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

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Operator on Function of Momentum (QM)

I have exactly 0 clue on how to start this problem, but I would be forever grateful for a hint in the right direction. Given the operators $\hat x=x$ and $\hat p=-i\hbar \frac{d}{dx}$, prove the ...
3
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2answers
450 views

Unitary spacetime translation operator

Srednicki writes: We can make this a little fancier by defining the unitary spacetime translation operator $$ T(a) \equiv \exp(-iP^\mu a_\mu/ \hbar) $$ Then we have $$ T(a)^{-1} \phi(x) T(a) = ...
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1answer
189 views

Time-ordered Derivative and Equal-time Commutator

In Green, Schwarz & Witten Superstring theory, Vol. I, page 141, I don't understand how pulling the derivative inside the Time-ordered product can give an Equal-time Commutator: $$\tag{3.2.44} ...
4
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1answer
715 views

What physical significance has the Heisenberg Group?

I read that the canonical commutation relation between momentum and position can be seen as the Lie Algebra of the Heisenberg group. While I get why the commutation relations of momentum and momentum, ...
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257 views

The correspondence between Poisson bracket and Commutators in Quantum Mechanics

I don't understand canonical quantization. In passing from classical to quantum, one replaces the Poisson brackets with the commutators. I don't really understand this. How can we generally show that ...
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1answer
691 views

Commutator of Lorentz boost generators : visual interpretation

I have always struggled to visualize the correctness of the commutation relation for the generators of the boost in the Lorentz group. We have $$[K_i,K_j] = i \epsilon_{ijk} L_k$$ I fail to picture ...
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Commutator with a square root

How to find the commutator $[a, \sqrt{a^\dagger a}]$? Here $a$ is a usual bosonic annihilation operator, and $[a, a^\dagger] = 1$. The first thing I tried is $$ [x,A] = [x, \sqrt{A}]\sqrt{A} + ...
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Imposing anti-commutation relations on fermionic quasi-particles

In many theories of CMT, we assume the nature of quasi-particles (without giving proper justifications). For example, we assume nature of quasi-particles to be fermionic in case of a interacting ...
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Are higher order mixed partial derivatives of wave function with different ordination equal?

For example, given two operators: $$A = \frac{\partial}{\partial x}+\frac{\partial}{\partial y},$$ $$B =\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2} + 1.$$ Deriving commutator ...
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Deriving the Angular Momentum Commutator Relations by using $\epsilon_{ijk}$ Identities

I've been trying to derive the relation $$[\hat L_i,\hat L_j] = i\hbar\epsilon_{ijk} \hat L_k $$ without doing each permutation of ${x,y,z}$ individually, but I'm not really getting anywhere. ...
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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 ...
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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 ...
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575 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} &= ...
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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 ...
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1answer
357 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 ...
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525 views

Matrix representation for fermionic annihilation operator

My guess it should look something like this: $ c_\sigma = ...
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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, ...
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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 ...
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1answer
585 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 ...
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1answer
400 views

Find $\hat{x}$ operator given $\hat{p}$ operator

This is problem $1.2$ of Molecular Quantum Mechanics by Atkins, 4th edition. I'm given the momentum operator $$p=\sqrt{\frac{\hbar}{2m}}(A+B)$$ with $$[A,B]=1$$ and I need to find $x$ in this ...
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379 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 ...
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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: ...
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1answer
675 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 ...
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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: ...
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420 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 ...
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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 $$\phi(x)\phi^\dagger(y)\ne0.$$ ...
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1answer
360 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 ...
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1answer
301 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 ...
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861 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 ...
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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 ...
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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)] = ...
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1answer
165 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 ...
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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 ...
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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? ...
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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 ...
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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 ...
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1answer
179 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 ...
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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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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 ...
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648 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 ...
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3answers
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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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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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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 ...
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Classical Limit of Commutator

In Dirac's book Principles of quantum mechanics (4th ed., pgs 87-88), he seems to give a very elementary argument as to how the commutator $[X,P]$ reduces to the Poisson brackets ${x,p}$ in the limit ...
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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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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 ...
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426 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 ...
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Generalizing Heisenberg Uncertainty Priniciple

Writing the relationship between canonical momenta $\pi _i$ and canonical coordinates $x_i$ $$\pi _i =\text{ }\frac{\partial \mathcal{L}}{\partial \left(\frac{\partial x_i}{\partial t}\right)}$$ ...
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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 ...
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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 $$[yp_z, zp_x] ~=~ y[p_z, z]p_x?$$