily figure out what it does on any ...
6
votes
1answer
243 views

Can I always find an unitary operator $B$ such that $\{A,B\}= 0$ for a given, unitary operator $A$?

Considering an arbitrary unitary operator $A$, what is the least criteria this operator must satisfy in order that it is possible to find at least another unitary operator $B$ that anti-commutes with ...
1
vote
1answer
341 views

Diagonalize a dot product with Pauli matrices

How can I diagonalize the following operator? $$\lambda \hat{\vec{\sigma}}\cdot\vec{r}$$ where $\lambda$ is a real constant, $\hat{\vec{\sigma}}=(\hat{\sigma_{x}},\hat{\sigma_{y}},\hat{\sigma_{z}}) ...
0
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0answers
28 views

Vaccuum expectations for a string of creation and annihilation operators

I want to determine the vaccum expectation value of a string of creation and annihilation operators. They have a very specific form: $$\langle \prod_{i=1}^n \hat{a}(k_i) \, \, \hat{N}_1 \prod_{j=1}^n ...
3
votes
1answer
92 views

BRST quantization (Green, Schwarz, Witten)

In Green, Schwarz, Witten Volume 1, section 3.2, BRST quantization is introduced in a general way. A Lie algebra $G$ is defined with elements $$ [K_i,K_j] = f_{ij}{}^k K_k \tag{3.2.1}$$ where ...
1
vote
1answer
84 views

Spectral functions in quantum mechanics

I'm a math student and a totally newcomer to quantum mechanics and I'm trying to teach myself this subject by studying Faddeev and Yakubovski's Lectures on Quantum Mechanics for Mathematics Students. ...
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0answers
85 views

Physical significance of non-commutativity of ladder operators in Quantum Harmonic Oscillator

If we apply the raising (creation) operator to $Ψ_n(x)$ and the apply to it the lowering (annihilation) operator, we get $Ψ_n(x)$ times a constant. Does it physically say something? Can we get any ...
0
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1answer
83 views

Does Operator Product Expansion form an algebra?

The operator product algebra in CFT is defined as $$\mathcal{O}_i(z,\bar{z})\mathcal{O}_j(\omega,\bar{\omega}) = \sum_{k} C^k_{ij}(z-\omega,\bar{z}-\bar{\omega})\mathcal{O}_k(\omega,\bar{\omega}).$$ ...
1
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1answer
61 views

Lorentz force derivation in quantum mechanics [closed]

In Sakurai and Napolitano, chapter 2, there's a derivation of the QM Lorentz force. Given $$H=\frac{1}{2m}\left(\mathbf{p}-\frac{e\mathbf{A}}{c}\right)^2+e\phi = \frac{\mathbf{\Pi}^2}{2m}+e\phi$$ ...
15
votes
2answers
577 views

The formal solution of the Schrodinger equation

Consider the Schrödinger equation (or some equation in Schrödinger form) written down as $$ \tag 1 i \partial_{0} \Psi ~=~ \hat{H} \Psi . $$ Usually, one likes to write that it has a formal solution ...
1
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1answer
42 views

Trace of operator defined by two state vectors in Quantum Mechanics

I'm studying QM from the book 'Quantum Mechanics. Concepts and Applications' by Zettili. There's an example which gives us two state vectors $$ | \psi \rangle = 9i \ | \phi_1 \rangle + 2 | \phi_2 ...
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0answers
50 views

What are some resources for Algebraic quantum mechanics for Physicists

I am interested in the GNS construction and other stuff. I am aware of Valter Moretti's book but I want something that is more inclined towards physicists.
4
votes
3answers
368 views

Relation between representations of boson operators?

I have a simple (I think !) question about the representations of boson operators and how they are related. First of all let's define two conjugate observables $Q$ and $P$ (i.e. $\left[Q,P\right]=i$ ...
0
votes
1answer
40 views

Matrix for Ladder Operators?

I found this website which shows how to derive the matrices for $L_{+}, L_{-}$ and while I understand the derivation of the equation for $<lm|L_{+}|lm'>$ and $<lm|L_{-}|lm'>$ I do not ...
1
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1answer
43 views

What is the relation between Hilbert space constructed from the GNS construction and the standard Hilbert space-state?

I recently started reading Algebraic quantum mechanics. So I have no knowledge of the subject. In the GNS construction we construct the Hilbert space of states as follows, We endow the algebra of ...