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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Parity operator eigenstates [closed]

I have a problem I cannot solve on my own. I have given two states $\psi_1$ and $\psi_2$ and an Operator $O$ such that $P \psi_1 = \epsilon_1 \psi_2$, $P \psi_2 = \epsilon_2 \psi_2$ and $POP^{-1} = ...
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52 views

Is the scalar field operator self-adjoint?

In A. Zee's QFT in a Nutshell, he defines the field for the Klein-Gordon equation as $$ \tag{1}\varphi(\vec x,t) = \int\frac{d^Dk}{\sqrt{(2\pi)^D2\omega_k}}[a(\vec k)e^{-i(\omega_kt-\vec k\cdot\vec ...
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39 views

State operator map in $R \times S^{D-1}$ to $R^D$

This question is relevant in my former question State-operator map, and scalar fields and State operator corrponding $i.e$ $S\times S$ to $R^2$. (which was wrong, corrected one was states in $R \times ...
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64 views

Parity operator with other operators

I need to show the following: $$P x P^{-1} = -x, \ P p P^{-1} = -p, \ P L P^{-1} = L$$ where $P$ is the parity operator and $x$, $p$ and $L$ are the position, momentum and angular momentum ...
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36 views

State-operator map, and scalar fields

Up so far, i have been studied state-operator correspondence, $i.e$, i have been questioned State operator corrponding $i.e$ $S\times S$ to $R^2$ which was wrong question. By studing Ginsparg's ...
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38 views

quantum mechanics probability of +1 spin between arbitrary directions

So there are two unit vectors $\hat{m}$ and $\hat{n}$ with arbitrary directions in 3D space. There is a spin operator along a particular direction in space, say that of $\hat{r}$, is: $\sigma_r= ...
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144 views

In nonrelativistic Quantum Mechanics, is the expectation value of a sum of operators always equal to the sum of the expectation values?

Suppose that $\lvert \psi_n \rangle$ are the eigenvectors of a Hamiltonian, $\hat{H}$, which span some Hilbert space $\mathcal{H}$ and satisfy $$\hat{H}\lvert \psi_n \rangle = E_n \lvert \psi_n ...
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213 views

Commutation of vector operators

I'm supposed to show that $\left[\mathbf A,\mathbf B\right]=0$ (for two vector operators $\mathbf A$ and $\mathbf B$) if and only if all components of $\mathbf A$ commute with all components of ...
3
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199 views

Real versus complex Hamiltonian

While a Hamiltonian must be a Hermitian matrix, it can either be real or complex. Is there a significance for having a real Hamiltonian? Does it have any additional physical symmetries? For ...
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298 views

Dirac notation - specific acting orientation for operators

I have this doubt: Imagine two operators $A$ and $B$ and the state $\psi$. I know that the following statement is true: $$\langle\psi| A|\psi\rangle^*=\langle\psi| A^\dagger|\psi\rangle$$ But ...
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107 views

Prove that a translation operator times a reflection operator is unitary and Hermitian [closed]

I am trying to prove some properties of the product of the (unitary) translation operator $\hat{T}(a)\psi(x) = \psi(x-a)$ and the (Hermitian) reflection operator $\hat{R} \psi(x) = \psi(-x)$. In ...
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76 views

Relativistic Commutation relation for momentum and position

We all know that the canonical commutation relation give you $$[x_i,p_j]=i\hbar\delta_ij,$$ is there a relativistic version such as $$[x^a,p_b]=i\hbar\delta_a^b?$$ If so what is the time ...
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281 views

Simple QFT simulation - how to do it

I would like to write a simple QFT simulation for a free scalar field with a cubic interaction term. However, I got stuck a bit. I will try to describe what I think I understand. I want to have a ...
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2answers
66 views

Expectation value of an imaginary operator acting on a real function

In a video (http://youtu.be/r_gBQ_qhg8U?t=9m58s) it's stated that a matrix element of an imaginary operator acting on a real wave function is zero, i.e. ...
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133 views

What does a line above a commutator, e.g. $\overline{[x, H]}$ mean?

What does this notation mean in relation to quantum mechanics? $$\overline{[x,H]}\qquad\text{or}\qquad\overline{[p,H]}\tag{1}$$ I know $[x,H]$ is just the commutator e.g $xH-Hx$, and the ...
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1answer
55 views

Calculate mean number of particles of time evolution coherent state [closed]

I seem to be missing some identities. I know you need to calculate P_n = |<n|alpha_t>|^2 and mean number of particles is the infinite sum of nP_n. However I ...
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65 views

Show that $|n\rangle$ is correctly normalized [closed]

Prove that $$|n\rangle = \frac1{\sqrt{n!}} (\hat a^\dagger)^n |0\rangle$$ is correctly normalized. I know I must show its bra-ket equals 1 but I don't know what bra-ket notation really means, so ...
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117 views

Canonical Commutation Relations in arbitrary Canonical Coordinates

If one were to formulate quantum mechanics in an arbitrary canonical coordinate system, does he impose canonical commutation relations using Dirac's recipe? ...
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115 views

Is there a physical significance to non-normal states of the algebra of observables?

Quantum theory may be formalized in several different ways. Generally, the physical discussion of different states of a quantum system distinguishes pure and mixed states, and then subsumes both in a ...
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93 views

Closure relation for degenerate eigenkets

Consider an observable in quantum mechanics, with a degenerate eigenvalue in a continuous spectrum. Is it possible for such an eigenvalue to have a finite degeneracy? If the degeneracy is infinite, ...
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89 views

Particle Hole Transformation of Hamiltonian

The particle hole transformation for a bipartite lattice $\Lambda$ (with sublattices $A$ and $B$) can be written as $$U^\dagger c_{i,\uparrow} U = \epsilon(i) c^\dagger_{i\uparrow} \\ U^\dagger ...
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183 views

Inverse Quantum Operator

In the quantum harmonic oscillator problem, how would one go about calculating $$\left\langle n\left|\frac{1}{X^2}\right|n\right\rangle$$ using raising and lowering operators $a^{\dagger}, a$ only, ...
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The momentum representation of $x$ and $ [x,p]$ [duplicate]

To deduce the momentum representation of $[x,p]$, we can see one paradom $$<p|[x,p]|p>=iℏ$$ $$<p|[x,p]|p>=<p|xp|p>−<p|px|p>=p<p|x|p>−p<p|x|p>=0$$ Why? If we ...
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Why Hamiltonian is Hermitian? [duplicate]

Everyone knows that this is needed to make eigenvalues real, but still why we enforcing such a structure at first place? An arbitrary operator can have as complex as real eigenvalues, we can simply ...
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431 views

Is there any theorem that suggests that QM+SR has to be an operator theory?

UPDATE To make my question more precise, I'll define what I mean by an operator theory: An operator theory is a theory in which the dynamical objects are operators, i.e., the equations of motion ...
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50 views

Show that translation and rotation operator are unitary

I have a problem understanding how to show that operators are unitary if they are not in the "normal" matrix form. The translation operator is defined as $$(T_v \psi)(x) = \psi(x-v)$$ and the ...
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44 views

Distinguishing degenerate states physically

Suppose there is a free particle on a circle with radius r. The energy spectrum is then $$E_n = \frac{n^2\hbar^2}{2mr^2} \,.$$ Thus, when $n \neq 0$, then the spectrum of energies is degenerate ...
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110 views

Why are eigenspaces of a Hermitian operator mutually orthogonal? [closed]

In Quantum Mechanics, from the properties of the solution of Schrodinger's Equation inside the infinite well, is that they are: Mutually orthogonal for different eigenvalues. Orthonormal. Complete. ...
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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 ...
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Relation of field creation operators to path integral?

Applying two field creation operators to a vacuum I get: $$\hat{\psi}^\dagger(x)\hat{\psi}^\dagger(y)|0\rangle = (\hat{\phi}(x)\hat{\phi}(y) - s^{-1}(x-y)) |0\rangle$$ where the quantum field ...
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Unitary Transfomation from One Basis to Another [closed]

So we have two orthonormal linearly independent basis $\{ |\phi_1 \rangle, \dots, |\phi_n \rangle \}$ and $\{ |\psi_1 \rangle, \dots, |\psi_n \rangle \}$. We can express the basis vectors of the ...
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337 views

Operator vs. Matrix in quantum formalism

We use in Dirac formalism of QM the tool of operators and kets in spatial and spin spaces to obtain eigenvalues and eigenkets. But the operation here is simply that of a matrix multiplication. Now ...
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$(H\Psi(x,t))^*=H\Psi^*(x,t)$?

In the solutions of an exercise I got confused about the following equality $$(H\Psi(x,t))^*=H\Psi^*(x,t).$$ Is this true in general? Or in special cases? It seems to imply that H is a real matrix ...
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What is the connection between Hilbert Space and path integrals?

Given a space of states $|\rangle$, $|x\rangle$, $|x,y\rangle$, with the creation operators such as $\hat{\phi}(x)|y,z\rangle=|x,y,z\rangle$ for creating a particle at position $x$ and so on. How ...
6
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What is meant by a “c-number”?

In Chapter 2 of David Tong's QFT notes, he uses the term "c-number" without ever defining it. Here is the first place. However, it's easy to check by direct substitution that the left-hand side ...
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Is there a simple expression for $[x,e^{ixp}]$?

I'm sure this exists somewhere, but somewhat surprisingly it is not that easy to google.* The commutators $$ \left[x,e^{i(ax^2+b(xp+px)+cp^2)}\right] $$ of position and the exponential of a quadratic ...
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59 views

OPE of parity even theories in CFT.

If I consider an OPE of some operators, which belong to a theory where parity is not violated, will I have a constraint on the kind of operators appearing in the right hand side ? For example, I ...
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1answer
64 views

Could you explain this flow of calculation?

I am reading this book, Quantum Optics by Walls and Milburn. I am working on Chapter 6 which is about the Stochastic Methods. I don't understand a calculation in this chapter. Let $w(t)$ be the ...
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1answer
291 views

Must bounded operators have normalisable eigenfunctions and discrete eigenvalues?

When we have bound states, to my knowledge, we have states that are normalisable and a discrete energy spectrum. However, in the case of scattering states that have a continuous energy spectrum, the ...
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1answer
86 views

Decompose a Hermitian Operator into Eigenvalues and Projectors

Quantum Computing - A Gentle Introduction by Eleanor Rieffell and Wolfgang Polak states on p57 : Any Hermitian operator $O$ with eigenvalues $\lambda_j$ can be written as $O = \sum_j \lambda_j ...
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1answer
385 views

The uncertainty in angular momentum

It is known that the different spatial components of the angular momentum don't commute with each other. $$ [L_x,L_y] \propto L_z \\ [L_y,L_z] \propto L_x \\ [L_z,L_x] \propto L_y $$ Also it is known ...
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1answer
44 views

Difference between operators used to represent quantum gates vs that to represent physical observables?

I have learnt that informations about a physical observable property is buried in the state vector of a quantum system. To get the possible value of a property all we need to do is multiply the state ...
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117 views

Gauge covariant derivative of a creation operator

Suppose we define the (gauge) covariant derivative or as $$\tilde{\nabla}=\nabla+ie\textbf{A},$$ where the vector potential $\textbf{A}$ has a matrix structure where only the diagonal has nonzero ...
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Definition of the “support” of the reduced density matrix

Some of the papers in condensed matter physics use the word "support" (space). For example, the following papers use the support especially for the reduced density matrix. ...
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114 views

Commutator of position and momentum

I'm reading Sakurai's Quantum Mechanics. One of the problem in the book asks to use the relation $$ \langle{x}|p\rangle=\frac{1}{\sqrt{2\pi\hbar}}e^{\frac{ipx}{\hbar}} $$ to evaluate ...
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289 views

Hermitian conjugate of differential operator

Help me find $\hat{B^\dagger}$, when we know that $$\hat{B}=i\frac{d}{dr}$$ with the condition that $\hat{B}$ is defined in spherical coordinates. My approach: $$ ...
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1answer
170 views

How/Why did Feynman relate the element of Hamiltonian matrix $H_{12}$ to the amplitude to go from $|1\rangle$ to $| 2\rangle$?

$$ \newcommand{\bk}[2]{\left\langle #1 | #2 \right\rangle} \newcommand{\ket}[1]{\left| #1 \right\rangle} \newcommand{\bra}[1]{\left\langle #1 \right|} \newcommand{\biik}[3]{\left\langle #1 | #2| ...
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235 views

Wick Theorem, ordering & CFT

I'm having a little trouble with correlation functions wick theorem and ordering in the context of OPE and CFT, for string theory. (1) My first question, the propagator is: $$<X(z) X(w)> = ...
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Free Vacuum vs Interacting Vacuum and Wick's theorem

I'm studying perturbation theory in QFT and I stumbled on a conceptual problem. My understanding of the interplay between LSZ reduction formula and the Gell-Mann & Low perturbation series is ...