Angular momentum operator Definition and 50 Threads

In quantum mechanics, the angular momentum operator is one of several related operators analogous to classical angular momentum. The angular momentum operator plays a central role in the theory of atomic and molecular physics and other quantum problems involving rotational symmetry. Such an operator is applied to a mathematical representation of the physical state of a system and yields an angular momentum value if the state has a definite value for it. In both classical and quantum mechanical systems, angular momentum (together with linear momentum and energy) is one of the three fundamental properties of motion.There are several angular momentum operators: total angular momentum (usually denoted J), orbital angular momentum (usually denoted L), and spin angular momentum (spin for short, usually denoted S). The term angular momentum operator can (confusingly) refer to either the total or the orbital angular momentum. Total angular momentum is always conserved, see Noether's theorem.

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  1. pixyl

    Trouble finding ##L^2## in function of ##x## and ##p##

    What I've done is $$\vec{L}^2 = \varepsilon_{ijk}x^jp^k\varepsilon_{imn}x^mp^n = (\delta_{jm}\delta_{kn} - \delta_{jn}\delta_{km})x^jp^kx^mp^n = x^jp^kx^jp^k - x^jp^kx^kp^j = $$ $$ = x^jx^jp^kp^k - i\hbar x^jp^j - x^jp^kx^kp^j = $$ $$ = x^jx^jp^kp^k - i\hbar x^jp^j - (x^jx^kp^kp^j - i\hbar...
  2. physical_chemist

    A Angular momentum uncertainty principle and the particle on a ring

    By considering a particle on a ring, the eigenfunctions of ##H## are also eigenfunctions of ##L_\text{z}##: $$\psi(\phi) = \frac{1}{\sqrt{2\pi}}e^{im\phi}$$ with ##m = 0,\pm 1,\pm 2,\cdots##. In polar coordinates, the corresponding operators are $$H =...
  3. G

    Tong QFT sheet 2, question 6: Normal ordering of the angular momentum operator

    My attempt/questions: I use ##T^{0i} = \dot{\phi}\partial^i \phi##, ##\dot{\phi} = \pi##, and antisymmetry of ##Q_i## to get: ##Q_i = 2\epsilon_{ijk}\int d^3x [x^j \partial^k \phi(\vec{x})] \pi(\vec{x})##. I then plug in the expansions for ##\phi(\vec{x})## and ##\pi(\vec{x})## and multiply...
  4. K

    I Position representation of angular momentum operator

    One of the component of angular momentum operator is ##\hat{L}_{x}=\hat{y} \hat{P}_{z}-\hat{z} \hat{P}_{y}## I want it's position representation. My attempt : I'll find the representation of the first term ##\hat{y} \hat{P}_{z}##. The total representation is the sum of two terms. The...
  5. Garlic

    Landau levels: Hamiltonian with ladder operators

    Dear PF, I hope I've formulated my question understandable enough. Thank you for your time, Garli
  6. Rabindranath

    Angular momentum operator for 2-D harmonic oscillator

    1. The problem statement I want to write the angular momentum operator ##L## for a 2-dimensional harmonic oscillator, in terms of its ladder operators, ##a_x##, ##a_y##, ##a_x^\dagger## & ##a_y^\dagger##, and then prove that this commutes with its Hamiltonian. The Attempt at a Solution I get...
  7. Gene Naden

    A Angular momentum operator derived from Lorentz invariance

    I am working through Lessons in Particle Physics by Luis Anchordoqui and Francis Halzen; the link is https://arxiv.org/PS_cache/arxiv/pdf/0906/0906.1271v2.pdf. I am on page 11, equation 1.3.20. The authors have defined an operator ##L_{\mu\nu} = i( x_\mu \partial \nu - x_\nu \partial \mu)##...
  8. Jezza

    I Adding types of angular momenta

    There are two types of angular momentum: orbital and spin. If we define their operators as pseudo-vectors \vec{L} and \vec{S}, then we can also define the total angular momentum operator \vec{J} = \vec{L}+\vec{S}. Standard commutation relations will show that we can have simultaneous well...
  9. B

    I Angular momentum operator commutation relation

    I am reading a proof of why \left[ \hat{L}_x, \hat{L}_y \right ] = i \hbar \hat{L}_z Given a wavefunction \psi, \hat{L}_x, \hat{L}_y \psi = \left( -i\hbar \right)^2 \left( y \frac{\partial}{\partial z} - z \frac {\partial}{\partial y} \right ) \left (z \frac{\partial \psi}{\partial x} -...
  10. V

    Eigenvectors of Ly and associated energies

    Homework Statement Consider a particle with angular momentum l=1. Write down the matrix representation for the operators L_x,\,L_y,\,L_z,for this particle. Let the Hamiltonian of this particle be H = aL\cdot L-gL_z,\,g>0.Find its energy values and eigenstates. At time t=0,we make a measurement...
  11. Y

    Time Inversion Symmetry and Angular Momentum

    Homework Statement Let ##\left|\psi\right\rangle## be a non-degenerate stationary state, i.e. an eigenstate of the Hamiltonian. Suppose the system exhibits symmetry for time inversion, but not necessarily for rotations. Show that the expectation value for the angular momentum operator is zero...
  12. AwesomeTrains

    Rayleigh–Ritz method - Yukawa coulomb potential

    Hello everyone Homework Statement I have been given the testfunction \phi(\alpha, r)=\sqrt{(\frac{\alpha^3}{\pi})}exp(-\alpha r) , and the potential V(r,\theta, \phi)=V(r)=-\frac{e^2}{r}exp(\frac{-r}{a}) Given that I have to write down the hamiltonian (in spherical coordinates I assume), and...
  13. S

    Commutation relations for angular momentum operator

    I would like to prove that the angular momentum operators ##\vec{J} = \vec{x} \times \vec{p} = \vec{x} \times (-i\vec{\nabla})## can be used to obtain the commutation relations ##[J_{i},J_{j}]=i\epsilon_{ijk}J_{k}##. Something's gone wrong with my proof below. Can you point out the mistake...
  14. K

    Eigenfunctions of the angular momentum operator

    Hi everyone, I tried to find the Eigenstate of the angular momentum operator myself, more specifically I tried to find a Function Y_{lm}(\theta,\phi) with L_zY_{lm}=mħY_{lm} and L^2Y_{lm}=l(l+1)ħ^2Y_{lm} where L_z=-iħ\frac{\partial}{\partial \phi} and...
  15. D

    Total angular momentum operator for a superposition

    Hi all, Quick quantum question. I understand the total angular momentum operation \hat{L}^2 \psi _{nlm} = \hbar\ell(\ell + 1) \psi _{nlm} which means the total angular momentum is L = \sqrt{\hbar\ell(\ell + 1)} But how about applying this to an arbitrary superposition of eigenstates such as...
  16. gfd43tg

    Spin angular momentum operator queries

    Hello, For the spin angular momentum operator, the eigenvalue problem can be formed into matrix form. I will use ##S_{z}## as my example $$S_{z} | \uparrow \rangle = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} 1 \\ 0 \end{pmatrix} = \frac {\hbar}{2} \begin{pmatrix} 1 \\ 0...
  17. gfd43tg

    How Do You Approach Angular Momentum Operator Algebra in Quantum Mechanics?

    Homework Statement Homework EquationsThe Attempt at a Solution This whole thing about angular momentum has me totally confused and stumped, but I am trying this problem given in a youtube video lecture I watched. I know of this equation ##L^{2} = L_{\pm}L_{\mp} + L_{z}^{2} \mp \hbar L_{z}##...
  18. A

    Angular momentum operator justification

    One can represent the mean of the angular momentum operator as a vector. But what is the (mathematical) justification to represent the operator by a vector which has a direction that the operator has not. Yet worse, l(l+1) h2 is the proper value of operator L^2 and from such result it is assumed...
  19. M

    Vector calculus: angular momentum operator in spherical coordinates

    Note: physics conventions, \theta is measured from z-axis We have a vector operator \vec{L} = -i \vec{r} \times \vec{\nabla} = -i\left(\hat{\phi} \frac{\partial}{\partial \theta} - \hat{\theta} \frac{1}{\sin\theta} \frac{\partial}{\partial \phi} \right) And apparently \vec{L}\cdot\vec{L}=...
  20. H

    What is the Time Evolution of the Angular Momentum Operator?

    Hi guys, this might be a stupid question but if I wanted a general expression for the time evolution of the angular momentum operator is it just the same as Hamiltonian? i.e ih ∂/∂t ψ = L2 ψ Solving this partial differential gives the time evolution of the angular momentum operator...
  21. CrimsonFlash

    What are the matrix elements of the angular momentum operator?

    What are the "matrix elements" of the angular momentum operator? Hello, I just recently learned about angular momentum operator. So far, I liked expressing my operators in this way: http://upload.wikimedia.org/math/8/2/6/826d794e3ca9681934aea7588961cafe.png I like it this way because it...
  22. L

    Eigenvalue of angular momentum operator

    Homework Statement I'm running through practice papers for my 3rd year physics exam on atomic and nuclear physics: This is the operator we found in the previous part of the question L = -i*(hbar)*d/dθ Next, we need to find the eigenvalues and normalised wavefunctions of L The...
  23. J

    The angular momentum operator acting on a wave function

    Hi guys, I need help on interpreting this solution. Let me have two wave functions: \phi_1 = N_1(r) (x+iy) \phi_2 = N_2(r) (x-iy) If the angular momentum acts on both of them, the result will be: L_z \phi_1 = \hbar \phi_1 L_z \phi_2 = -\hbar \phi_2 My concern is, \phi_1 and \phi_2...
  24. T

    Prove angular momentum operator identity

    Homework Statement Using the operator identity: \hat{L}^2=\hat{L}_-\hat{L}_+ +\hat{L}_z^2 + \hbar\hat{L}_z show explicitly: \hat{L}^2 = -\hbar^2 \left[ \frac{1}{\sin^2\theta} \frac{\partial^2}{\partial\phi^2} + \frac{1}{\sin\theta} \frac{\partial}{\partial\theta}...
  25. S

    Prove that the angular momentum operator is hermitian

    Greetings, My task is to prove that the angular momentum operator is hermitian. I started out as follows: \vec{L}=\vec{r}\times\vec{p} Where the above quantities are vector operators. Taking the hermitian conjugate yields \vec{L''}=\vec{p''}\times\vec{r''} Here I have used double...
  26. X

    Angular momentum operator identity J²= J-J+ + J_3 + h*J_3 intermediate step

    Homework Statement I do not understand equal signs 2 and 3 the following Angular momentum operator identity: Homework Equations \hat{J}^2 = \hat{J}_1^2+\hat{J}_2^2 +\hat{J}_3^2 = \left(\hat{J}_1 +i\hat{J}_2 \right)\left(\hat{J}_1 -i\hat{J}_2 \right) +\hat{J}_3^2 + i...
  27. USeptim

    Wave function collapse by orbital angular momentum operator Lz

    I have some doubts about the implications of the orbital angular operators and its eigenvectors (maybe the reason is that I have a weak knowledge on QM). If we choose the measurement of the z axis and therefore the Lz operator, the are the following spherical harmonics for l=1...
  28. P

    Angular momentum operator acting on |j,m>

    Homework Statement Prove that e^{-i \pi J_x} \mid j,m \rangle =e^{-i \pi j} \mid j,-m \rangle Homework Equations J_x \mid j,m \rangle =\frac{\hbar}{2} [\sqrt{(j-m)(j+m+1)} \mid j,m+1 \rangle + \sqrt{(j+m)(j-m+1)} \mid j,m-1 \rangle] The Attempt at a Solution Expanding e^{-i \pi J_x}...
  29. J

    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...
  30. H

    Solving the Angular Momentum Operator for j=1

    Homework Statement Consider the angular momentum operator \vec{J_{y}} in the subspace for which j=1. Write down the matrix for this operator in the usual basis (where J^{2} and J_{z} are diagonal). Diagonalize the matrix and find the eigenvalues and orthonormal eigenvectors. Homework...
  31. E

    Showing that (x+iy)/r is an eigenfunction of the angular momentum operator

    Homework Statement I know that,if (operator)(function)=(value)(samefunction) that function is said to be eigenfunction of the operator. in this case i need to show this function to be eigenfunction of the Lz angular momentum:Homework Equations function: ψ=(x+iy)/r operator: Lz= (h bar)/i (x...
  32. Z

    Angular momentum operator eigenvalues in HO potential.

    Homework Statement Find wave functions of the states of a particle in a harmonic oscillator potential that are eigenstates of Lz operator with eigenvalues -1 h , 0, 1 h and have smallest possible eigenenergies. Check whether these states are also the eigenstates of L^2 operator. Eventually...
  33. A

    How Do Angular Momentum Operators Satisfy Algebraic Equations?

    Homework Statement Using matrix representations find L^{3}_{x},L^{3}_{y},L^{3}_{z} and from these show that L_{x},L_{y},L_{z} satisfy the same algebraic equations. What are the roots of the algebraic equations? 2. The attempt at a solution My problem is that I'm not sure what this...
  34. D

    For Angular Momentum Operator L, prove [Lx,Ly] = ihLz

    Homework Statement For an angular momentum operator ~L =ˆiLx +ˆjˆLy + ˆkˆLz = ˆr × ˆp, prove that [ˆLx, ˆLy] = i\hbarˆLz, [ˆLx, ˆLz] = −i\hbarˆLy, [ˆL2, ˆLx] = 0, [p^{2}, ˆLx] = 0, [r^{2}, ˆLx] = 0, [ˆLx, ˆy] = i\hbarˆz, [ˆLx, ˆpy] = i\hbarˆPz. **Note: I'm really only looking for help for the...
  35. I

    Angular Momentum Operator Eigenfunction

    Homework Statement Let the angular part of a wave function be proportional to x2+y2 Show that the wave function is an eigenfunction of Lz and calculate the associated eigenvalue. Homework Equations Lz = xpy-ypx px = -i\hbar\frac{\partial}{\partialx} py =...
  36. E

    Commutator of square angular momentum operator and position operator

    can someone please help me with this. it's killing me. Homework Statement to show \left[\vec{L}^{2}\left[\vec{L}^{2},\vec{r}\right]\right]=2\hbar^{2}(\vec{r}\vec{L}^{2}+\vec{L}^{2}\vec{r})Homework Equations I have already established a result (from the hint of the question) that...
  37. A

    Representation of Angular Momentum Operator in the (j,j')

    Hello All, I'm trying to understand how the (j,j') representation of the Lorentz group. Following Ryder, I can see why we define A=J+iK and B=J-iK, which each form an SU(2) group. So it's clear to me what the rep of these generators is when acting on a state (j,j'): Rep(A)\otimes1+1\otimes...
  38. C

    Expectation value of the angular momentum operator

    Homework Statement Hey forum, I copied the problem from a pdf file and uploaded the image: http://img232.imageshack.us/img232/6345/problem4.png What is the probability that the measurement of L^{2} will yield 2\hbar^{2} Homework Equations \left\langle L^{2} \right\rangle = \left\langle \Psi...
  39. I

    QM question, angular momentum operator and eigen functions

    For the operator L(z) = -ih[d/d(phi)] phi = azimuthal angle 1) write the general form of the eigenfunctions and the eigenvalues. 2) a particle has azimuthal wave function PHI = A*cos(phi) what are the possible results of a measurement of the observable L(z) and what is the...
  40. P

    Angular Momentum Operator in terms of ladder operators

    Homework Statement http://img716.imageshack.us/i/captur2e.png/ http://img716.imageshack.us/i/captur2e.png/ Homework Equations Stuck on the last part The Attempt at a Solution http://img689.imageshack.us/i/capturevz.png/ http://img689.imageshack.us/i/capturevz.png/
  41. L

    Do Lx and Lz Angular Momentum Operators Exhibit an Uncertainty Relation?

    The operators used for the x and y components of angular momentum are: Show that Lx and Lz obey an uncertainty relation 2. No relevant equations. The Attempt at a Solution I'm going on that the assumption that if LxLy - LyLz does not equal zero then they don't...
  42. H

    Does Linear momentum operator and angular momentum operator

    Homework Statement Does Px Lx operators commute? Homework Equations This is just me wondering The Attempt at a Solution I tried doing this and I got something weird, my friend said that when you take a derviative with respect z or something that when you try to take the derivative of...
  43. A

    What's the total angular momentum operator for a system of two particles?

    Suppose we're in two dimensions, and both particles have mass 1. Particle 1's location is given by its polar coordinates (r_1,\theta_1); likewise for Particle 2 (r_2,\theta_2). Is it true that the total angular momentum \vec{L} is just the sum of the individual angular momenta of the...
  44. Q

    Components for the angular momentum operator L

    Homework Statement Consider wavefunction psi (subscript "nlm") describing the electron in the stationary state for the hydrogen atom with quantum numbers n,l,m and the third component L3 for the orbital angular momentum operator L. What is the expectation value of L3 and of L3^2 for the...
  45. J

    Matrix represntation of angular momentum operator (QM)

    Homework Statement The matrix R(q) for rotating an ordinary vector by q around the z-axis is given by@ cosq -sinq 0 sinq cosq 0 0 0 1 From R calculate the matrix J(z). Homework Equations -The Attempt at a Solution All I know is that U(q) = exp[-iJ(z)q] is the unitary...
  46. V

    Question regarding eigenvalues of angular momentum operator

    Hello! First of all let me wish you a happy new year! This is not a homework problem, but rather a curiosity of mine. In Schwabl's Quantum Mechanics, one can find the proof of the fact that all eigenvalues of the angular momentum Lz are either integers or half-integers, raging from -l to l(l...
  47. P

    Problem with angular momentum operator math

    Homework Statement I am basically trying to show that LxLy-LyLx=i(hbar)LzHomework Equations Lx=yPz-zPy Ly=zPx-xPzThe Attempt at a Solution I get to the end where I have i(hbar)Lz-z*y*PxPz+z*xPyPz. How do I get these last two terms to cancel out? I am not too strong in operator math (it hasnt...
  48. L

    Why Does the Momentum Operator Yield Different Results in Rotations?

    Hello, sorry I am new to this forum, I hope I found the right category. I have a question about the momentum operator as in Sakurai's "modern quantum mechanics" on p. 196 If I let 1-\frac{i}{\hbar} d\phi L_{z} = 1-\frac{i}{\hbar} d\phi (xp_{y}-yp_{x}) act on an eigenket | x,y,z...
  49. T

    Matrix of angular momentum operator

    as known to all, we can find a matrix representation for every operator in quantum mechanics. for example for total angular momentum of one particle j(square) the elements are j(j+1)(square)h(bar) δmm' However I have stucked in two particle systems. for example I could not find the...
  50. E

    Fourier Analysis of Angular Momentum Operator

    Okay, if I want to do a Fourier Analysis of a wavefunction, I can use the following transform pairs for real space and momentum space. Ψ(x) = (2π hbar)^(-1/2) * ∫ dp Φ(p) exp(ipx/hbar) Φ(p) = (2π hbar)^(-1/2) * ∫ dx Ψ(x) exp(-ipx/hbar) So, what I want to...
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