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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What exactly implies the need of quantum mechanics for self-adjoint and not only symmetric operators? [duplicate]

We know that quantum mechanics requires self-adjoint operators, not only symmetric. Can we say that this follows ONLY from the two following axioms of quantum mechanics, namely that each observable ...
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206 views

Deriving cross product from angular momentum algebra

Is it possible to derive: \begin{equation} \hat{L}=\hat{r}\times \hat{p} \end{equation} from the angular momentum algebra: \begin{equation} [\hat{L}_i,\hat{L}_j]=i\ \hbar\ \epsilon_{ijk}\hat{L}_k\ ? ...
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Does the time ordering operator have a rigorous definition?

In quantum field theory, the time ordering operator (TOO) appears in the formal expressions for the scattering amplitudes. It acts upon a product of operators that each depends on time, and returns ...
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2answers
148 views

What is the meaning of commuting Hamiltonians?

I have two quantum mechanical Hamiltonians such that \begin{equation} [\hat{H}_1,\hat{H}_2] = 0, \end{equation} where $\hat{H}_1$ and $\hat{H}_2$ act on the same set of states. What is there to ...
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186 views

Tensor product of operators in QM

If I wanted to find the coefficients of a linear transformation between 2 vectors in the basis for 2 spin $1/2$ paticles (let's say for starters we are not even looking for a unitary transform): ...
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76 views

Why can we not choose the stress tensor in a CFT to be identically symmetric?

The stress tensor for a conformal field theory (or any quantum field theory) can be derived from the action $S$ by the functional derivative $$T^{\mu \nu} ~=~ -\frac{2}{\sqrt{|g|}}\frac{\delta ...
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122 views

What is the idea behind canonical quantization?

From what I understand, canonical quantization of a classical theory consists of replacing the observables by abstract operators, of which only the commutation rules, which have to correspond to the ...
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45 views

About shift operators

The question is this: Does $$L_+ L_- Y_{lm} $$ ,where $Y_{lm}$ is a spherical harmonic function, equals to zero. If so, why? The two operators above are defined as $$L_+ ={L_x + iL_y } $$ $$L_-={L_x ...
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Physical significance of Cayley Transform

In the book on Quantum Mechanics by Capri (in Chapter 6), its said that an operator $A$ is self adjoint if the operator, $U$ given by $$ U = (A - i I)(A + i I)^{-1} = -(I+iA)(I-iA)^{-1} = -\text ...
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Why do $S_x$ and $S_y$ flip up/down spin states but $S_z$ does not?

By using the notation $S\lvert s,m_s\rangle$, such that $\bigl\lvert\frac{1}{2},\frac{1}{2}\bigr\rangle=\lvert+\rangle$ and $\bigl\lvert\frac{1}{2},-\frac{1}{2}\bigr\rangle=\lvert-\rangle$ we can ...
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55 views

Why is it “disconcerting” if the components of an operator do not commute?

A symmetrized operator is given by $$\hat{R}=\frac{1}{2\hat{H}}\hat{N}+\hat{N}\frac{1}{2\hat{H}}.$$ With $\hat{H}$ the Hamiltonian and $\hat{N}$ the first moment of energy. The defined $\hat{R}$ is ...
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Cayley transform to von Neumann theorem

Self-ajointness of an operator can be found using the Cayley transform of the operator, if its unitary, $$ U = (A - i I)(A + i I)^{-1} $$ From this we can go about finding the deficiency subspaces ...
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71 views

What really is the self-adjoint extension?

Going through the Quantum mechanics book by Capri, am time and again held with some stupid doubts on this topic of self-adjointness. We have for the momentum operator in finite domain, $$ p = ...
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67 views

Quantum state of a system after measurements with non-commutative operators

a) Assume two operators $A$ and $B$. 1) Assume $$[A,B]=0 $$ and $$ ψ= \sum c_n u_n ~~~~\text a~ wavefunction~ describing~ the~ state~ of~ the~ system $$ with $$Aψ=a_n u_n $$ $$Bψ=b_n u_n$$ If we ...
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49 views

Using creation and annihilation operators to prove the expression for the $n$th excited state in terms of the vacuum state

How does one prove that the $n^{th}$ excited state of a quantum harmonic oscillator (QHO) can be obtained by applying the creation operator $a^{\dagger}$ $n$-times to the vacuum state $\vert ...
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1answer
59 views

Eigenvectors of $p_x$ in a particular domain

Defining the $p_x$ operator for the problem of particle in a infinite well. In the book by Capri on Quantum mechanics, the domain of the operator is given by, $$ p = -i\hbar \frac{\partial ...
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87 views

Commutator of fermionic operators

The fermionic creation/annihilation operators are defined by the anti-commutation relations: $$ \{a_k^{\dagger},a_q^{\dagger}\} = 0 = \{a_k,a_q \} $$ $$ \{a_k^{\dagger},a_q\} = \delta_{kq} \, .$$ I ...
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90 views

Has anyone published the procedure to generalize ladder operators for any potential in Schrodinger's equation?

I know that the ladder operator for the quantum harmonic oscillator \begin{align} H\psi_m = \left(\dfrac{p^2}{2m}+\dfrac{1}{2}m\omega^2x^2\right)\psi_m=E_m\psi_m \end{align} is \begin{align} A = ...
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How to do time evolution of operators in the Heisenberg Picture while staying in the Heisenberg Picture

Consider the time evolution of an operator in the Heisenberg picture: $$\tag{1}i\hbar \frac{d}{d t} \hat{A}_{H}(t) = \left([ \hat{A}_S(t), \hat H_S (t)] + i\hbar \frac{d}{d t} \hat{A}_S(t) ...
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What is the physical interpretation of the automorphism on bounded operators induced by an S matrix?

In a QFT, the S-matrix $S$ is a unitary operator, that fixes the vacuum and commutes with the unitary operators implementing the action of the Poincare group on an appropriate Hilbert space $H$. ...
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685 views

Does it mean anything if the commutator of an operator with the Hamiltonian is equal to the Hamiltonian?

Question says it all, really. I have $[\hat{H},\hat{O}]=-2i\hbar\hat{H}$. Does this mean that the operator $\hat{O}$ (an observable) is special in some way?
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Does Peskin & Schroeder Eq. (4.26), $U(t_1,t_2)U(t_2,t_3) = U(t_1,t_3)$ imply $[H_0,H_{int}] = 0$?

Peskin & Schroeder equation (4.17) define the operator, \begin{equation} U(t,t_{0})~=~e^{i(t-t_{0})H_{0}}e^{-i(t-t_{0})H} \tag{4.17} \end{equation} where $$H~=~H_0+H_{\text{int}}\tag{4.12}$$ is ...
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51 views

Are measurement results only orthogonal?

Are all measurement operators on a quantum mechanical system defined by a Hilbert space, such that all possible post-measurement states are orthogonal? For example measuring a qubit in some ...
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58 views

Precisely when is a matrix representation of Hermitian operator also Hermitian?

I asked a question on math exchange Are properties of linear operators reflected in matrix representations with different output and input basis?. In that question I asked: if I had a Hermitian ...
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1answer
256 views

Why does replacing bra and ket basis vectors by their row and column representations give the wrong matrix representation in a non-orthogonal basis?

I have a Hermitian operator (for a 2D Hilbert space) given by $$H=|\psi\rangle \langle \psi|+|\phi\rangle \langle \phi|$$ where $|\psi\rangle$ and $|\phi\rangle$ are normalized but not necessarily ...
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158 views

Expectation value of Commutator of Hermitian operators [closed]

Assume $\hat{A},\hat{B},\hat{C}$ are Hermitian. $$[\hat{A},\hat{B}]=i\hat{C}$$ and $$\hat{A}|a\rangle=a|a\rangle.$$ Then $$\langle a|i\hat{C}|a\rangle=\langle a|[\hat{A},\hat{B}]|a\rangle =0 .$$ ...
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92 views

Why is the electric field operator normalized by a volume?

I came across the following definition of the electric field operator: But I am not sure what this $V$, the "volume of a box", is about. It seems to enter the discussion in order to have standing ...
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When is matrix representation of a Hermitian operator invertible?

If I have a Hermitian operator $H:V \to V$ on a finite-dimensional vector space $V$, and I write down its matrix representation in some basis $B$ with matrix representation being $[H]_B$, then in what ...
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Ballentine's proof of (one half of) Stone's theorem

Reading Ballentine's "Quantum Mechanics; A Modern Development" I got stuck on his really short proof of what I think is Stone's theorem. On page 65 (paperback, reprint of 2008) he writes about about a ...
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$p^4$ in radial coordinates not Hermitian

Griffiths' quantum textbook claims in question 6.15 that "$p^2$ is Hermitian, but $p^4$ is not, for hydrogen states with $l=0$." First off, I am puzzled at his use of terminology. An operator is ...
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Translation Operators

Show that if the wave function $\langle x|\psi\rangle$ is modified by a position-dependent phase $\langle x|\psi\rangle \to e^{\frac{ip_ox}{\hbar}}\langle x|\psi\rangle$ then $\langle x\rangle ...
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62 views

Translation Operator

Let $|\psi\rangle \to |\psi'\rangle = \hat{T}(\delta x)|\psi\rangle$ for infinitesimal $\delta x.$ Show that $\langle x \rangle' = \langle x \rangle + \delta x$ and $\langle p_x \rangle' = \langle ...
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125 views

How do I prove that the del squared operator commutes with the angular momentum operator? [closed]

I need to prove in Cartesian coordinates that $[\nabla^{2},\hat{L_{z}}]= 0$ I know that the angular momentum operator is defined as: $\hat{L_{z}}=x\hat{p_{y}}-y\hat{p_{x}}$ And the del squared is ...
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67 views

Anticommutator difference [closed]

What is the value of this difference of anticommutators $$\{x^2,p^2\}-(\{x,p\}^2)/2$$ if the commutator $$[x,p]=i\hbar ~?$$ I have tried and obtained a value $$-3\hbar^2/2 - 2i\hbar px.$$ But ...
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Prove that $e^{\frac{i\lambda}{\hbar} S_x}S_ze^{-\frac{i\lambda}{\hbar}S_x}=S_z\cos(\lambda)+S_y\sin(\lambda)$ [closed]

Prove using Hadamard's lemma that $$e^{\frac{i\lambda}{\hbar} S_x}S_ze^{-\frac{i\lambda}{\hbar}S_x}=S_z\cos(\lambda)+S_y\sin(\lambda) $$ where $\lambda$ is a complex number. I get: ...
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96 views

Prove that $(e^{i\lambda A})^\dagger=e^{-i\lambda A^\dagger}$ [closed]

Prove $$(e^{i\lambda A})^\dagger=e^{-i\lambda A^\dagger}$$ where $A$ is an operator. Can anyone explain how to go about this question? Writing it as a power series gets confusing. So basically I ...
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91 views

How to calculate the expectation value of position/momentum using path integrals?

We have the formula: \begin{equation} \langle F \rangle = \frac{\int Dx \times F[\phi] exp\{i/\hbar S[\phi]\}}{\int Dx \times exp\{i/\hbar S[\phi]\}} \end{equation} Now, I am wondering how a change ...
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1answer
82 views

How to calculate the expectation value of position vector?

$$\psi (\vec{x})=Ae^{-(1/4a^2)|\vec{x}-\vec{x}_0|^2}e^{i\vec{p}_0\cdot \vec{x}/\hbar}$$ The wave function is like this, then how is the expectation value of position vector (not position) calculated? ...
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The equivalence between Heisenberg and Schroedinger pictures

In quantum mechanics, the two pictures of Schroedinger and Heisenberg are taken as equivalent, where in the former wavefunctions are time variants and operators are not, and in the latter it is the ...
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What is the Hermitian adjoint of operator $\hat{K}\psi = \psi^*$?

For many operators, their adjoint can be expressed as a function of other known operators, for example $$\hat{T}_a^\dagger = \hat{T}_{-a} \\ \hat{p}_x^\dagger = \hat{p}_x$$ where $\hat{T}_a \psi (x) = ...
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1answer
36 views

How to pull out the momentum operator?

In the equation (1.7.17), how does operator $p$ get out of the bracket without any operation though $<a | $, $| x'>$ are function of $x'$? How to prove this?
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313 views

Understanding the Quantum Vacuum State [duplicate]

In terms of the creation and annihilation operators $a_{j}$ and $a_{j}^{\dagger}$ (fermionic or bosonic, doesn't matter): Is the vacuum state $\mid\mathrm{vacuum}\rangle$ exactly the zero vector on ...
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211 views

Binomial expansion of non-commutative operators

I would like to determine the general expansion of $(A+B)^n$, where $[A,B]\neq0$, i.e. A and B are two generally no-commutative operators. How could I express this in terms of summations of the ...
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155 views

Calculating $\langle x | \hat{x} | p \rangle$ in $p$ basis

I am trying to calculate $\langle x\ |\ \hat{x}\ |\ p\rangle$. I can work in the $x$-basis like so: $$\langle x\ |\ \hat{x}\ |\ p\rangle=\int dx'\langle x\ |\ \hat{x}\ |\ x'\rangle\langle x'\ |\ ...
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56 views

Total angular momentum operator

How do the eigenfunctions of the total angular momentum operator analytically look like? I mean the operator is given by $J = L+S$ so the eigenfunctions have to be tensor-product states, right? Can ...
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46 views

Apply Hamiltonian to position eigenstates

Let $\hat{H}$ be the free Hamilton operator, is it then true that $$\langle {\bf r}| \hat{H} ~=~ - \frac{\hbar^2}{2m} \Delta \langle {\bf r}|~?$$ Where $\Delta\equiv \nabla^2$. I currently don't see ...
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1answer
110 views

Prove that this operator is unitary

$\hat{O}\equiv(1/\sqrt{2\pi})\int e^{-iNz}dz$ $\hat{O}^\dagger\equiv(1/\sqrt{2\pi})\int e^{iN'x}dx$ We have the operator $\hat{O}$ and its Hermitian adjoint $\hat{O}^\dagger$, in the one dimensional ...
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39 views

Can someone clarify what should and should not be an operator in my verification of the 1D solution to the SE for a free particle?

I just worked out the 1D free particle solution to the Schrödinger equation. My wave function was \begin{equation} \psi(x,t) = Ae^{i(px-Et)/\hbar} \end{equation} So I plugged this into both sides ...
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75 views

Simultaneous eigenket

J. J. Sakurai states in his "Modern Quantum Mechanics", this fact as a theorem ($\pi$ is the parity operator): Suppose $$[H,\pi]=0$$ and $| n>$ is a nondegenerate eigenket of $H$ with ...
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2answers
122 views

Unitary operator algebra and multiplying by identity

If $\hat{H}$ is Hermitian, with eigenvalues $a_k$, then $$\hat{H} = \sum_k a_k \left|\psi_k\right> \left<\psi_k\right|.$$ I read that it then follows that $$\begin{align*} e^{i\hat{H}} = ...