What is total angular momentum: Definition and 64 Discussions
In quantum mechanics, the total angular momentum quantum number parametrises the total angular momentum of a given particle, by combining its orbital angular momentum and its intrinsic angular momentum (i.e., its spin).
If s is the particle's spin angular momentum and ℓ its orbital angular momentum vector, the total angular momentum j is
The associated quantum number is the main total angular momentum quantum number j. It can take the following range of values, jumping only in integer steps:
where ℓ is the azimuthal quantum number (parameterizing the orbital angular momentum) and s is the spin quantum number (parameterizing the spin).
The relation between the total angular momentum vector j and the total angular momentum quantum number j is given by the usual relation (see angular momentum quantum number)
The vector's z-projection is given by
where mj is the secondary total angular momentum quantum number, and the
ℏ
{\displaystyle \hbar }
is the reduced Planck's constant. It ranges from −j to +j in steps of one. This generates 2j + 1 different values of mj.
The total angular momentum corresponds to the Casimir invariant of the Lie algebra so(3) of the three-dimensional rotation group.
I went and checked is energy conserved the answer is no since the bullet velocity is not the same as it was initially ,
I am not sure how to check Linear momentum conservation and angular momentum conservation , at first I assumed that the system linear momentum would be conserved during...
Hi, I'm struggling with understanding the idea of tensor product and direct sum beyond the very basics. I know that direct sum of 2 vectors basically stacks one on top of another - I don't understand more than this . For tensor product I know that for a product of 2 matrices A and B the tensor...
I have read Classical Mechanics book by David Morin, and there are some parts that I do not understand from its derivation.
Note :
## V## and ##v## is respectively the velocity of CM and a particle of the body relative to the fixed origin , while ##v'## is velocity of the particle relative to...
I made a new version of the falling cat video, with narration. It explains how cats turn around while having zero net angular momentum during the fall:
To show that when ##[J^2, H]=0 ## the propagator vanishes unless ##j_1 = j_2## , I did (##\hbar =1##)
$$ K(j_1, m_1, j_2 m_2; t) = [jm, e^{-iHt}]= e^{iHt} (e^{iHt} jm e^{-iHt}) - e^{-iHt} jm $$
$$ = e^{iHt}[jm_H - jm] $$
So we have
$$ \langle j_1 m_1 | [jm, e^{-iHt} ] | j_2 m_2 \rangle $$
$$ =...
I understand that in a system composed of two articles, the total angular momentum is:
J = J1 + J2
From the operators: J^2, Jz, J1z, J2z，J^21z，J^22z,
I get two possible sets of operators that commute:
{J^2, Jz, J^21z, J^22z} and {J^21z, J^22z, J1z, J2z}
What I don't understand is why the...
I have a 5.0 m tractrix and am trying to work out angular momentum and total angular momentum for two hitchpoint speeds 60 & 70 km/h.
My result shows a higher total angular momentum for the lower speed.
This is not what I expected.
Here are my equations
Positions:
Derivatives
Angular velocity...
Homework Statement
A certain odd-parity shell-model state can hold up to a maximum of 4 nucleons. What are its values of J and L? What about an odd-parity shell-model state with a maximum of 6 nucleons?
Homework Equations
Parity = (-1)L
J = L+S
Total angular momentum, J, is equal to orbital...
I need websites or books that has quantum mechanical exercises in particular that finds the total angular momentum eigenvalues (for example two spin 1/2 systems). Do you know where I can train?
Hello.
Let's have two electrons with same orbital quantum number li and these electrons are in antiparallel; one electron has magnetic quantum number mi = a and and other electron has mi = -a (but we don't know which one has ml = a as we're in coupled representation to talk about total angular...
Consider a flat 2D rigid body rotating about an axis perpendicular to the body passing through a point P that is
(1) in the same plane as the body and
(2) different from the body's center of mass (CM).
In this case does Theorem 7.1 (eqn 7.9) still apply?
In the last step of the derivation of...
Consider addition of two angular momenta J = J1 + J2 , with j1=j2=1. Find the eigenstates of the total angular momentum I jm > in terms of the product states I j1 m1 j2 m2 > in two ways
(a) Make use of the tables of the Clebech _Gordan coefficients
(b) The state with m1 = m2 = 1 must be a...
Homework Statement
The problem deals with a charged (Q) rotating sphere around its axis (Ω_0) z^^ (z hat) of radius a.
I'm asked to find the total angular momentum of the EM fields.
2. The attempt at a solution
There is a solution posted to this question and I was just wondering why my...
Say, I have two spin-1/2 particles in the states characterized by ##(n=2, l=1, m_l=1, m_s=1/2)##and##(n=2, l=1, m_l=1, m_s=-1/2)##. Now, using something like the jj coupling scheme, I first find out the (orbital+spin)angular momentum for the individual particles:(i) ##| 11\rangle...
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...
Hi,
I'd like to know how to calculate parity and total angular momentum of nuclei which have even Z and even N and also Z and N are magic numbers, such as 8O8 or 20Ca20 (the number before the element is Z and the after one is N).
I don't know how to insert LaTeX formulas.
Thank you in andvance
Homework Statement
I'm just stuck on one part of a larger problem. I need to find the range of total angular momentum values for an electron in a j-j coupling scheme.
Homework Equations
j= l + and - 1/2
The Attempt at a Solution
The electrons here are in a 5d 6s configuration. So for the...
Homework Statement
Consider a particle with orbital momentum ##l=1## and spin ##s = 1/2## to be in the state described by
$$\Psi = \frac{1}{\sqrt{5}}| 1,1\rangle|\downarrow\rangle+\frac{2}{\sqrt{5}}|1,0\rangle|\uparrow\rangle$$
If the total angular momentum is measured what would be the...
Sometimes the concept of angular momentum is presented using the idea of total angular momentum J. In those cases, its always said that we have \vec{J}=\vec L + \vec S . But I can't understand how that's possible. Because orbital angular momentum operators are differential operators and so are...
I'm trying to understand why the eigen value of the total angular momentum L^{2} is \hbar ^2 l(l+1). The proofs I have seen go like this. Using the ladder operators L_{\pm} = L_x \pm iL_y we can see and the |l, m \rangle state with maximum value of m (eigen value of L_z )
\langle l,m_{max} |...
Homework Statement
The question is to find the total angular momentum of the following atoms in their ground state - Na (11 electrons), and Rb (37 electrons). That's all the info given.
Homework Equations
I have no idea - that's what I can't find!
The Attempt at a Solution
I've...
Why, for states with angular momentum l >0, do states with smaller total angular momentum J have a higher binding energy? For example, why does the 2p1/2 state have a higher binding energy than the 2p3/2 state? If the 2p orbital is filled (2p6), wouldn't Hund's third rule indicate that the...
In the section "Angular momentum in quantum mechanics" of the angular momentum page in wikipedia,one can find the following:
But we know that spin is an intrinsic property of a particle,a property that can be used for its identification.So how is it that it is not conserved?
I mean,we say...
hi all, these days i m going through Arther Beiser's modern physics book where i read about the total angular momentum in many electron atoms..now i could understand the LS coupling and spin orbit effect how they combine to form total angular momentum but if i m given the magnitude of L=2...
I am considering the occurence of an incident circularly polarised EM wave on a ground state hydrogen atom. The result is that the final state of the atomic electron transition will be at m= ±1 depending on the orientation of polarisation (LCP or RCP).
I understand that this is due to the...
Hey,
I'm not exactly sure how much this question wants, however the two in question are parts a) and b) below.
So part a) asks to write the expression for the total angular momentum J, I though this was just:
\hat{J}=\hat{J}^{(1)}+\hat{J}^{(2)}
but when we come to showing it...
Hey,
My question is on determining an 'uncertainty' quantity using total angular momentum operators in the x,y and z directions where we know the commutation relations between the x,y and z directions of the total angular momentum operators.
I'm not really sure where to go with this at...
Hey,
Also a bit confused on this one, the question is displayed below
The possible total spin values are 1,0,-1 and I know that
\mid l-s\mid\leq j\leq l+s\: ,\: -j\leq m_{z}\leq j
Where the latter inequality is in integer steps, so I'm not really sure if it is as simple as...
Hey,
My question is on the probability of attaining a particular eigenvalue for the total angular momentum operator squared for a particular state ψ, the question is shown in the image below:
I believe the eigenvalue of the total angular momentum operator squared is given by j(j+1)...
Homework Statement
How to prove that for any representation of the spin, the state e^{-i{\pi}J_x/\hbar}|j,m\rangle
is proportional to |j,-m\rangle
The exponential term is the rotation operator where J_x is the x-component of the total angular momentum operator,
and |j,m\rangle is an...
HI,i am aiming to show that 1/(2)^1/2(|spin up>|spin down> + |spin down>|spin up>) is an eigenvalue to the total angular momentum operator in a two-electron system.
I know that i should end up with getting the eigenvalues of the separate spins; L1|spin up> and
L2|spin down> and so...
I'm reading about the derivation of the lande' g-factor which comes about when one considers an electron moving about a nucleus which is put in an external magnetic field. This gives rise to a perturbative hamiltonian
H = - (\vec \mu_s + \vec \mu_s) \cdot \vec B_{ext} = \frac{e}{2m}...
Hey,
I've been able to do most of these problems but at this point I stopped because it gives 5 values for L and I just wanted to double check it is correct.
L = 2+2,...,2-2 = 4,3,2,1,0
S = 1,0
then you have to evaluate J for each L and S which will give 10 possibilities.
It's...
An electron has intrinsic angular momentum(spin) and orbital angular momentum, which gives rise to the total angular momentum of the electron, let's call it pj. When the electron is placed in an external magnetic field, the pj vector precesses around the magnetic field in one of two states(with...
Homework Statement
Calculate the allowed total angular momentum quantum numbers J for 2 protons in a nuclear shell model state j = 3/2.
Homework Equations
J = j1 + j2 where j are the total angular momenta of each proton. Protons are spin 1/2, orbital angular momentum L is not given...
Homework Statement
There are 2 electrons, one with n=1, l=0 and the other with n=2, l=1. The question asks what is the dimensionality of total angular momentum space.
Homework Equations
(2j_{1}+1)(2j_{2}+1)The Attempt at a Solution
I know for 2 electrons (spin 1/2 each) the possible values of...
I have two 2 rigid bodies with masses m1 and m2 and Moments of Inertia I1 and I2, they are connected by a free rotational joint at some point, their coms lie at c1 and c2. There's gravity.
In the beginning both have some angular velocity \omega_i
Questions:
- Total angular momentum of the...
Hi
It's easy to see that for addition of 2 angular momenta l1 and l2 , the space l1 m1 , l2 m2 is equivalent to the space of l1 l2 l m (where l is the total angular momentum).
Counting the total number of states is usually a convenient way to make sure you got the addition right.
But what...
Larmor Frequency...spin, orbital or total angular momentum??
For an electron bound in an atom, its orbital angular momentum L precesses around the magnetic field applied B, at the Larmor Frequency...or is it that its TOTAL L precesses around
B, while L precesses around this, and its spin...
Homework Statement
Positronium is formed by stopping anti-electrons in matter. It is found that the bound system is formed in two distinct states, both of which have orbital angular momentum $ L = 0 $ . Consider the possible spin configuration of this system, to determine the expected total...
Homework Statement
a) Identify the different total angular momentum states possible for the case l = 3, s = ½.
b) What is the minimum angle the angular momentum vector may make with the z-axis in the case of i) l = 3 and ii) l = 1?
c) A hydrogen atom in its ground state is subjected...
Homework Statement
Find net angular momentum for a soccer ball (moment of inertia=2/3mr^2) that's going 3.6 m/s to the right and 28.5 radians per second clockwise at the same time.
R=0.142 m
m=0.678 kg
Homework Equations
L=(perpendicular component of r)mv
L=Iw
The Attempt at a...
1. List all the possible values of total angular momentum for the following
1. 3P
2. 2D
Homework Equations
j = s + lThe Attempt at a Solution
is this problem as simple as, for 1.
for 3p, n = 3, l = 0,1,2
since it's a p orbital, l = 1
thus j = l + s, all possible values are 3/2, and 1/2
but...
Homework Statement
Given the semimajor axis of Jupiter's orbit: 5.2 AU, and the eccentricity: .048 and the period: 11.86 years, find the total angular momentum of the Jupiter-Sun system. Assume it is an isolated system - ignore interactions from other planets etc.
Homework Equations
The...
Homework Statement
How long should the day be so that the total angular momentum of the Earth
(including its rotation about its own axis and its (approximately) circular orbit around the
sun) is zero (Note: the magnitude of the angular velocity is 2pi/T where T is the period of rotation?)...
The problem statement
Icm=\frac{2}{5}mr2
\omega=v/r
m=mass of object
There's some things I don't really get about total angular momentum in a rigid body. Suppose a perfectly spherical object is rolling in uniform circular motion (ie. in a loop). Find the total angular momentum.
Homework...
Homework Statement
The spin of each of the two nucleons is 3/2 with Sz=0 . The orbital angular momentum b/w the two nucleons is zero . What values can the total angular momentum have ?
Homework Equations
The Attempt at a Solution
This was a question in my exam and i got stuck , so...
Homework Statement
consider a system with total angular momentum, l=1 in the state
|\psi>=\frac{1}{\sqrt{2}}|1>-\frac{1}{2}|0>+\frac{1}{2}|-1>
find |^{^}L_{\psi}>
Homework Equations
^{^}L_{z}|\psi>=\hbar m|\psi>The Attempt at a Solution
the basis in the wavefunction given are|1> , |0>, |-1>...
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...