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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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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49 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.
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236 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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39 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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42 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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104 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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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 ...
2
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0answers
71 views

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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2answers
71 views

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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56 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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82 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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48 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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101 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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48 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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57 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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53 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 &= ...
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106 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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69 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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91 views

Individual terms in a Hamiltonian matrix

Reference to Problem 2, Chapter 2 in Modern Quantum Mechanics by JJ Sakurai, Consider the following Hamiltonian of a two state system $$ ...
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43 views

Spherical Polar Proof for Non-Commutativity of Indivdual Quantum Angular Momentum Operators

How can the following commutation relation be solved through spherical polar coordinates $[\hat{L}_{z}$,$\hat{L}_{x}]$ = $\imath\hbar\hat{L}_{y}$ I understand the derivation through partial ...
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1answer
87 views

Ladder operators - commutation relations and their properties

At the beginning of Fetter, Walecka "Many body quantum mechanics" there is a statement, that every property of creation and annihilation operators comes from their commutation relation (I'm ...
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1answer
49 views

Explicit form of the translation operator generators in the Poincare group?

Let $P_0$ be the generator for temporal translation and $P_1, P_2, P_3$ be for spatial translations. Let $p_μ$ be the momentum operator in the $x_μ=x^μ$ direction. I watched a lecture where the guy ...
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Where does the $i$ come from in the Schrödinger equation?

I am currently trying to follow Leonard Susskind's "Theoretical Minimum" lecture series on quantum mechanics. (I know a bit of linear algebra and calculus, so far it seems definitely enough to follow ...
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1answer
58 views

How does this quantisation relation come about?

I'm currently doing a course in string theory and in the lecture notes it is stated: $$ [x^-, p^+]~=~-i \tag{1}$$ I am fine with this. However, after trying (and failing) a question, I looked at ...
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56 views

how can act fractional operator on kets?

Knowing $\hat{A}\left|ψ\right\rangle$ and $\hat{B}\left|ψ\right\rangle$ , how to find answer of ($\hat{A}+\hat{B} )^ {1/2} \left|ψ\right\rangle $ Note : may $\hat{A}$ and $\hat{B}$ are not in the ...
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33 views

Advection Operator shift in scalar product

Can someone help me with advection operator shifts? I can't figure out the rule for the shift inside of a scalar product. The terms $(u,(v\cdot \nabla)\delta v)_\Omega$ and $(u,(\delta v\cdot ...
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1answer
96 views

Sakurai Exercise 2.17 (Harmonic Oscillator, Ladder operators) [closed]

I the field of the harmonic oscillator and ladder operators I am trying to solve exercise 2.17 from Sakurai and want to proof the following relation $$ \langle x^{2n} \rangle = (2n - 1)!! \langle ...
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1answer
89 views

What is the Quantum Mechanical Operator for Electric Potential?

I understand that charge and electric potential are conjugate observables in QM. See https://en.wikipedia.org/wiki/Conjugate_variables The quantum mechanical operator for charge, q, is simply equal ...
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why we dont have “direct” velocity operator just as $p$? ( as use $p$ space not $v$? ) in quantum?

why there is no direct velocity operator on quantum mechanic while there is for mumentum ( $p_{x}=d/dx$ ) Also why use mumentum space not velocity ?
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77 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, ...
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1answer
46 views

Orbital angular momentum eigenstates in the $|\mathbf{r}\rangle$ representation

Consider the orbital angular momentum operators $L^2$ and $L_z$. In the $|\mathbf{r}\rangle$ representation using spherical coordinates those operators actions are given by ...
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82 views

Can I *always* decompose a normalizable function into the discrete Hydrogen spectrum?

This question has been bothering me for a while now: can one reconstruct an arbitrary (normalizable) function $\phi(\mathbf r)$ in $\mathbb R^3$, with only the discrete set of Hydrogen wavefunctions ...
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172 views

Momentum operator representation

If $\hat{p}$ acts on position eigenstate, it is $$\tag{1}\hat{p}\left|x\right\rangle=+i\hbar\frac{\partial }{\partial x}\left|x\right\rangle .$$ But in general $$\tag{2}\hat{p} = -i\hbar ...
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173 views

Confusion with Weinberg's QFT book, volume 1, chapter 3: time translation and Heisenberg picture

Sorry if this is a naive question, but I am new to QFT. In the treatment of scattering in section 3.1 of The quantum theory of fields, vol.1, Weinberg first presented the general transformation rule ...
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48 views

Proof that a Hermitian Matrix is not defective?

I am taking an introductory course into Quantum Mechanics. To me to seems pretty simple to prove most properties of Hermitian operators. However, I am stuck at an edge case, proving that if an ...
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62 views

QFT: Ground State Momentum - Normalisation of States

In my notes I have, $$ \left\langle \mathbf{p} \left| \mathbf{q} \right.\right\rangle = \left\langle 0 \left| {a(\mathbf{p})}\ {a(\mathbf{q})}^{\dagger} \right| 0 \right\rangle $$ I am not sure how ...
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1answer
86 views

Conservation of momentum in infinite square well

This is inspired by Griffiths QM section 2.2, on the infinite square well, which is about how far I've gotten (so, sorry if this is addressed later in the book). For any given starting wavefunction, ...
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56 views

Commutation of two vector operator [closed]

Consider vectors $\overrightarrow { A } $ and $\overrightarrow { B } $ as operators or vector of operators. If this commutation holds$$[\overrightarrow { A },\overrightarrow { B }]=0$$ Then, is that ...
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95 views

Galilean relativity in QM

Intro I've been trying to show that the generator of boosts can be written in operator form as can be seen here, as: $$ B = \sum_i m_i x_i(t) - t \sum_i p_i $$ As a reminder the transformation ...
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2answers
59 views

Why do $\hat{X}$ and $\hat{P}$ have to correspond to position and momentum?

As far as I understand, in QM we treat observables as operators, and the eigenvalues of these operators are the possible values we can measure of the observables. It is usually simpler to work in the ...
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2answers
92 views

Manipulation of operators in quantum mechanics

I'm reading some notes on quantum mechanics that state the following. $$\langle x\rvert \left( \hat{x} + \frac{i\hat{p}}{m\omega}\right) \lvert E \rangle = 0 \Rightarrow \left( x+ ...
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67 views

Why aren't $\hat{x}$ and $\hat{p}$ considered functions of time in the expectation value?

In Griffiths Intro to QM (2nd edition), he gives the equation $$ \frac{d}{dt} \langle Q \rangle = \frac{i}{\hbar}\langle [\hat{H},\hat{Q}] \rangle + \left\langle ...
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21 views

Why should an generator acts on an operator with the Lie bracket?

When we deal with ordinary symmetries which form a Lie group, we have an corresponding Lie algebra with a structure of Lie bracket $[,]$. A infinitesimal transformation can act on a state or an ...
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1answer
127 views

Kronecker sum or direct sum?

When we write $$H=\sum_k H_k$$ in condensed matter physics, are we using Kronecker sum or direct sum? I think this is direct sum. However, Wikipedia says it is Kronecker sum. Can anyone give some ...
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75 views

Vector operators in quantum mechanics

Vector operators $\vec{V}$ in quantum mechanics are usually defined as those that commute in a particular way with the spatial Angular Momentum $\vec{L}$: $[L_i,V_j]=i\hbar\varepsilon_{ijk}V_k$. I am ...
2
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1answer
77 views

Proof that trace is independent of representation [closed]

$$\begin{align} \sum_{a'} \langle a'|X|a'\rangle &=\sum_{a',b',b''} \langle a'|b'\rangle \langle b'|X|b''\rangle\langle b''|a'\rangle \\ &=\sum_{b',b''} \langle b''|b'\rangle \langle ...
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1answer
48 views

Lorentz covariant completeness relation

Let be $$ P^\mu |p> = p^\mu |p> $$ i.e. $|p>$ is the eigen-vector of the 4-momentum operator. Where does the following Lorentz-covariant completeness relation come from? $$ \int d^4p ...
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38 views

Determine adjoint operator [closed]

I need to determine the adjoint operator to $a_i$: $a_i |n_0, n_1, ... \rangle_S = \sqrt{n_i} |n_0, n_1, ..., n_{i-1}, n_i - 1, n_{i+1}, .. \rangle_S$ where the $S$ should denote the symmetrizer. I ...
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28 views

Matrix representation of a fermionic creation and annihilation operator in graphene nanoribbons?

From the other question Matrix representation for fermionic annihilation operator, what if we have to find the matrix representation for the operators $a_{\sigma}^{\dagger}(k,n)$ and $b_{\sigma}(k,n)$ ...
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35 views

Heisenberg picture transition amplitudes

I want to calculate the transition amplitude for a particle to start at position $q_1$ at time $t_1$ to position $q_2$ at time $t_2$ in the Heisenberg picture. As we are in the Heisenberg picture, ...