What is Commutation: Definition and 220 Discussions

In law, a commutation is the substitution of a lesser penalty for that given after a conviction for a crime. The penalty can be lessened in severity, in duration, or both. Unlike most pardons by government and overturning by the court (a full overturning is equal to an acquittal), a commutation does not affect the status of a defendant's underlying criminal conviction.
Although the concept of commutation may be used to broadly describe the substitution of a lesser criminal penalty for the original sentence, some jurisdictions have historically used the term only for the substitution of a sentence of a different character than was originally imposed by the court. For example, the substitution of a sentence of parole for the original sentence of incarceration. A jurisdiction that uses that definition of commutation would use another term, such as a remission, to describe a reduction of a penalty that does not change its character.A commutation does not reverse a conviction and the recipient of a commutation remains guilty in accordance with the original conviction. For example, someone convicted of capital murder may have their sentence of death commuted to life imprisonment, a lessening of the punishment that does not affect the underlying criminal conviction, as may occur on a discretionary basis or following upon a change in the law or judicial ruling that limits or eliminates the death penalty.In some jurisdictions a commutation of sentence may be conditional, meaning that the convicted person may be required to abide by specified conditions or may lose the benefit of the commutation. The conditions must be lawful and reasonable, and will typically expire when the convicted completes any remaining portion of his or her sentence. For example, the pardon may be conditioned upon the person's being a law-abiding citizen, such that if the beneficiary of the commutation commits a new crime before the condition expires the original sentence may be restored.

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  1. M

    Quantum Mechanics Commutation Problem

    [Li, Lj]=ih εijk Lk the problem is , show that this equation.Can you help me to solve this problem with levi-civita symbol?
  2. C

    Derivation of creation and annihilation operator commutation relations

    Hi, I'm hopng someone can help me. I've begun working my way through Lahiri's "A first book of quantum field theory". In chapter 3 he shows the Fourier decomposition of the free field is given by \phi(x) = \int \frac{d^3 P}{\sqrt{(2\pi)^3 2E_p}} (a(p) e^{-ip\cdot x} + a^D(p) e^{ip...
  3. B

    Transients(RC) elements in the circuit after commutation?

    Homework Statement Find I1,I2,I3,Uc Homework Equations I was trying to make this task but my result was not correct Before commutation I=E/(R1+R2)=0.1 A U=0.1*R2=100 V After commutation I=0 A In the moment equatations I1*R1+I1*R2-I3*R2=E I3*R2-I1*R2=Uc I3=C*dUc/dt But when I had got result...
  4. L

    Hi,In Mandl&Shaw, when we calculate the covaiant commutation

    Hi, In Mandl&Shaw, when we calculate the covaiant commutation relations for a scalar field we obtain : [\phi(x),\phi(y)]= i\Delta(x-y)=0 and the last equality stands if x-y is a space-like interval. But I don't understand why. We know that it is zero if the time component is zero and we...
  5. Pengwuino

    Fine structure of H, momentum-potential commutation

    I'm going through Maggiore's QFT text and he is doing the fine structure constant for the Hydrogen atom. While trying to get to the Schrodinger-like Hamiltonian, he comes to a point where he has a term proportional to \epsilon * p^2 where \epsilon = (E - m) which eventually is the energy...
  6. dextercioby

    Commutation relations (maths)

    One of my dilemmas about <standard> quantum mechanics is spelled out in the sequel: If the position and momentum observables of a single-particle quantum system in 3D are described by the self-adjoint linear operators Q_i and P_i on a seperable Hilbert space \mathcal{H} subject to the...
  7. jfy4

    Generalized commutation relations

    I would like to work out the following commutation relations (assuming I have the operators right...:-p) (1) \left[\hat{p}^{\alpha},\hat{p}_{\beta}\right] (2) \left[\hat{p}_{\alpha},\hat{L}^{\beta\gamma}\right] (3) \left[\hat{L}^{\alpha\beta},\hat{L}_{\gamma\delta}\right] where...
  8. C

    What is the commutation between Xk and Xl?

    Homework Statement The question asks for [Xi^2, Lj] I can get to the line: ih(bar) Xk ekjl Xl + ih(bar) ekjl Xk Xl this line must be zero but I don't see how it is. It looks like an expansion of a commutation between Xk and Xl but not quite right
  9. M

    Why isn't commutation transitive?

    I know this is really basic, but can anyone explain why commutation isn't transitive? (Eg in the case of invariance of the Hamiltonian under a non-abelian group, all the transformations of the group commute with H but don't all commute with each other.) I thought there was only one basis in...
  10. P

    Commutation of differentiation and averaging operations

    I've been studying Turbulence, and there's a lot of averaging of differential equations involved. The books I've seen remark offhandedly that differentiation and averaging commute for eg. < \frac{df}{dt} > = \frac{d<f>}{dt} Here < > is temporal averaging. If...
  11. C

    Commutation and Eigenfunctions

    My first question is, does any operator commute with itself? If this is the case, is there a simple proof to show so? If not, what would be a counter-example or a "counter-proof"? My second question has to do with the properties of an eigenvalue problem. If you have an operator Q such that...
  12. E

    Dirac spinors and commutation

    Hey guys, i'm stuck (yet again! :) ) I am somewhat confused by Dirac spinors u,\bar{u}. Take the product (where Einstein summation convention is assumed): u^r u^s\bar{u}^s Is this the same as u^s\bar{u}^s u^r? Probably not because u^r is a vector while the other thing is a matrix...
  13. I

    Proving Tensor Commutation: T^abc S_b vs S_b T^abc

    Homework Statement How would you show that T^{abc}S_{b} = S_{b}T^{abc} but T^{abc} S_{bd} \neq S_{bd} T^{abc} in general? The Attempt at a Solution If I write out the sums explicitly, they appear totally the same to me. Any hints or ideas please?
  14. R

    Grassmann Numbers & Commutation Relations

    If you have a Grassman number \eta that anticommutes with the creation and annihilation operators, then is the expression: <0|\eta|0> well defined? Because you can write this as: <1|a^{\dagger} \eta a|1>=-<1| \eta a^{\dagger} a|1> =-<1|\eta|1> But if \eta is a constant, then...
  15. L

    Supercharges commutation rules

    Hi all I'm new on this forum. I'm here since I'm working with n-extended susy and R-sym and I don't know how to calculate a commutator. First of all I introsuce my notation: \mu_A T^A is a potential cupled to R-sym generator \mu_{\alpha i} is a superpotential cupled to supercharges...
  16. C

    How Are Commutation Relations Derived in Quantum Field Theory?

    In Srednicki's book, he discusses quantizing a non-interacting spin-0 field \phi(x) by defining the KG Lagrangian, and then using it to derive the canonical conjugate momentum \pi(x) = \dot{\phi}(x). Then, he states that, by analogy with normal QM, the commutation relations between these fields...
  17. F

    Do Commuting Operators Always Share a Common Basis of Eigenvectors?

    Hey guys, I'm studying some quantum physics at the moment and I'm having some problems with understanding the principles behind the necessary lineair algebra: 1) If two operators do NOT commutate, is it correct to conclude they don't have a similar basis of eigenvectoren? Or are there more...
  18. S

    QFT: Proving Commutation of Fields & Momentum

    Hi everyone My question is about QFT I'm reading mandle and shaw in chapter 2 as you know there is a question (2.4) about the commutations between the field and momentum. [ [P]^{}[j], \phi ] as momentum is in integral form I don't know how to prove them! I tried to open the terms...
  19. Amith2006

    Does Measurement in the X Direction Affect Y Coordinate in Quantum Systems?

    Does x & y directions commute? Seem trivial! Just wondering whether any measurement made in the x direction affect it's y coordinate.
  20. O

    Commutation relations of P and H

    Can we always calculate the commutation relations of two observables? If so, what’s the commutator of P (momentum) and H (Hamiltonian) in infinite square well, considering that the momentum is not a conserved quantity?
  21. M

    Commutation relation of operators

    Im reading in a quantum mechanics book and need help to show the following relationship, (please show all the steps): If A,B,C are operators: [A,BC] = B[A,C] + [A,B]C
  22. D

    Trace of the fundamental commutation relation

    Hi. So I have learned that this holds for the trace if A and B are two operators: \text{Tr}(AB)=\text{Tr}(BA). Now I take the trace of the commutator between x and p: \text{Tr}(xp)-\text{Tr}(px)=\text{Tr}(xp)-\text{Tr}(xp)=0. But the commutator of x and p is i\hbar. Certainly the trace of...
  23. R

    Commutation relationsl, angular momentum

    Homework Statement calculate the following commutation relations [L_{x}L_{y}]= [L_{y}L_{z}]= ][L_{z}L_{x}]= Homework Equations [L_{x},L_{y}]= -\hbar^2[y\frac{\partial}{\partial x}-x\frac{\partial}{\partial y}] where the expression in the parentheses describes L_{z} but i...
  24. I

    Significance of Commutation Relations

    I am aware that the commutation relation between conjugate variables shows that one quantity is the Fourier transform of the other, and so to imply the Heisenberg Uncertainty condition. So for example, the commutation relation between x, p (position and momentum respectively) leads to a non-zero...
  25. S

    How Does DC Motor Commutation Influence Rotation and Torque?

    i am unable to understand dc motor commutation can anybody help please
  26. P

    Commutation relation of angular momentum

    Just wondering if any1 would by any chance know how to calculate the commutation relations when L = R x P is the angular momentum operator of a system? [Li, P.R]=? P
  27. U

    What properties does Baym use to derive the L commutation relation?

    In Baym's Lectures on Quantum Mechanics he derives the following formula [n.L,L]=ih L x n (Where n is a unit vector) I follow everything until this line: ih(r x (p x n)) + ih((r x n) x p) = ih (r x p) x n I can't seem to get this to work out. What properties is he using here?
  28. M

    Commutation of (L^2)op and (Lz)op

    Homework Statement It has been shown that the operators (Lx)op and (Ly)op do not commute but satisfy the following equation: (Lx)op(Ly)op - (Ly)op(Lx)op = i(hbar)(Lz)op (a) Use this relation and the two similar equations obtained by cycling the coordinate labels to show that...
  29. J

    QM Angular Momentum Commutation Question

    Homework Statement Consider a state | l, m \rangle, an eigenstate of both \hat{L}^{2} and \hat{L}_{z}. Express \hat{L}_{x} in terms of the commutator of \hat{L}_{y} and \hat{L}_{z}, and use the result to demonstrate that \langle \hat{L}_{x} \rangle is zero. Homework Equations [...
  30. V

    Dirac Gamma Matricies and Angular Momentum Commutation Relations

    Homework Statement This isn't really the problem, but figuring this out will probably help me with the rest of the problem. I want to know what [\gamma^0, L_x] is. Homework Equations I know the commutation (or rather anticommutation) relations between the gamma matricies, and I know the...
  31. J

    Commutation of operators in QM

    Can somebody please explain the following? Given the measurements of 2 different physical properties are represented by two different operators, why is it possible to know exactly and simultaneously the values for both of the measured quantities only if the operators commute?
  32. N

    Recovering the generator of rotation from canonical commutation relations

    I'm having a course in advanced quantum mechanics, and we're using the book by Sakurai. In his definition of angular momentum he argues from what the classical generator of angular momentum is, and such he defines the generator for infitesimal rotations as...
  33. K

    Commutation Relations and Ehrenfest

    Homework Statement Let \psi(\vec{r},t) be the wavefunction for a free particle of mass m obeying Schrodinger equation with V=0 in 3 dimensions. At t=0, the particle is in a known initial state \psi_0(\vec{r}). Using Ehrenfest's theorem, show that the expectation value <x^2> in the state...
  34. S

    Intuitive Understanding for Commutation

    I am trying to get a grip on the commutation properties of operators. Different authors get to those differently: some start from translator operators, some relate those to Poisson brackets, etc... My objective is to get a good intuitive feeling of what commuting and not commuting observables...
  35. N

    How do commutation relations help in solving quantum mechanics?

    Homework Statement Suppose the operators P and Q satisfy the commutation relation [P,Q]=a, where a is a constant(a number, not an operator). a)Reduced the commutator [P,Q^n] where Q^n means the product of n Qs, to the simplest possible form. b) Reduce the commutator...
  36. A

    Commutation Relations and Symmetries for SU(2)

    Homework Statement I'm working through a bit of group theory (specifically SU(2) commutation relations). I have a question a bout symmetries in the SU(2) group. It's something I'm trying to work through in my lecture notes for particle physics, but it's a bit of a mathsy question so I thought...
  37. facenian

    Is the Commutation Relation for Angular Operators in Hilbert Space Valid?

    There must something wrong with my understanding of this relations because I think the usual way they are derived in many textbooks makes no sense. It goes like this, first assume that to every rotation O(a) in euclidean space there exists a rotation operator R(a) in Hilbert space,second: the...
  38. snoopies622

    The significance of commutation

    I'm in chapter two of H. S. Green's Matrix Mechanics and at a sticking point. In section 2.2 he gives the following scenario: An atom emits a photon with angular velocity ω, it has energy Ei before the emission and Ef after, so Ei - Ef = ħω. (That I can understand.) ψi and ψf are...
  39. W

    Raising and lowering operators & commutation

    Homework Statement Show [a+,a-] = -1, Where a+ = 1/((2)^0.5)(X-iP) and a- = 1/((2)^0.5)(X+iP) and X = ((mw/hbar)^0.5)x P = (-i(hbar/mw)^0.5)(d/dx)2. The attempt at a solution It would take forever to write it all up, but in summary: I said: [a+,a-] = (a+a- - a-a+) then...
  40. M

    Commutation of angular momentum operators

    Homework Statement None of the operators Lz, Ly and Lx commute. Show that [Lx,Ly] = i(h-cross)Lz Homework Equations Where Lx is defined as Lx=-ih ( y delta/delta x - z delta/delta y) Where Ly is defined as Ly=-ih ( z delta/delta x - x delta/delta z) Where Lz is defined as Lz=-ih (...
  41. E

    Can tensors always commute with each other or are there exceptions?

    Hello, I am still having a hard time with tensors... The answer is probably obvious, but is it always the case (for an arbitrary metric tensor g_{\mu \nu} that g_{ab}g_{cd}=g_{cd}g_{ab} ? I was trying to find a formal proof for that, and was wondering if we could use the relations: (1)...
  42. P

    Commutation of 2 operators using braket notation?

    How do you work out the commutator of two operators, A and B, which have been written in bra - ket notation? alpha = a beta = b A = 2|a><a| + |a><b| + 3|b><a| B = |a><a| + 3|a><b| + 5|b><a| - 2|b><b| The answer is a 4x4 matrix according to my lecturer... Any help much appreciated...
  43. K

    Question about commutation and operator

    I am reading the book by J.J.Sakurai, in chapter 3, there is a relation given as \langle \alpha', jm|J_z A |\alpha, jm\rangle Here, j is the quantum number of total angular momentum, m the component along z direction, \alpha is the third quantum number. J_z is angular momentum operator, A...
  44. L

    Show operator can be an eigenfunction of another operator given commutation relation

    Homework Statement Suppose that two operators P and Q satisfy the commutation relation: [P,Q]=P. Suppose that psi is an eigenfunction of the operator P with eigenvalue p. Show that Qpsi is also an eigenfunction of P, and find its eigenvalue. Homework Equations The Attempt at a...
  45. kreil

    Proving e^Ae^B=e^{A+B} for Commuting Matrices

    Homework Statement Show, by series expansion, that if A and B are two matrices which do not commute, then e^{A+B} \ne e^Ae^B, but if they do commute then the relation holds. Homework Equations e^A=1+A e^B=1+B e^{A+B}=1+(A+B) The Attempt at a Solution Assuming that the first 2...
  46. C

    [PhD Qualifier] Commutation relation

    Homework Statement Two quantum mechanical operators obey the following commutation relation. [\hat{A},\hat{B}]=i Given this commutation relation which of the following are true or false? Justify your answers. a) The two observables are simultaneously diagonalizable. b) The two satisfy a...
  47. J

    Lim inf and product commutation

    If f_1,f_2,f_3,\ldots and g_1,g_2,g_3,\ldots are some arbitrary real sequences, is it true that \underset{n\to\infty}{\textrm{lim inf}}\;(f_n g_n) = (\underset{n\to\infty}{\textrm{lim inf}}\; f_n) (\underset{n\to\infty}{\textrm{lim inf}}\; g_n)? For arbitrary \epsilon >0 there exists...
  48. S

    Does Commuting with Hamiltonian Ensure Observables' Commutator is Constant?

    [SOLVED] commutation of observables Homework Statement Prove: If the observables (operators) Q1 and Q2 are both constant of the motion for some Hamiltonian H, then the commutator [Q1, Q2] is also a constant of the motion. okay, first question.. am i being asked to prove that [[Q1, Q2], H] =...
  49. V

    Prove Commutation Property for 2x2 Matrices D

    Let D = [d11 d12] [d21 d22] be a 2x2 matrix. Prove that D commutes with all other 2x2 matrices if and only if d12 = d21 = 0 and d11 = d22. I know if we can prove for every A, AD=DA should be true, but I really don't know how to proceed from there. I tried equating elements of AD with...
  50. N

    Commutation relations in relativistic quantum theory

    Given the Hamiltonian H = \vec{\alpha} \cdot \vec{p} c + \beta mc^2, How should one interpret the commutator [\vec{x}, H] which is supposedly related to the velocity of the Dirac particle? \vec{x} is a 3-vector whereas H is a vector so how do we commute them. Is some sort of tensor product in...
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