# What is Commutators: Definition and 90 Discussions

In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.

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1. ### I Commutators, operators and eigenvalues

Hi I just wanted to check my understanding of something which has come up when first studying path integrals in QM. If x and px are operators then [ x , px ] = iħ but if x and px operate on states to produce eigenvalues then the eigenvalues x and px commute because they are just numbers. Is...
2. ### I Why do brushed DC motors only use commutators, and not slip rings?

Commutators always generate unwanted sparks and bad EM interference. If slip ring can replace commutator in DC motor, then electric vehicle industry will love it more than multi-phase AC induction motor.
3. ### Requesting guidance on Commutators & Intro QM

I have approached this question step by step as shown in the image attached. I request someone to please guide if I have approached the (incomplete) solution correctly and also guide towards the complete solution, by helping me to rectify any mistakes I may have made. I'm still unsure how to...
4. ### I Can someone explain why momentum does not commute with potential?

My assumption is that knowing potential can lead to knowing the position, but I don't know how this can be.
5. ### B Commutators of functions of operators

I would like to ask whether if operators ##A## and ##B## commute also operators ##e^A## and ##e^B## commute? Also I have a question is it possible that ##e^A## is matrix where all elements are ##\infty## so that ##e^A \cdot e^B-e^B\cdot e^A## has all elements that are ##\infty##?
6. ### A Exploring the Conditions for Evaluating Commutators with Fermionic Operators

I found a theorem that states that if A and B are 2 endomorphism that satisfies $$[A,[A,B]]=[B,[A,B]]=0$$ then $$[A,F(B)]=[A,B]F'(B)=[A,B]\frac{\partial F(B)}{\partial B}$$. Now I'm trying to apply this result using the creation and annihilation fermionics operators $$B=C_k^+$$ and $$A=C_k$$...
7. ### I Inner Product Between States of Multiple Particles

$$<p_1 p_2|p_A p_B> = \sqrt{2E_1 2E_2 2E_A 2E_B}<0|a_1 a_2 a_{A}^{\dagger} a_{B}^{\dagger} |0>$$ $$=2E_A2E_B(2\pi)^6(\delta^{(3)}(p_A-p_1)\delta{(3)}(p_B-p_2) + \delta^{(3)}(p_A-p_2)\delta^{(3)}(p_B-p_1))$$ The identity above seemed easy, until I tried to prove it. I figured I could work this...
8. ### I Is my reasoning about commutators of vectors right?

Hello guys, I have a question regarding commutators of vector fields and its pushforwards. Let me define a clockwise rotation in the plane \,\phi:\mathbb{R}^2\rightarrow\mathbb{R}^2 \,.\; [\,\partial_x\,,\,\partial_y\,]=0 \,, \;(\phi_{*}\partial_x) = \partial_r and \,(\phi_{*}\partial_y) =...
9. ### Prove formula for the product of two exponential operators

Homework Statement Consider two operators A and B, such that [A,[A, B]] = 0 and [B,[A, B]] = 0 . Show that Exp(A+B) = Exp(A)Exp(B)Exp(-1/2 [A,B]) Hint: define Exp(As)Exp(Bs) as T(s), where s is a real parameter, differentiate T(s) with respect to s, and express the result in terms of T(s)...
10. ### Bosonic annihilation and creation operators commutators

Homework Statement After proving the relations ##[\hat{b}^{\dagger}_i,\hat{b}^{\dagger}_j]=0## and ##[\hat{b}_i,\hat{b}_j]=0##, I want to prove that ##[\hat{b}_j,\hat{b}^{\dagger}_k]=\delta_{jk}##, however I'm not sure where to begin. 2. The attempt at a solution I tried to apply the...
11. ### Spacetime and Geometry: Vanishing commutators#2

This is a refinement of a previous thread (here). I hope I am following correct protocol. Homework Statement I am studying Spacetime and Geometry : An Introduction to General Relativity by Sean M Carroll and have a question about commutators of vector fields. A vector field on a manifold can...
12. ### Spacetime and Geometry: Vanishing commutators

Homework Statement I am studying Spacetime and Geometry : An Introduction to General Relativity by Sean M Carroll and have a question about commutators of vector fields. A vector field on a manifold can be thought of as differential operator which transforms smooth functions to smooth functions...
13. ### I Why do you need infinite size matrix which commute....

...to give a number? https://ocw.mit.edu/courses/physics/8-04-quantum-physics-i-spring-2016/lecture-notes/MIT8_04S16_LecNotes5.pdf On page 6, it says, " Matrix mechanics, was worked out in 1925 by Werner Heisenberg and clarified by Max Born and Pascual Jordan. Note that, if we were to write xˆ...
14. ### Strings, Virasoro Operators&constraints, commutator algrebra

Homework Statement [/B] Question: (With the following definitions here: ) - Consider ##L_0|x>=0## to show that ##m^2=\frac{1}{\alpha'}## - Consider ##L_1|x>=0 ## to conclude that ## 1+A-2B=0## - where ##d## is the dimension of the space ##d=\eta^{uv}\eta_{uv}## For the L1 operator I am...
15. ### I Introducing QM via commutators

I understand there exists some way of teaching QM via postulating commutation relation between coordinate and momentum operator. May be even not simply postulating but bringing some reasons why such a commutator should be equal to "i"? Could you recommend some good book or article which teaches...
16. ### I Why Don't L2 and Ly Commute When L2 and Lx Do?

Hi. To show that [ L2 , L+ ] uses the following commutators [ L2 , Lx ] = 0 and [ L2 , Ly ] = 0 . But if [ L2 , Lx ] = 0 this shows that L2 and Lx have simultaneous eigenstates ; but then should L2 and Ly not commute ? Thanks
17. ### Compute Commutator: JxJy, Jz | i ħ Result

Homework Statement Find the resul of [Jx Jy , Jz] where J is the angular momentum operator. Possible answers to this multiple chioce question are A) 0 B) i ħ Jz C) i ħ Jz Jx D) i ħ Jx Jz E) i ħ Jx Jy Homework Equations [AB,C]=A [B,C]+[A,B] B [Ji , Jj]=i ħ εijk Jk where εijk is the Levi-Civita...
18. ### I Do operators that commute also indicate if their powers commute?

If ##A## and ##B## are two operators that commute (i.e. [##A##,##B##] = 0), does that indicate if ##A^m## and ## B^n## more generally commute where m and n are not necessarily non-negative integers?
19. ### Show that [A, F(B)] = [A, B]F'(B)

Homework Statement A and B commute with their commutator, C=[A,B] Show that [A, F(B)] = [A, B]F'(B) F(B) = ∑n=0∞fnBn Homework Equations [A,B] = AB - BA [A,BC] = [A,B]C + B[A,C]The Attempt at a Solution So all I can think to do is: [A,Bn] = [A,BBn-1] = [A,B]Bn-1 + B[A,Bn-1]...
20. ### Can commutation relations prove <Lx^2> = <Ly^2> in an eigenstate of L^2 and Lz?

Homework Statement Show that ##<L_x^2> = <L_y^2>## using the commutation relations. The system is in the eigenstate |l,m> of ##L^2## and ##L_z##. Homework Equations ##[L_x, L_y] = i \hbar L_z## ##[L_y, L_z] = i \hbar L_x## ##[L_z, L_x] = i \hbar L_y## ##[L_x, L^2] = 0## ##[L_y, L^2] = 0##...
21. ### Using Mathematica to solve for Jacobi Identity

Hi everyone, I'm new to Physics Forums and to Mathematica, as well as Jacobi Identity. In any case, I was wondering on how I may use Mathematica to solve various Quantum Mechanics related problems through commutators. Like if it's possible to find out what is the form of a particular commutator...
22. ### What is the exact connection between Poisson brackets and commutators

I'm not perfectly clear on the connection between Poisson brackets in classical mechanics and commutators in quantum mechanics. For any classical mechanical system, if I can find the Poisson bracket between two physical observables, is that always the value of the corresponding commutator in the...
23. ### Questions about operators and commutators

so I have an expression here: [P,g(r)]= -ih dg/dr P is the momentum operator working on a function g(r). Is this true because when you expand the left hand side the expression g(r)P is zero because the del operator has nothing to work on?
24. ### Commutators and cevi levita symbol

Homework Statement The problem statement can be seen in the picture i uploaded. Homework Equations - The Attempt at a Solution The attempt to the solution can be seen in the picture i uploaded. I reached to the A and i don't know how to proceed to the solution that is given below. How does...
25. ### Could we calculate the number of Planck flops in a human lifetime?

just finished Susskind's QM. Definitely blew my mind. Can't stop wondering about commutators. Trying synthesize and remember with self quiz I can't self grade. Super quiz: Could one in principle calculate the number of Planck flops (commutaor operations or bits) in a human lifetime? My...
26. ### Show [J2, J+] = 0 - Homework Solution

Homework Statement To show [J2, J+] = 0 2. Homework Equations J+ = Jx + i Jy [J2, Jx ] = 0 [J2, Jy ] = 0The Attempt at a Solution Step 1: L.H.S. = [J2, J+] Step 2: L.H.S. = [J2, Jx + i Jy ] Step 3: L.H.S. = [J2, Jx ] + i [J2, Jy ] Step 4: L.H.S. = 0 + 0 Step...
27. ### Relationship between commutators and observables

Homework Statement Suppose A^ and B^ are linear quantum operators representing two observables A and B of a physical system. What must be true of the commutator [A^,B^] so that the system can have definite values of A and B simultaneously? Homework Equations I will use the bra-ket notation for...
28. ### Commutators, Traces & [x,p] = I(hbar) Explained

I read the following as a model solution to a question but I don't understand it - " there is no possible finite dimensional representation of the operators x and p that can reproduce the commutator [x,p] = I(hbar)(identity matrix) since the LHS has zero trace and the RHS has finite trace. My...
29. ### Deriving commutator for angular momentum components

Homework Statement Prove that ## [L_a,L_b] = i \hbar \epsilon_{abc} L_c ## using Einstein summation convention. I think I have achieved the solution but I am not sure of my last steps, since this is one of my first excersises using this convention. Homework Equations [/B] ## (1)...
30. ### Commutatos and angular momentum

Homework Statement Let e and f be unit vectors. Le = eL is the definition of the component of angular momentum in direction e. Calculate the commutator [Le,Lf ] in terms of e, f and L Homework Equations [A,B]=(AB-BA) The Attempt at a Solution we know that L=r x p, in classical mechanics, and...
31. ### A simple proof involving degeneracy and commutators

Homework Statement In the absence of degeneracy, prove that a sufficient condition for the equation below (1), where \left|a'\right> is an eigenket of A, et al., is (2) or (3). Homework Equations \sum_{b'} \left<c'|b'\right>\left<b'|a'\right>\left<a'|b'\right>\left<b'|c'\right> = \sum_{b',b''}...
32. ### Understanding Commutators in Quantum Mechanics: General and Specific Questions

Hi all, My motivation is understanding some derivations in Quantum Mechanics, but I think my questions are purely algebraic. I have a general question and then a specific one: General Question - when writing the commutator of commuting vector and a scalar operators (for instance angular...
33. ### Commutators, mutual eigenkets, and observables

I have two quick questions: 1. Why if say [x,y] = 0, it implies that there is a mutual complete set of eigenkets? where x and y can be anything, like momentum, position operators. 2. If an operator is not hermitian, why isn't it an observable? (More specifically, why isn't its...
34. ### Commutators; matrices? numbers? both?

The commutator of two operators A and B, which measures the degree of incompatibility between A and B, is AB - BA (at least according to one textbook I have). But multiplying/substracting matrices just yields matrices! (http://en.wikipedia.org/wiki/Matrix_multiplication). So firstly, how can a...
35. ### Commuting Pauli Matrices: A Tricky Homework Challenge

Homework Statement Express the product where σy and σz are the other two Pauli matrices defined above. Homework Equations The Attempt at a Solution I'm not sure if this is a trick question, because right away both exponentials combine to give 1, where the result is...
36. ### Commutators of Operators and Constants: A Quantum Mechanics Exercise

Homework Statement Let ## \hat{A} = x ## and ## \hat{B} = \dfrac{\partial}{\partial x} ## be operators Let ## \hat{C} ## be defined ## \hat{C} = c ## with c some complex number. A commutator of two operators ## \hat{A} ## and ## \hat{B} ## is written ## [ \hat{A}, \hat{B} ] ## and is...
37. ### Commutators: solving with jacobian

Homework Statement Hello:) My problem is as follows: Determine the following commutators: [px2,x],[pxx2],[px2,x2],[]. The calculation can be done in two ways, either by inserting a test function, and using the explicit expressions for the operators, or by utilizing Jacobi identity and using...
38. ### Commutators of angular momentum, linear momentum squared and H

Homework Statement For the linear momentum operator ##\hat{\mathbf{p}}## and angular momentum operator ##\hat{\mathbf{L}} ##, prove that ##\begin{eqnarray} [\hat{\mathbf{L}},\hat{\mathbf{p}}^2]&=&0\end{eqnarray}##: [Hint: Write ##\hat{\mathbf{L}}## as the ##x##-component of the angular...
39. ### Griffiths Quantum Mechanics: Solving Angular Momentum with Commutators

I'm having a lot of trouble following Griffith's quantum mechanics text. I'm in section 4.3 which discusses angular momentum using commutators. The text proceeds as follows: [L_x, L_y] = [yp_z - zp_y, zp_x - xp_z]\\ =[yp_z, zp_x] - [yp_z, xp_z] - [zp_y, zp_x] + [zp_y, xp_z]\\ =[yp_z, zp_x] +...
40. ### Uncertainty, Symmetry, and Commutators

It seems the uncertainty principle, the commutator between operators, and the symmetry of the action integral are all related. And I wonder how universal this is. For example, the action integral is invariant with respect to time, and this leads to conserved quantity of energy. This means...
41. ### Operator algebra with commutators

Homework Statement Consider the operator A and its Hermitian adjoint A*. If [A,A*] = 1, evaluate: [A*A,A] Homework Equations standard rules of linear algebra, operator algebra and quantum mechanics The Attempt at a Solution [A,A*] = AA* - A*A = 1 A*A = (1+AA*) [A*A,A] =...
42. ### Do commutators hold in interacting theory?

Do the Fourier coefficients of an interacting field obey the commutation relations? I think I was able to show once that they do if the Fourier coefficients are taken at equal times (the coefficients are time-dependent in the interacting theory), but my proof felt shaky. In any case, does...
43. ### Commutators, propagators, and measurement.

Hi all, Reading through Peskin and Schroeder, I came across the following statement, with regards to propagators: Could someone explain how the commutator is related to the measurement of the field in this context? Searching online, the only thing that crops up is the usual uncertainty...
44. ### Commutators and Expectation Values

Homework Statement Here's a link to an image of the exam question. It appears in the exam every couple of years, and it's due in my exam this coming week. I've looked in both the textbook and the course notes, and they simply *state* the conclusion, so I don't have a way of proving it, and...
45. ### Commutators for functions of operators (momentum/position)

Homework Statement show that [x,f(p_x)] = i \hbar d/d(p_x) f(p_x) Homework Equations x is the position operator in the x direction, p_x is the momentum operator; i \hbar d/dx [x, p_x]=xp-px The Attempt at a Solution I'm stuck. maybe chain rule for d/dx and d/d(p_x)...? But I...
46. ### Fermion Current Commutators in 2 dimensions

Homework Statement Given the current: J^{\epsilon}_{0} (t,x) = \overline{\psi_{L}}(t,x + \epsilon) \gamma^{0} \psi_{L}(t,x - \epsilon) = \psi_{L}^{\dagger} (x + \epsilon) \psi_{L}(x - \epsilon) with \psi_{L} = \frac{1}{2} (1 - \gamma^{5}) \psi_{D}. Use the canonical equal time...
47. ### Commutators of Angular momentum operator

The letters next to p and L should be subscripts. [Lz, px] = [xpy − ypx, px] = [xpy, px] − [ypx, px] = py[x, px] −0 = i(hbar)py 1.In this calculation, why is [x, px] not 0 even they commute? 2.Why is py put on the left instead of the right in the second last step? i thought it should be...
48. ### Solving Commutators & Calculations: W^2, p^α=0

Hi you guys! I am having a hard time understanding some stuff in this context. I was not able to find any guidance in books or anything. say I want to calculate: [W^2,p^\alpha]=0, W=\frac{m}{2}\varepsilon^{\mu\nu\lambda\varepsilon}M_{\mu\lambda}p_\delta How do I do that? I can't...
49. ### Symmetry Groups Algebras Commutators Conserved Quantities

Symmetry, Groups, Algebras, Commutators, Conserved Quantities OK, maybe this is asking too much, hopefully not. I'm trying to understand the connection between all of these constructions. I wonder if a summary about these interrelationship can be given. If I understand what I'm reading, there...
50. ### What is the commutator between S_{z}^{n} and S_{y}?

Homework Statement The entire problem is quite in depth. But what I am having trouble with is just a small part of it, and it boils down to finding the following commutator: \left[ S_{z}^{n},S_{y}\right] where S_{z} and S_{y} are the quantum mechanical spin matrices. The reason is that I have...