What is Angular momentum operator: Definition and 49 Discussions
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.
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 =...
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...
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...
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...
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)##...
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...
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} -...
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...
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...
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...
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...
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...
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...
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...
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}##...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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 =...
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...
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...
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...
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...
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/
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...