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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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} = \sqrt{\frac{\bar{n}^{n_\lambda}e^{-\...
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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 = -\dfrac{\hbar^...
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67 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 $$H_f=\omega_ca^{\...
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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 ...
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75 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 $\mid R\rangle$, $\mid L\rangle$....
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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 ...
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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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59 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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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 ...
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85 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 ...
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2answers
152 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 ...
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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 ...
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3answers
83 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 ...
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49 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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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 ...
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421 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 ...
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1answer
43 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 ...
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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 $$\vec{\sigma}\,,...
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1answer
291 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 ...
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2answers
117 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 ...
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245 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 ...
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353 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}}) $...
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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 ...
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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 $f_{ij}{}...
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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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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 ...
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87 views

Does Operator Product Expansion form an algebra?

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

Lorentz force derivation in quantum mechanics [closed]

In Sakurai and Napolitano, chapter 2, there's a derivation of the QM Lorentz force. Given $$H=\frac{1}{2m}\left(\mathbf{p}-\frac{e\mathbf{A}}{c}\right)^2+e\phi = \frac{\mathbf{\Pi}^2}{2m}+e\phi$$ ...
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608 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 ...
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45 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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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.
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370 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. $\left[Q,P\right]=i$ ...
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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 ...
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49 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 ...
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3answers
186 views

Does quantum mechanics only deal with Hermitian/self-adjoint operators?

Whatever matrix I am seeing in quantum mechanics are all Hermitian matrices. We are using their eigenvalues for different types of work. Fortunately all their eigenvalues are real. Have you ever seen ...
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1answer
110 views

The state space is somehow defined by the observables?

In Quantum Mechanics states of a system are described by vectors in a Hilbert space called the state space while the physical quantities associated to the system are described by hermitian operators ...
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110 views

Commutation relations in quantum mechanics

As we know, simple harmonic oscillator can be solved only by commutation relations between creation and annihilation operators, and the Hamiltonian expression. The spin energy is either solved only ...
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1answer
105 views

What is the origin of the quantum operators for $p$ and $E$? [closed]

It is always stated the quantum operators for p and E are the ones we´re familiar with (the operator for energy, $H=i\hbar\frac{\partial}{\partial t}$ and the momentum operator, $\mathbf{p}=-i\hbar\...
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Part of a Wigner theorem [closed]

I was trying to understand why there should exist operator in Hilbert space to correspond to any symmetry transformation and found about Wigner's theorem. In it, I can see that any transformed vector ...
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509 views

What is the physical meaning of weak expectation values?

In the two-state formalism of Yakir Aharonov, the weak expectation value of an operator $A$ is $\frac{\langle \chi | A | \psi \rangle}{\langle \chi | \psi \rangle}$. This can have bizarre properties. ...
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2answers
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Commutator relationships and the exponential

I am currently trying to prove that the two following commutator relationships are equivalent (for an operator $\hat{A}(s)$ that depends on a continuous parameter $s$), so if one holds the other one ...
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1answer
60 views

Should state vectors be considered constant?

By the principle of superposition, a state vector can be defined as $$\begin{align} \psi(x) &= c_1 \psi_1(x) + c_2 \psi_2(x) + \cdots + c_n \psi_n(x) \\ \lvert\psi\rangle &= \begin{pmatrix}c_1 ...
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1answer
103 views

Relationship between Quantum superposition and Uncertainty principle

I'm an amateur in quantum mechanics. I am confused after reading the following in the wikipedia article about quantum superposition: If the operators corresponding to two observables do not ...
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3answers
108 views

Eigenstates of Ladder Operators

According ot Griffith's Intro to Quantum Mechanics (page 147), if some function $f$ is an eigenfunction of $L^{2}$, then $L_{-}f$ is also an eigenfunction of $L^{2}$. Is $f$ also an eigenfunction ...
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49 views

Sum and Different and angular momentum operators

Why is $\overrightarrow{L_{1}}+\overrightarrow{L_{2}}$ an angular momentum operator, but not $\overrightarrow{L_{1}}-\overrightarrow{L_{2}}$? What does this show about the applicability of the vector ...
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50 views

Expansion operator for quantum mechanics

As a counterpart to the quantum mechanical translation operator (see for example this post) is there a unitary operator which describes the stretching of a line. That is consider I have a chain of ...
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1answer
72 views

Dirac equation from a vierbein operator?

Klein-Gordon equation can be derived straightforwardly by getting the mass-energy relation from special relativity in tensorial form, $$\eta^{\mu\nu}p_{\mu}p_{\nu} = m^2c^2$$ and promoting the ...
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1answer
59 views

General expectation value

I have a basic question related to finding expectation values of an operator $\hat{Q}$ We know that the expectation of $\hat{Q}$ (in the position space) is given by $$\langle Q \rangle=\int {\Psi^* ...
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2answers
109 views

Derivation of Schrodinger's wave equation

To derive $$i \hbar \frac{\partial}{\partial t} \psi = H \psi,$$ we start with $$i \hbar \frac{\partial}{\partial t} |\alpha \rangle = H| \alpha \rangle$$ and then multiply by $\langle x|$ on the ...
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1answer
54 views

Why do I get negative expectationvalues when I use ladder operators? [closed]

I'm trying to find the expectationvalue for $p^2$ where $p = i\sqrt{\frac{hmw}{2}}(a_{+} - a_{-})$ and i end up with the following result \begin{align*} \langle \psi_0|p^2|\psi_0\rangle &= -\frac{\...