The commutator tag has no wiki summary.
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0answers
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Why is the expectation value of the Hamiltonian/position commutator equal to zero?
I know that the commutator $[H,x]$ is $-i \hbar \,p/m$, but how do I show that the modulus of the expectation value of this is zero?
-1
votes
1answer
65 views
Proof $\left[ {\hat H,{{\hat p}_i}} \right] = - \frac{\hbar }{i}\frac{{\partial \hat H}}{{\partial {{\hat q}_i}}}$ [closed]
I have a problem with the Hamiltonian, I don't think anything to solve it!!
So could you give me some hints!
Knowing that:
$$\left[ {{{\hat p}_i},{{\hat q}_k}} \right] = \frac{\hbar }{i}{\delta ...
1
vote
1answer
46 views
Energy eigenvalues of a Q.H.Oscillator with $[\hat{H},\hat{a}] = -\hbar \omega \hat{a}$ and $[\hat{H},\hat{a}^\dagger] = \hbar \omega \hat{a}^\dagger$
I just finished deriving the commutators:
\begin{align}
[\hat{H}, \hat{a}] &= -\hbar \omega \hat{a}\\
[\hat{H}, \hat{a}^\dagger] &= \hbar \omega \hat{a}^\dagger\\
\end{align}
On the ...
4
votes
2answers
94 views
Proof for commutator relation $[\hat{H},\hat{a}] = - \hbar \omega \hat{a}$
I know how to derive below equations found on wikipedia and have done it myselt too:
\begin{align}
\hat{H} &= \hbar \omega \left(\hat{a}^\dagger\hat{a} + \frac{1}{2}\right)\\
\hat{H} &= ...
4
votes
2answers
74 views
Translator Operator
In Modern Quantum Mechanics by Sakurai, at page 46 while deriving commutator of translator operator with position operator, he uses $$\left| x+dx\right\rangle \simeq \left| x \right\rangle.$$ But for ...
2
votes
1answer
62 views
Moyal Product in Non Commutative Quantum Mechanics
Can someone please explain me what is a Moyal product?
Also, how does putting $$X_a(\psi) ~=~ x_a\star\psi$$ realise $$[X_a,X_b]=i\theta_{ab}{\bf 1}?$$
Ref: Quantum mechanics on non-commutative ...
0
votes
1answer
84 views
Matrix representation for fermionic annihilation operator
My guess it should look something like this:
$ c_\sigma = ...
1
vote
0answers
37 views
QFT basics for Klein-Gordon fields
I am teaching myself QFT from Peskin for next years maths course and I have two questions:
What is a c-number? Is it a complex number, and if so why does it mean, ...
3
votes
2answers
116 views
Quantum commutator
I'm given this commutator:
$$\left[PXP,P\right]$$
Being $P\psi=-i\hbar\partial_x\psi$, and $X\psi=x\psi$
I've solved it in two ways, the first one is just aplying the commutator to some function ...
2
votes
1answer
58 views
Quantum mechanical analogue of conjugate momentum
In classical mechanics, we define the concept of canonical momentum conjugate to a given generalised position coordinate. This quantity is the partial derivative of the Lagrangian of the system, with ...
1
vote
1answer
96 views
Klein-Gordon Canonical Commutation Relation (CCR)
In the complex Klein-Gordon field we regard as dynamical variables the field $\phi$, the complex conjugate of the field $\phi^*$, and the momenta $\pi$, $\pi^*$. I can't see how should arise the ...
3
votes
1answer
83 views
State space of QFT, CCR and quantization, and the spectrum of a field operator?
In the canonical quantization of fields, CCR is postulated as (for scalar boson field ):
$$[\phi(x),\pi(y)]=i\delta(x-y)\qquad\qquad(1)$$
in analogy with the ordinary QM commutation relation:
...
0
votes
1answer
131 views
Evaluate Commutator with Partial Derivatives
I need to evaluate the following commutator...
$[x(\frac{\partial}{\partial y})-y(\frac{\partial}{\partial x}),y(\frac{\partial}{\partial z})-z(\frac{\partial}{\partial y})]$
i tried applying an ...
5
votes
2answers
208 views
Causality and Quantum Field Theory
I have a problem with proof of causality in Peskin & Schroeder, An Introduction to QFT, page 28. To avoid confusion I use three vectors notation, rewriting the Eq. (2.53) for $y=0$ as follows:
...
3
votes
1answer
123 views
What conservation law corresponds to this local $U(1)$ symmetry of the CCR?
It is known that canonical commutation relations do not fix the form of momentum operator. That means that if canonical commutation relations (CCR) are given by
...
8
votes
2answers
243 views
In QFT, why does a vanishing commutator ensure causality?
In relativistic quantum field theories (QFT),
$$[\phi(x),\phi^\dagger(y)] = 0 \;\;\mathrm{if}\;\; (x-y)^2<0$$
On the other hand, even for space-like separation
$$\phi(x)\phi^\dagger(y)\ne0.$$
...
7
votes
1answer
193 views
Canonical quantization in supersymmetric quantum mechanics
Suppose you have a theory of maps
$\phi: {\cal T} \to M$
with $M$ some Riemannian manifold,
Lagrangian
$$L~=~ \frac12 g_{ij}\dot\phi^i\dot\phi^j + \frac{i}{2}g_{ij}(\overline{\psi}^i ...
7
votes
1answer
144 views
Theories with non-vanishing commutators outside the lightcone
I'm reading Weinberg's new book on Quantum Mechanics, and in Chapter 8.7 "Time-Dependent Perturbation Theory" he derives the usual Dyson series for the $S$ matrix when the interaction Hamiltonian ...
3
votes
2answers
289 views
Commutator with expontential [A, exp(B)]
How can I tell if $A$ and $\exp(B)$ commute?
For $[A, B]$ it's simply $AB-BA$ and for $[\exp(A), \exp(B)]$ I think it'd be $\exp(A)\exp(B) - \exp(B)\exp(A) = \exp(A+B) - \exp(B+A) = 0$. Update: it's ...
1
vote
1answer
117 views
Canonical transformation and Hamilton's equations
I was trying to prove, that for a transformation to be Canonical, one must have a relationship:
$$
\left\{ Q_a,P_i \right\} = \delta_{ai}
$$
Where $Q_a = Q_a(p_i,q_i)$ and $P_a = P_a(p_i,q_i)$.
Now ...
3
votes
1answer
99 views
QED Commutation Relations Implications
In Brian Hatfield's book on QFT and Strings there is the following quote:
In particular $$ [A_i (x,t), E_j(y,t)] = -i \delta_{ij}\delta(x-y) $$
implies that $$ [A_i(x,t),\nabla \cdot E(y,t)] = ...
1
vote
1answer
98 views
Commutation relation of $J^2$ and $R(\alpha,\beta,\gamma)$
If $R(\alpha,\beta,\gamma)$ is the Rotation operator and $\alpha,\beta,\gamma$ are Euler angles and $J$ is the total angular momentum then how to get to this:
$$[J^2,R]~=~0?$$
This is stated in ...
4
votes
3answers
550 views
Canonical Commutation Relations
Is it logically sound to accept the canonical commutation relation (CCR)
$$[x,p]~=~i\hbar$$
as a postulate of quantum mechanics? Or is it more correct to derive it given some form for $p$ in the ...
1
vote
2answers
189 views
Example of two linearly independent, nowhere vanishing vector fields in $\mathbb{R}^{2}$
I knew that two linearly independent and nowhere-vanishing vector fields provide a basis for the tangent space at each point in $\mathbb{R}^{2}$.
Is it necessary that these two vector fields commute? ...
3
votes
2answers
154 views
Does the canonical commutation relation fix the form of the momentum operator?
For one dimensional quantum mechanics $$[\hat{x},\hat{p}]=i\hbar $$
Does this fix univocally the form of the $\hat{p}$ operator? My bet is no because $\hat{p}$ actually depends if we are on ...
3
votes
1answer
305 views
Momentum as Generator of Translations
I understand from some studies in mathematics, that the generator of translations is given by the operator $\frac{d}{dx}$.
Similarly, I know from quantum mechanics that the momentum operator is ...
4
votes
1answer
128 views
Why Must Conserved Currents of Lorentz Symmetry Satisfy the Lorentz Algebra
I've seen it written many times that the commutation relation
$[M^{I-},M^{J-}]=0$
is required for Lorentz invariance in the light cone gauge quantisation of the bosonic string. This follows ...
5
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4answers
287 views
Does uncertainty imply noncommutativity?
We already know that non-commutativity of observables leads to uncertainty in quantum mechanics cf. e.g. this and this Phys.SE post. What about the opposite: Does uncertainty imply noncommutativity?
...
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331 views
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 ...
2
votes
2answers
276 views
Non-commuting operators can't share any eigenvector
In an introductory Quantum Mechanics textbook, I found the following statement:
For two Hamiltonians $H$ and $H'$, non commuting with each other, but commuting with the same group of translations ...
4
votes
3answers
1k views
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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2answers
161 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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2answers
225 views
Symmetries, Generators, Commutators and Observables
I'm learning about generators and conservation laws and have derived the equation (1)
$$[Q,A]=-i\hbar f(A)$$
which is satisfied by the observable generator $Q$ for a transformation group with ...
1
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1answer
101 views
transformations with commutators and anticommutators that generate displacements
is well known that composition of point reflections generate pure displacements. This implies that the commutator of two point reflections will be a pure displacement. Are there similar elemental ...
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1answer
439 views
Operators and Commutator Definitions
I have several problems with General Definitions of an Operator and Commutator :
the product of operators is generally not commutative:
$$\hat A \hat B \not= \hat B\hat A .$$
what is this means ...
1
vote
1answer
180 views
Commutators with a density matrix
The equation describing the evolution of our system is as follows:
$ \dot{\rho} = u_1(t)(a^\dagger a \rho - 2a\rho a^\dagger +\rho a^\dagger a) + u_2(t)(a a^\dagger \rho - 2a^\dagger\rho a +\rho a ...
4
votes
3answers
348 views
Generalizing Heisenberg Uncertainty Priniciple
Writing the relationship between canonical momenta $\pi _i$ and canonical coordinates $x_i$
$$\pi _i =\text{ }\frac{\partial \mathcal{L}}{\partial \left(\frac{\partial x_i}{\partial t}\right)}$$
...
1
vote
1answer
161 views
Multiplication of 3-vector operators
I've started reading "Quantum Mechanics: A Modern Development" by Leslie E. Ballentine and have some trouble understanding how to handle 3-vector operators (i.e. an operator $\mathbf{A}$ with ...
0
votes
1answer
562 views
Derivation of angular momentum commutator relations
I'm trying to understand the derivation of the angular momentum commutator relations. How is
$$[zp_y, zp_x] ~=~ 0?$$
How is
$$[yp_z, zp_x] ~=~ y[p_z, z]p_x?$$
4
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1answer
858 views
Compatible Observables
My QM book says that when two observables are compatible, then the order in which we carry out measurements is irrelevant.
When you carry out a measurement corresponding to an operator $A$, the ...
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4answers
2k views
What is the connection between Poisson brackets and commutators?
The Poisson bracket is defined as:
$$\{f,g\} = \sum_{i=1}^{N} \left[
\frac{\partial f}{\partial q_{i}} \frac{\partial g}{\partial p_{i}} -
\frac{\partial f}{\partial p_{i}} \frac{\partial ...
3
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3answers
572 views
Index Manipulation and Angular Momentum Commutator Relations
I have been trying for hours and cannot figure it out. I am not asking anyone to do it for me, but to understand how to proceed.
We have the relations
$$[L_i,p_j] ~=~ i\hbar\; \epsilon_{ijk}p_k,$$
...
2
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1answer
409 views
The implication of anti-commutation relations in quantum mechanics
All the textbooks I saw are very clear about the implications of commutating operators in quantum mechanics. However, much less is said about anti-commutation relations. Does it have a general ...
2
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1answer
371 views
How far can you get (in quantum mechanics) with just commutation relations?
Clearly it is possible to derive a set of commutation relations from some Hamiltonian, and certainly they give useful and interesting invariants when investigating the behavior of quantum systems. ...
2
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1answer
180 views
Expectation of a commutation relation
Is there any significance to: $\langle[H,\hat{O}]\rangle =0$ (which can easily be shown) where $H$ is the Hamiltonian, $\hat{O}$ is an arbitrary operator? Thanks.
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2answers
368 views
Operator relation involving the logarithm of an operator?
Dirac gives the relation: $\exp(iaq)f(q,p) = f(q, p - a\hbar)\exp(iaq)$ where $\hbar$ is Planck's constant. Can anybody give me the corresponding relation when the $\exp$ function is a $\ln$?
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4answers
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Trace of a commutator is zero - but what about the commutator of $x$ and $p$?
Operators can be cyclically interchanged inside a trace:
$${\rm Tr} (AB)~=~{\rm Tr} (BA).$$
This means the trace of a commutator of any two operators is zero:
$${\rm Tr} ([A,B])~=~0.$$
But what about ...
3
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2answers
909 views
Momentum-Representations in Quantum Mechanics
Why do we get information about position and momentum when we go to different representations. Why is momentum, which was related to time derivative of position in classical physics, now in QM just a ...
