Why a particle with spin=0 can't posses a magnetic dipole moment?

[happyparticle]

Homework Statement:

Why a particle with spin=0 can't posses a dipole moment?

Relevant Equations:

$\langle j'|| \vec{J}|| j \rangle = \hbar \sqrt{j(j+1)} \delta_{jj'}$

Hi,
I would like to know why a particle with spin=0 can't posses a magnetic dipole moment?

Using Wigner-Eckart theorem for $\langle j,1,m,0|j,m \rangle$ I get $\langle j'|| \vec{J}|| j \rangle = \hbar \sqrt{j(j+1)} \delta_{jj'}$

It seems like the right hand side is the magnetic dipole moment. Thus, j must be 0 for a particle with spin =0.

However, I'm not so sure to understand what j means.
I'll try to explain what I understand.
$\vec{J}$ is the total angular momentum and $\vec{J} = \vec{L} + \vec{S}$, where $\vec{S}$ is the spin.
Furthermore, we have 2 systems (2 particles) $\langle j'|| \vec{J}|| j \rangle = \hbar \sqrt{j(j+1)} \delta_{jj'}$
where $j_1 = j, j_2 = 1 , m_1 = m, m_2 =0$ and $j = j_1 + j_2, m = m_1 + m_2$

Are $j_1, j_2$ the total angular momentums for each particle and what exactly are $m_1, m_2$? Are they quantum number $m_s$ ?

If the spin of the particle is null then $\vec{S} = 0$ which mean $\vec{J} = \vec{L}$

This is all I know. I can't show that j=0.

I hope this is clear...

thank you

 

[topsquark]

Homework Statement:: Why a particle with spin=0 can't posses a dipole moment?
Relevant Equations:: $\langle j'|| \vec{J}|| j \rangle = \hbar \sqrt{j(j+1)} \delta_{jj'}$

Hi,
I would like to know why a particle with spin=0 can't posses a magnetic dipole moment?

Using Wigner-Eckart theorem for $\langle j,1,m,0|j,m \rangle$ I get $\langle j'|| \vec{J}|| j \rangle = \hbar \sqrt{j(j+1)} \delta_{jj'}$

It seems like the right hand side is the magnetic dipole moment. Thus, j must be 0 for a particle with spin =0.

However, I'm not so sure to understand what j means.
I'll try to explain what I understand.
$\vec{J}$ is the total angular momentum and $\vec{J} = \vec{L} + \vec{S}$, where $\vec{S}$ is the spin.
Furthermore, we have 2 systems (2 particles) $\langle j'|| \vec{J}|| j \rangle = \hbar \sqrt{j(j+1)} \delta_{jj'}$
where $j_1 = j, j_2 = 1 , m_1 = m, m_2 =0$ and $j = j_1 + j_2, m = m_1 + m_2$

Are $j_1, j_2$ the total angular momentums for each particle and what exactly are $m_1, m_2$? Are they quantum number $m_s$ ?

If the spin of the particle is null then $\vec{S} = 0$ which mean $\vec{J} = \vec{L}$

This is all I know. I can't show that j=0.

I hope this is clear...

thank you

Who told you that a spin 0 particle could not have a magnetic moment? It cannot have a spin magnetic moment (intrinsic magnetic moment) but, as you showed above, it can have an orbital magnetic moment.

-Dan

 

[happyparticle]

Why exactly it cannot have a spin magnetic moment?

Can I show it from $\langle j'|| \vec{J}|| j \rangle = \hbar \sqrt{j(j+1)} \delta_{jj'}$ ?

 

[topsquark]

Why exactly it cannot have a spin magnetic moment?

Can I show it from $\langle j'|| \vec{J}|| j \rangle = \hbar \sqrt{j(j+1)} \delta_{jj'}$ ?

Because s = 0...

-Dan

 

[happyparticle]

I mean, is it a relationship between s and j ?

 

[PeroK]

I mean, is it a relationship between s and j ?

$j = l + s$

 

[topsquark]

I mean, is it a relationship between s and j ?

The spin magnetic moment of a particle is given by
$\boldsymbol{ \mu } = g \dfrac{e}{2 m} \textbf{S}$

If s = 0 then $\boldsymbol{ \mu } \mid \psi \rangle = g \dfrac{e}{2 m} \textbf{S} \mid \psi \rangle = \textbf{0} \mid \psi \rangle$

-Dan

 

[happyparticle]

Ah I see. I thought that S was the eigenvector and the eigenvalue wasn't necessarily s. thus if s=0 then S wasn't necessarily 0. I think that as usual I had misunderstood.
Thank you

 

[PeroK]

Ah I see. I thought that S was the eigenvector and the eigenvalue wasn't necessarily s. thus if s=0 then S wasn't necessarily 0. I think that as usual I had misunderstood.
Thank you

I can't make any sense of this. Are you sure you understand the concepts of operators, eigenvectors and eigenvalues.

 

[happyparticle]

I thought so, but now you make me doubt.

 

[malawi_glenn]

I mean, is it a relationship between s and j ?

You wrote it in the original post

 

[happyparticle]

You wrote it in the original post

I was replying to topsquark. I meant if s = 0 why j is automatically 0, that kind of relation.

 

[topsquark]

I was replying to topsquark. I meant if s = 0 why j is automatically 0, that kind of relation.

I never said j = 0 because s = 0. j = l + s. If s = 0 then j = l. If l is not zero then the state has an angular magnetic moment, just not a spin angular magnetic moment. Only if j = 0 does the state have no angular magnetic moment.

-Dan

 

[kuruman]

I would like to know why a particle with spin=0 can't posses a magnetic dipole moment?

When I first saw this question, I thought it was about a free particle with zero spin. Then @happyparticle brought in the Wigner-Eckart theorem stated talking about orbital angular momentum. Orbital angular momentum presupposes a nucleus which the supposedly zero-spin particle must be orbiting. $\mathbf{S}$ as in $\mathbf{J}=\mathbf{L}+\mathbf{S}$ is the total spin in a many-electron atom and $\mathbf{S}=0$ has nothing to do with a particle with spin = 0 in the original question.

I think the simplest answer to the original question is that spin is an intrinsic property of particles and so is the magnetic moment associated with the spin of the particle. Asking why a particle with zero spin has no magnetic moment is like asking why a bald man has no hair on his head.

 

[Vanadium 50]

Tell me. If S=0, in what direction does the magnetic moment point?

 

[kuruman]

Tell me. If S=0, in what direction does the magnetic moment point?

If a man is bald, what color is his hair?

 

[topsquark]

If a man is bald, what color is his hair?

Fish.

-Dan

 

[jim mcnamara]

Something is fishy here. Let's lock the thread for a while until the smell dissipates....