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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Can I prove boundedness of an operator without checking it for its whole domain?

(I don't have a direct reference so this is a little fishy and I'll delete it if nobody recognises what I'm talking about, but I though for starters I'll ask anyway) I've heard at university that if ...
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48 views

Why there is no operator for time in QM? [duplicate]

Is there one central reason why there is no "Time" operator in QM? I know this question has been asked before, but I thought I would try to stimulate some fresh thinking.
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Derivatives of operators

How do derivatives of operators work? Do they act on the terms in the derivative or do they just get "added to the tail"? Is there a conceptual way to understand this? For example: say you had the ...
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340 views

Evaluate Commutator with Partial Derivatives

I need to evaluate the following commutator... $[x(\frac{\partial}{\partial y})-y(\frac{\partial}{\partial x}),y(\frac{\partial}{\partial z})-z(\frac{\partial}{\partial y})]$ i tried applying an ...
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Sum of two density matrices: $\rho=p_1\rho_1+p_2\rho_2$

Suppose we have $$\rho=p_1\rho_1+p_2\rho_2$$ Where $\rho_1$ and $\rho_2$ are density matrices with $p_1+p_2=1$ I'm trying to show this is also a density matrix If we let $$\rho_1=\sum_i^n ...
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186 views

Once I have the eigenvalues and the eigenvectors, how do I find the eigenfunctions?

I am using Mathematica to construct a matrix for the Hamiltonian of some system. I have built this matrix already, and I have found the eigenvalues and the eigenvectors, I am uncertain if what I did ...
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260 views

Physical Significance of Operator Norm/Spectral Norm of a Quantum Operator

Is there any physical significance of operator norm/spectral norm of a hermitian operator?
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586 views

“An operator is hermitian”. Implications?

Alastair Rae states that there are 4 postulates of Quantum Mechanics in his text on the subject matter. The first part of his second postulate can be stated as: Every dynamical variable may be ...
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436 views

How to evaluate spin operators in second quantization for spin symmetry-broken Slater determinants?

Suppose we have the following Slater determinant: \begin{equation} | \Psi \rangle = \prod \limits_{i,i'} a^+_{i\alpha} a^+_{i'\beta} | \rangle \end{equation} where $a^+_{i\alpha}$ creates an electron ...
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Matrix elements of momentum operator in position representation

I have two related questions on the representation of the momentum operator in the position basis. The action of the momentum operator on a wave function is to derive it: $$\hat{p} ...
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2answers
503 views

Constructing the exponential form of a unitary operator

I think I've got this figured out but wanted to make sure I'm doing this right. Working with operators that satisfy bosonic commutation relations $[b,b^\dagger] = 1$, I define a very general unitary ...
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224 views

Time-ordering in QFT

In Srednicki QFT page 37. In the derivation of LSZ reduction formula, he introduces the time-order operator $T$, so no time-dependent creation/annihilation operators are left in the transition ...
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311 views

What is the nature of the correspondence between unitary operators and reversible change?

Why does the formalism of QM represent reversible changes (eg the time evolution operator, quantum gates, etc) with unitary operators? To put it another way, can it be shown that unitary ...
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191 views

Why must quantum logic gates be linear operators?

Why must quantum logic gates be linear operators? I mean, is it just a consequence of quantum mechanics postulates?
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603 views

Intuitive meaning of Hilbert Space formalism

I am totally confused about the Hilbert Space formalism of Quantum Mechanics. Can somebody please elaborate on the following points: The observables are given by self-adjoint operators on the ...
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452 views

Eigenvalues of a quantum field?

Fields in classical mechanics are observables. For example, I can measure the value of the electric field at some (x,t). In quantum field theory, the classical field is promoted to an operator-valued ...
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359 views

What is the value of a quantum field?

As far as I'm aware (please correct me if I'm wrong) quantum fields are simply operators, constructed from a linear combination of creation and annihilation operators, which are defined at every point ...
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345 views

When you apply the spin operator, what exactly is does it tell you?

The example I'm trying to understand is: $ \hat{S}_{x} \begin{pmatrix} \frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}} \end{pmatrix} = 1/2 \begin{pmatrix} \frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}} ...
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834 views

A Short way to show Conservation of Quantum Laplace–Runge–Lenz Vector?

I had been asked to prove the conservation of Quantum Laplace–Runge–Lenz Vector: ...
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309 views

Operator Ordering Ambiguities

I have been told that $$[\hat x^2,\hat p^2]=2i\hbar (\hat x\hat p+\hat p\hat x)$$ illustrates operator ordering ambiguity. What does that mean? I tried googling but to no avail.
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Constructing the space of quantum states

I want to learn how to construct spaces of quantum states of systems. As an exercize, I tried to build the space of states and to find hamiltonian spectrum of the quantum system whose Hamiltonian is ...
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654 views

Adjoint of momentum operator

In position basis, we have, $$\langle x \mid \hat p \mid \Psi(t) \rangle = -\imath \hbar \frac{\partial{\langle x \mid \Psi(t) \rangle}}{\partial{x}} $$ Now i know $\hat{p}$ is a hermitian operator ...
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160 views

Notation for differential operators and wave function math

I know that $[\frac {d^2}{dx^2}]\psi$ is $\frac {d^2\psi}{dx^2}$ but what about this one $[\frac {d^2\psi}{dx^2}]\psi^*$? Is it this like $\frac {d^2\psi\psi^*}{dx^2}$ or this like $\frac ...
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Is H=H* sloppy notation or really just incorrect, for Hermitian operators?

I saw it in this pdf, where they state that $P=P^\dagger$ and thus $P$ is hermitian. I find this notation confusing, because an operator A is Hermitian if $\langle \Psi | A \Psi \rangle=\langle A ...
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4answers
687 views

Is the momentum operator well-defined in the basis of standing waves?

Suppose I want to describe an arbitrary state of a quantum particle in a box of side $L$. The relevant eigenmodes are those of standing waves, namely $$ \left<x|n\right>=\sqrt{\frac{2}{L}}\cdot ...
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Evolution operator for time-dependent Hamiltonian

When i studyed QM I'm only working with non time-dependent Hamiltonians. In this case unitary evolution operator has the form $$\hat{U}=e^{-\frac{i}{\hbar}Ht}$$ that follows from this equation $$ ...
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371 views

Existence of adjoint of an antilinear operator, time reversal

The time reversal operator $T$ is an antiunitary operator, and I saw $T^\dagger$ in many places (for example when some guy is doing a "time reversal" $THT^\dagger$), but I wonder if there is a ...
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449 views

Hamiltonian in position basis

Let $ H = \frac{-h^2}{2m}\frac{\partial^2 }{\partial x^2}$. I want to find the matrix elements of $H$ in position basis. It is written like this: $\langle x \mid H \mid x' \rangle = ...
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351 views

The cleverest way to calculate $\left[\hat{a}^{M},\hat{a}^{\dagger N}\right]$ with $\left[\hat{a},\hat{a}^{\dagger}\right]=1$

Who can provide me some elegant solution for $$\left[\hat{a}^{M},\hat{a}^{\dagger N}\right]\qquad\text{with} \qquad\left[\hat{a},\hat{a}^{\dagger}\right]~=~1$$ other than brute force calculation? ...
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2answers
316 views

Dirac notation question

I don't understand this equality $$\int \!d^3p~\langle\textbf{x}|e^{-i(\hat{\textbf{p}}^2/2m)t}|\textbf{p}\rangle\langle\textbf{p} | \textbf{x}_0 \rangle ~=~\int\! ...
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3answers
332 views

What does it mean to apply an operator to a state?

Let's say I have an operator $\hat{A}$ and a state $|\psi\rangle$. What exactly is the state $\hat{A}|\psi\rangle$? Is it just another different state that I am describing using my $\hat{A}$ and ...
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1answer
167 views

What is the operator for the edge current of a fracional quantum Hall state?

The edge of a fractional quantum Hall state is a chiral conformal field theory. In the Laughlin case it corresponds to the chiral boson, $$ S = \frac{1}{4\pi} \int dt dx ...
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Difficulties with bra-ket notation

I have started to study quantum mechanics. I know linear algebra,functional analysis, calculus, and so on, but at this moment I have a problem in Dirac bra-ket formalism. Namely, I have problem with ...
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What does the Canonical Commutation Relation (CCR) tell me about the overlap between Position and Momentum bases?

I'm curious whether I can find the overlap $\langle q | p \rangle$ knowing only the following: $|q\rangle$ is an eigenvector of an operator $Q$ with eigenvalue $q$. $|p\rangle$ is an eigenvector of ...
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1answer
336 views

Holstein-Primakoff and Dyson-Maleev representation

In Holstein-Primakoff and Dyson-Maleev representation, spin operators are represented by bosonic operators. Roughly speaking, a state with $S^z=S-m$ corresponds to a state containing $m$ bosons. In ...
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201 views

Proof of $Dq-qD=1$ where $D=\frac{\partial }{\partial q}$ is the differential operator

Can anyone provie me the proof of $Dq-qD=1$ where $D=\frac{\partial }{\partial q}$ refers to the differential operator? Or if it's something special to quantum mechanics, why is it? Is this ...
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241 views

Physical meaning of some operators formed by $|Q\rangle \langle Q|$

In Dirac's formulation of quantum mechanics, Suppose that $q$ represents position observable. About $|q\rangle \langle q|$: what does this operator mean? I do get that it results in an operator, but ...
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334 views

State normalization in Dirac's formulation of quantum mechanics

Let us divide the time $T$ into $N$ segments each lasting $δt = T/N$. Then we write $\langle q_F | e^{−iHT} |q_I \rangle = \langle q_F | e^{−iHδt} e^{−iHδt} . . . e^{−iHδt} |q_I \rangle $ Our ...
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3answers
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Proof of Canonical Commutation Relation (CCR)

I am not sure how $QP-PQ =i\hbar$ where $P$ represent momentum and $Q$ represent position. $Q$ and $P$ are matrices. The question would be, how can $Q$ and $P$ be formulated as a matrix? Also, what is ...
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229 views

Why does $i ( LK-KL )$ represent a real quantity?

According to my textbook, it says that $i( LK-KL )$ represents a real quantity when $K$ and $L$ represent a real quantity. $K$ and $L$ are matrices. It says that this is because of basic rules. ...
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190 views

Ordering Ambiguity in Quantum Hamiltonian

While dealing with General Sigma models (See e.g. Ref. 1) $$\tag{10.67} S ~=~ \frac{1}{2}\int \! dt ~g_{ij}(X) \dot{X^i} \dot{X^j}, $$ where the Riemann metric can be expanded as, $$\tag{10.68} ...
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208 views

How to express continuous values as a matrix

Usually a quantity of a matrix is defined as the eigenvalues of the matrix. If so, how can anyone express continuous values, as in Schrodinger picture, into a matrix?
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1answer
66 views

A physical quantity that is a real combination and commutability

Suppose that a matrix $$A ~=~ x_1 B + x_2 C$$ is a linear combination of two self-adjoint matrices $B$ and $C$. I'm interested in when $A$ represents a physical quantity. When the linear ...
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164 views

Matrix representing the quantity - why can some matrices not be physical quantity?

In Heisenberg picture, my textbook says that the following matrix $A = \frac{5}{3}\Sigma_1 + i\frac{4}{3}\Sigma_2$ cannot represent physical quantity. the book says this is because ...
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562 views

The Hermiticity of the Laplacian (and other operators)

Is the Laplacian operator, $\nabla^{2}$, a Hermitian operator? Alternatively: is the matrix representation of the Laplacian Hermitian? i.e. $$\langle \nabla^{2} x | y \rangle = \langle x | ...
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296 views

Derivative of a Position Eigenket

I was flicking through Zettili's book on quantum mechanics and came across a 'derivation' of the momentum operator in the position representation on page 126. The author derived that ...
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Why are the inner products of the eigenfunctions of an operator with a discrete eigenvalue spectrum guaranteed to exist?

I was reading through a textbook, and the statement was made that the inner products are guaranteed to exist if the eigenvalue spectrum of the operator is discrete. I have come across no support for ...
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145 views

The notion of bounded states in quantum mechanics and their characterization with operators

Is there any case of potential $V$, such that the continuity of the operator $H=c\ \Delta+V$ is not spoiled? And I don't know any non-differnetial operator examples for continous spectra. I ...
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522 views

Existence of creation and annihilation operators

In a multiple particle Hilbert space (any space of any multi-particle system), is it sufficient to define creation and annihilation operators by their action (e.g. mapping an n-particle state to an ...
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306 views

What is the physical meaning of weak expectation values?

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