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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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 ...
9
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142 views

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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334 views

Ladder Operator killing state

So in my last question, @joshphysics showed me how to prove $K_\pm$ were ladder operators. Now I need to show that there is a lowest state, i.e $$\langle m_0|K_+=K_-|m_0\rangle=0$$ I am not ...
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64 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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135 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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63 views

Proof that trace is independent of representation [on hold]

$$\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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130 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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36 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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53 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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65 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, ...
5
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268 views

Wick's theorem for calculating OPE

I am trying to understand a calculation using Wick's theorem. Let $T(z)$ be the analytic part of a stress-energy tensor, and $\phi(z)$ a free boson field. Now, ...
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83 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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156 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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44 views

Commutation of two vector operator

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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119 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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2answers
79 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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49 views

Transforming operators and minus signs

If I have an operator $A_H$ in the Heisenberg picture, then it obeys the equation $-i \frac{\partial}{\partial t}A_H=[H,A_H]$. However, if I plug in the expression $H=i\frac{\partial}{\partial t}$ I ...
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50 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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59 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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2answers
422 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 non-commutative operators. How could I express this in terms of summations of ...
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17 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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331 views

How does an operator transform under time reversal?

We know that a time-reversal operator $T$ can be represented as $$T=UK$$ where $U$ is some unitary operator and $K$ is the complex conjugation operator. Then under time-reversal operation, a quantum ...
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93 views

Radial quantum number for infinite circular well

For completeness, I will sketch the solution of a particle in an infinite circular well first and then get to my question. I apologize in advance since the introduction is standard undergraduate ...
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393 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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Weyl Ordering Rule

While studying Path Integrals in Quantum Mechanics I have found that [Srednicki: Eqn. no. 6.6] the quantum Hamiltonian $\hat{H}(\hat{P},\hat{Q})$ can be given in terms of the classical Hamiltonian ...
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423 views
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158 views

Proof for Negele and Orland equation (2.34)

The equation (2.34) of Negele and Orland has $$\mathcal H_\text{A}(\hat{\mathbf p},\hat{\mathbf x}) = \frac{1}{2m}\left(\hat {\mathbf p} - \frac e c \mathbf A(\hat{\mathbf x})\right)^2.\tag{2.34a}$$ ...
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234 views

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 ...
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72 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 ...
3
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2answers
299 views

$\hat{L}_{x}$ and $\hat{L}_{y}$ do not commute… or do they?

So $\hat{L}_{x}$ and $\hat{L}_{y}$ do not commute: $$ [ \hat{L}_{x}, \hat{L}_{y}] = i\hbar \hat{L}_{z}$$ But, what if we perform this operation on a state such that: $$\hat{L}_{z} \phi_{l, m_{l}} ...
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136 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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152 views

Uncertainty relation for non-simultaneous observation

Heisenberg's uncertainty relation in the Robertson-Schroedinger formulation is written as, $$\sigma_A^2 \sigma_B^2 \geq |\frac{1}{2} \langle\{\hat A, \hat B\}\rangle -\langle \hat A\rangle\langle ...
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45 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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Why is normal ordering a valid operation?

Why is normal ordering even a valid operation in the first place? I mean it can give us some nice results, but why can we do the ordering for the operators like that? Is its definition motivated by ...
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288 views

Shape of the state space under different tensor products

I am currently studying generalized probabilistic theories. Let me roughly recall how such a theory looks like (you can skip this and go to "My question" if you are familiar with this). Recall: In a ...
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1answer
224 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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1answer
45 views

Why is the angular momentum added for two independent electron system? (no problem)

There is no problem now. But somebody may be confused by the same analysis when studying QM or Group theory. (actually my motivation for asking this question comes from the SU(5) Grand Unification ...
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35 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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20 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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1answer
30 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, ...
3
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27 views

Baryonic operators in ${\cal N}=1$ $U(N)$ SQCD in four dimensions

Seiberg's duality is usually considered as a duality for $SU(N_c)$ theories with $N_f$ flavors. In his case, the vacuum for $N_f \geq N_c$ is parameterized by mesons $M$ and baryons ${\bar B}$ and ...
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2answers
66 views

Negative powers of operators

This may sound like a strange question, but just to be sure: Suppose I have a general Hermitian operator in Hilbert space whose action on an eigenvector is given by $R|r\rangle = r|r\rangle$. Then, I ...
9
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412 views

Eigenstate of field operator in QFT

Why don't people discuss the eigenstate of the field operator? For example, the real scalar field the field operator is Hermitian, so its eigenstate is an observable quantity.
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284 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}}) ...
4
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1answer
202 views

Virasoro operators commutation relations

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 ...
6
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220 views

Equation 2.27 from Pachos's introduction to topological quantum computing

http://quince.leeds.ac.uk/~phyjkp/Files/IntroTQC.pdf above is the PDF that is hosted on his website. The equation is on page 22 (pg 30 in the pdf). In chapter 2. It is the second equation of the ...
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54 views

Limit of the position and momentum commutator [closed]

The commutator of position and momentum operator, $\hat{p}$ and $\hat{x}$, respectively is derived as $[\hat{x},\hat{p}]=i\hbar$. Let $\lim_{x\rightarrow x_{o}} [\hat{x},\hat{p}]=\lim_{x\rightarrow ...
2
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3answers
1k views

Commutators involving functions

I am looking for the commutator: $$[e^{aq},p]$$ My approach is to Taylor expand the function: $$[\sum_n \frac{1}{n!}(aq)^n,p]$$ I know that $[q^n,p]=ni\hbar q^{n-1}$ So how do I account for $n$ ...
3
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1answer
111 views

Is the energy always discrete?

In the von Neumann axioms for quantum mechanics, the first postulate states that a quantum state is a vector in a separable Hilbert space. It means it is assumed the Hilbert space has a basis with at ...
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41 views

Spin-1/2 rotation operator: rotation about an angle of $\pi$

The spin-1/2 rotation operator: $$ R_{n}(\alpha) = \begin{pmatrix} cos(\frac{\alpha}{2})-in_{z}sin(\frac{\alpha}{2}) & (-in_{x}-n_{y})sin(\frac{\alpha}{2}) \\ ...