A mathematical construct used to study the effect of applying two operators in succession.

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3
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1answer
89 views

Proving $[a_k^\dagger, a_q^\dagger]=0$

I am trying to prove the commutation relations between the creation and annihilation operators in field theory. I was already able to show that $[a_k, a_q^\dagger]=i\delta(k-q)$. I want to show that ...
0
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0answers
28 views

How to derive a new form of Hamiltonian operator in quantum mechanics using canonical commutation relation?

How does one derive $$\hat{H} = \frac{1}{2}\hat{p}^2m(\hat{q}) - \frac{i}{2}\hat{p}\frac{m'(\hat{q})}{m^2(\hat{q})} + V(\hat{q})$$ from hamiltonian $$\hat{H} = \hat{p}\frac{1}{2m(\hat{q})}\hat{p} + ...
1
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0answers
50 views

Commutation relation of a operator with Hamiltonian [duplicate]

Given that the eigenvalues of a Hamiltonian operator $H$ are bounded below, will a Hermitian operator $T$ exist such that $[T, H] = i\hbar{\bf 1}$ identity operator?
0
votes
1answer
65 views

Commutation relation of position and momentum using Dirac notation

This is likely a very trivial/silly question, but in following a derivation of the position and momentum commutation relation using the dirac notation, I am having trouble justifying a certain step. ...
6
votes
1answer
128 views

Understanding Poisson brackets

In quantum mechanics, when two observables commute, it implies that the two can be measured simultaneously without perturbing each other's measurement results. Or in other words, the uncertainty in ...
-1
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0answers
28 views

Quantum Liouville condition?

I understand there is the quantum-Liouville equation, namely the Von-Neumann equation, but does Liouville's theorem apply to the Wigner distribution too? What about the Moyal bracket, is that a ...
0
votes
1answer
68 views

Quantum mechanics, operator commutes with Hamiltonian

My textbook said, if an operator $\hat{O}$ commutes with the Hamiltonian, then we can use the eigen vectors of the Hamiltonian as a basis of the Hilbert space, then express the operator $\hat{O}$ in ...
0
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2answers
30 views

Quantization conditions/ Real Scalar field

It is often written in books, the quantization conditions for classical field theory leading to Lagrangian of a real scalar field and thus to Klein Gordon equation. And these are introduced by ...
4
votes
1answer
84 views

Apparent spacetime dependence of creation and annihilation operators

I'm currently going through An Introduction to Quantum Field Theory by Hartmut Wittig I've stumbled upon. Having trouble with equation (2.29), I'm asking the question: Do creation and annihilation ...
2
votes
1answer
245 views

Generalization of canonical commutation relation

The canonical commutation relation $$[x,p] = i\hbar$$ can be generalized to $$[p_i,F(\vec{x})] = -i\hbar\frac{\partial F(\vec{x})}{\partial x_i}, \ [x_i, F(\vec{p})] = i\hbar\frac{\partial ...
0
votes
1answer
81 views

Are the Pauli matrices closed under commutation?

I tried to make a group multiplication table for the Pauli matrices, but I keep getting multiples in front of the elements. What am I doing wrong? I thought the Pauli matrices formed a group that was ...
5
votes
2answers
110 views

Motivating Complexification of Lie Algebras?

What is the motivation for complexifying a Lie algebra? In quantum mechanical angular momentum the commutation relations $$[J_x,J_y]=iJ_z, \quad [J_y,J_z] = iJ_x,\quad [J_z,J_x] = iJ_y$$ become, on ...
0
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0answers
79 views

How to prove $\hat{p}|x\rangle=i\hbar\frac{\partial}{\partial x}|x\rangle$,using $[\hat{x},\hat{p}]=i\hbar$? [duplicate]

How to prove $$\hat{p}|x\rangle=i\hbar\frac{\partial}{\partial x}|x\rangle,$$ using $$[\hat{x},\hat{p}]=i\hbar~?$$ The question seems to be uncomplete because for any $f(x)$ ...
8
votes
2answers
594 views

Does the commutator of anything with itself not vanish?

In a quantum mechanics exam one question was to write the commutator of a couple of operators. Everybody got points taken away since they did not write $[Q_i, Q_i] = 0$ for all the operators $Q_i$ in ...
2
votes
3answers
210 views

How to derive $[x_i, F(\vec p)] = i \hbar \frac {\partial F(\vec p)}{\partial p_i}$

Wikipedia indicates that the following relation is "easily shown": $[x_i, F(\vec p)] = i \hbar \frac {\partial F(\vec p)}{\partial p_i}$, however I'm having some trouble showing it. I think I'm just ...
3
votes
2answers
208 views

Measuring non-commuting observable at once

Given an Hilbert space $H$ (finite dimensional for sake of clarity), and two non-commuting operators $$A = \sum_a a |a\rangle\langle a|$$ and $$B=\sum_a b |b\rangle\langle b|,$$ is it possible to find ...
0
votes
1answer
60 views

Commutativity of Position Operators

Does the position operator $q_{i}$ of one harmonic oscillator commute with the position operator $q_{j}$ of another different harmonic oscillator? In other words, is $q_{i} q_{j} = q_{j} q_{i}$ true? ...
1
vote
2answers
66 views

What is the commutator of the exponential derivative operator and the exponential position operator?

What is the commutator of the exponential derivative operator and the exponential position operator? \begin{align} \left[\exp(\partial_x),\exp(x)\right] = \exp(\partial_x)\exp(x) - ...
1
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0answers
57 views

Why is commutation relations the first step in quantization?

Why is commutation relations the first step in quantization?
2
votes
1answer
121 views

In QFT, why do fermions have to anticommute in order to insure causality?

I have seen this question and I believe I understand the answer to it. However, AFAIK, only for bosons the causality condition is a vanishing commutator. For fermions we expect the anticommutator ...
3
votes
5answers
429 views

Commutator algebra in exponents

Considering $X$ and $Y$ such that $[X,Y]=\lambda$, which is complex, and $\mu$ is another complex number, prove: $$e^{\mu(X+Y)}=e^{\mu X} e^{\mu Y} e^{-\mu^2\lambda/2}$$ My attempt (so far) is: ...
1
vote
1answer
50 views

kitaev-honeycomb : can't get wilson loop squared to yield +1

I'm new here, loving this website and I'm having some difficulty with the wilson-loop operator in kitaev's honeycomb model. problem statement The Kitaev model (Kitaev, 2006 is the original paper) ...
2
votes
3answers
116 views

Measuring position and momentum at the same time?

In a non-relativistic quantum mechanical system in an infinite potential well. I try to measure the energy and the position of the system simultaneously. Since, the respective operators do commute ...
22
votes
5answers
2k views

What is the physical meaning of commutators in quantum mechanics?

This is a question I've been asked several times by students and I tend to have a hard time phrasing it in terms they can understand. This is a natural question to ask and it is not usually well ...
1
vote
1answer
91 views

What is the physical importance of the commutation relations of angular momentum?

What is the physical meaning of these commutation relations: $$[L_{z},L_{\pm}]=\pm\hbar L_{\pm}\tag{1}$$ and $$[L_{+},L_{-}]=2\hbar L_{z} ~?\tag{2}$$
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0answers
106 views

Link between Quantum and Classical Mechanics [duplicate]

In classical mechanics we have momentum as generator of translation by following definition: $$f(x+\delta x)=f(x)+[f(x),p]\delta x+....$$ I was wondering whether using this relation and commutation ...
2
votes
2answers
77 views

Can we correctly define momentum operator only by means of position operator and their commutation relation?

In "J.M. Ziman. Electrons and Phonons: The Theory of Transport Phenomena in Solids" the author formally introduces the position (displacement) operator and then defines the momentum operator with the ...
7
votes
2answers
135 views

Dilation operator in CFT viewed as 'hamiltonian'?

From the commutation relations for the conformal Lie algebra, we may infer that the dilation operator plays the same role as the Hamiltonian in CFTs. The appropriate commutation relations are ...
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vote
2answers
111 views

Why angular momentum about three independent axes?

The generic commutation relations for the angular momentum operator are $[J_x, J_y] = i \hbar J_z$, where the $J_i$, $i = x,y,z$ are the components of the angular momentum vector operator, $\mathbf ...
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0answers
51 views

Noncanonical commutation relation and noncanonical wave mechanics [closed]

Consider noncanonical operators $\hat{x}_1,\hat{x}_2,\hat{p}_1,\hat{p}_2$ satisying the following condition in the $q_1,q_2$ - basis ($\psi=\langle q_1,q_2|\Psi\rangle$)(similar to wave mechanics): ...
2
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0answers
83 views

Spinor Commutator in Peskin and Schroeder

In (3.87, page. 53) Peskin and Schroeder write $$\psi(\vec{x}) = \int\frac{d^{3}p}{(2\pi)^{3}} \frac{1}{\sqrt{2E_{\vec{p}}}} e^{i\vec{p} \cdot \vec{x}} \sum_{s=1,2} (a_{\vec{p}}^{s}u^{s}(\vec{p}) + ...
0
votes
1answer
143 views

Derivation of (2.45) in Peskin and Schroeder

I'm having trouble understanding the step $$\left[\pi (\vec{x},t),\int d^{3}y ~(\frac{1}{2} \pi (\vec{y},t)^{2}+\frac{1}{2}\phi (\vec{y},t)(-\nabla^{2} +m^{2})\phi (\vec{y},t)) \right]$$ $$ =\int ...
6
votes
2answers
229 views

How to replace $T$-product with retarded commutator in LSZ formula?

I am reading Itzykson and Zuber's Quantum Field Theory book, and am unable to understand a step that is made on page 246: Here, they consider the elastic scattering of particle $A$ off particle $B$: ...
0
votes
2answers
124 views

Determine $p_x$ from $[x,p_x]=i\hbar $ [closed]

With $[x,p_x]=i\hbar $, how to determine the form of the operator $p_x$?
1
vote
1answer
101 views

Trace operation on dynamic equation: physical meaning?

Suppose we have Heisenberg equation of motion for some observable $A$, $$ i\hbar\frac{dA}{dt}= -[H,A] $$ since the trace of any finite dimensional commutator structure vanish(not something like ...
0
votes
2answers
72 views

How can $J_1^2, J_2^2, J_{1z}, J_{2z}$ commute mutually?

I'm reading through J. J. Sakurai's Modern Quantum Mechanics book and currently looking at the "Angular-momentum addition" part. Here, it says you have two options and that one option is to ...
4
votes
2answers
181 views

Commutation relations of the generators of the conformal group

My question is from P.98 of the book by Di Francesco on Conformal Field theory. He gives the six non-vanishing commutation relations between the elements $P_{\mu}, D, L_{\mu \nu}$ and $K_{\mu}$ ...
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votes
1answer
55 views

Calculations with operators - Proof: Equation of Operators [duplicate]

I have a problem in Quantum mechanics 1 with Operators. I have to prove the following equation. I tried it for about 4 hours without any result: Condition: $[[\hat A,\hat B],\hat A]=[[\hat A,\hat ...
0
votes
0answers
66 views

QM: Commutation relations between irreducible vectors and angular momentum $[J^2,T_q^k]$

reading about the irreducible tensors and its commutation relations with the angular momentum one can find relations for $J_{z}$, $J_{+}$, $J_{-}$, but I was wondering, what about $J^2$ ? from ...
2
votes
1answer
142 views

Commutation Relations for Creation & Annihilation Opertors of Two Different Scalar Fields

Let us consider two different scalar fields $\phi$ and $\chi$. The commutation relations for the creation and annihilation operators of the scalar field $\phi$ are given by $$ [a(\textbf{k}), ...
4
votes
1answer
148 views

A question about causality and Quantum Field Theory from improper Lorentz transformation

Related post Causality and Quantum Field Theory In Peskin and Schroeder's QFT p28, the authors tried to show causality is preserved in scalar field theory. Consider commutator $$ [ \phi(x), \phi(y) ...
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votes
1answer
100 views

How does $p_x$ commute with $p_y$, i.e. $[p_x,p_y]=0$? [closed]

I know it's a simple and basic question but would someone show me how to evaluate $[\hat{p}_x,\hat{p}_y]$?
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votes
2answers
445 views

Why uncertainty principle is not like this?

In Griffiths' QM, he uses two inequalities (here numbered as $(1)$ and $(2)$) to prove the following general uncertainty principle: $$\sigma_A^2 \sigma_B^2\geq\left(\frac{1}{2i}\langle [\hat A ,\hat ...
3
votes
1answer
294 views

Finding the creation/annihilation operators

Using Minkowski signature $(+,-,-,-)$, for the Lagrangian density $${\cal L}=\partial_{\mu}\phi\partial^{\mu}\phi^{\dagger}-m^2\phi \phi^{\dagger}$$ of the complex scalar field, we have the field ...
3
votes
1answer
108 views

Is obtaining the coordinate representation of momentum operator from commutator more fundamental than generator of translation

Related post: What is the most general expression for the coordinate representation of momentum operator? There are two methods of obtaining the coordinate representation of momentum in quantum ...
0
votes
2answers
132 views

Projection operators and their subspaces (of Hilbert space)

I've been watching Susskind's lectures on Quantum Entanglement, and something he said regarding (non-)commuting projection operators confused me. Consider two subspaces {$|a>$} and {$|b>$} of ...
0
votes
1answer
108 views

How to derive the commutation relationship between $\hat{L}^2$ and $\hat{\textbf{p}}$ [closed]

How to prove that $$[\hat{L}^2,\hat{\textbf{p}}] = i\hbar(\hat{\textbf{p}}\times\hat{\textbf{L}} - \hat{\textbf{L}} \times \hat{\textbf{p}})$$ I tried to expand $\hat{L}^2$: ...
1
vote
1answer
53 views

Position and potential Energy

Why are the position and potential energy of a particle able to be measured precisely in Quantum Mechanics? I mean why do they commute with each other?
0
votes
1answer
71 views

Is $\langle k \vert k_1k_2\rangle=0$

Using that $$ \vert k_1k_2\rangle = a^\dagger({\bf k_1})a^\dagger({\bf k_2})\vert 0 \rangle$$ and the commutation relations $$[a({\bf k}),a^\dagger({\bf k'})]=(2\pi)^32\omega\delta^3(\bf {k}- \bf ...
2
votes
1answer
174 views

The Physical Meaning behind a Commutator [duplicate]

I've just been introduced to the idea of commutators and I'm aware that it's not a trivial thing if two operators $A$ and $B$ commute, i.e. if two Hermitian operators commute then the eigenvalues of ...