Matrix Elements: J_+*J_- = J^2 - J_z^2+hbarJ_z?

[Nusc]

Well, $\langle j,m|S^2_{-}|j,m\rangle=\langle j,m|(S^2_{-}|j,m\rangle)$ and $S^2_{-}|j,m\rangle=S_{-}(S_{-}|j,m\rangle)=$___?

Normally J_+*J_- is substituted as J^2 - J_z^2+hbarJ_z

As described above, J_+(J_-J_+), wouldn't the content in parenthesese just cancel out to 1 ?

 

[SpectraCat]

Well, $\langle j,m|S^2_{-}|j,m\rangle=\langle j,m|(S^2_{-}|j,m\rangle)$ and $S^2_{-}|j,m\rangle=S_{-}(S_{-}|j,m\rangle)=$___?

Normally J_+*J_- is substituted as J^2 - J_z^2+hbarJ_z

As described above, J_+(J_-J_+), wouldn't the content in parenthesese just cancel out to 1 ?

I am going to need you to define some terms there. What is S_-, the lowering operator for spins? What is j? Is it the generic total angular momentum quantum number, or is it for a specific case, such as an atom, where you are considering the coupling of orbital angular momentum (l) to spin (s)? Are you dealing specifically with states of spin 1/2 particles, or is it a generic case?

 

[Nusc]

What is S_-, the lowering operator for spins?
Yes.

What is j? Is it the generic total angular momentum quantum number, or is it for a specific case, such as an atom, where you are considering the coupling of orbital angular momentum (l) to spin (s)?
j is the generic total ang mom

Are you dealing specifically with states of spin 1/2 particles, or is it a generic case?

Spin 1 particle

 

[SpectraCat]

What is S_-, the lowering operator for spins?
Yes.

What is j? Is it the generic total angular momentum quantum number, or is it for a specific case, such as an atom, where you are considering the coupling of orbital angular momentum (l) to spin (s)?
General

Are you dealing specifically with states of spin 1/2 particles, or is it a generic case?

Spin 1 particle

So what happens in general when the spin lowering operator is applied to a state |j,m>? What are the properties of the scalar product of two angular momentum states?

Answer those questions and you will have your answer.

 

[Nusc]

S_-|j,m>=|j,m-1>

S_+|j,m-1> = |j,m> Is that right? Back to my question, J_+*J_- is substituted as J^2 - J_z^2+hbarJ_z which isn't 1. why is this?



<j',m'|S_+^3|j,m> = c_jm^+c_jm+1^+c_jm+2^+<j',m'|j,m+3> delta_j',j delta_m',m+3

 

[Nusc]

MY secnod question is the following
S_-S_+|+-> = h^2 |-+>

Where does the eigenvalue h^2 come from?

 

[SpectraCat]

S_-|j,m>=|j,m-1>

S_+|j,m-1> = |j,m> Is that right?

Close, there should be a constant multiplying the state on the rhs of each equation above. But anyway, that isn't what you posted originally .. your expression from your OP has two applications of the lowering operator ...

Back to my question, J_+*J_- is substituted as J^2 - J_z^2+hbarJ_z which isn't 1. why is this?

just to clarify, I guess you mean:
$$J_{+}J{-}=J^{2}-J_{z}^{2} + \hbarJ_{z}$$

That is correct, but why would you think it equals one in the general case? That expression is an operator, so you should look at what it does to a state ... since any angular momentum eigenstate of each of the operators in that expression, this is pretty trivial:

$$[J^{2}-J_{z}^{2} + \hbarJ_{z}]|j,m> = J^{2}|j,m>-J_{z}^{2}|j,m> + \hbarJ_{z}|j,m> = \hbar^{2}[j(j+1) - m^{2} + m]|j,m>$$

<j',m'|S_+^3|j,m> = c_jm^+c_jm+1^+c_jm+2^+<j',m'|j,m+3> delta_j',j delta_m',m+3

Not sure what that last expression has to do with anything, but FWIW it looks basically correct (although it could do with some tex formatting .

 

[Nusc]

What constant are you referring to?

 

[SpectraCat]

MY secnod question is the following
S_-S_+|+-> = h^2 |-+>

Where does the eigenvalue h^2 come from?

I don't really understand that ... what is |+->?

In any case, hbar is the quantum mechanical unit of angular momentum. So the result of applying any angular momentum operator to a quantum state will always be multiplied by hbar. In the case above, the operator is a product of two angular momentum operators, so you get hbar*hbar from the application of that operator to a state.

 

[SpectraCat]

What constant are you referring to?

Perhaps you should check out a website where all of this is worked out sequentially in some detail .. for example, check out: http://farside.ph.utexas.edu/teaching/qmech/lectures/node72.html

 

[Nusc]

I don't really understand that ... what is |+->?

In any case, hbar is the quantum mechanical unit of angular momentum. So the result of applying any angular momentum operator to a quantum state will always be multiplied by hbar. In the case above, the operator is a product of two angular momentum operators, so you get hbar*hbar from the application of that operator to a state.

+ spin up

- spin down



Do you know how S_z would act on such a state given what I said earlier?

S_-S_+|+-> = h^2 |-+>

In addition

S_+S_- |+-> = 0

S_z^-*S_z^+|+-> = ?

it supposed to be -hbar^2/4 not sure why.

 

[Nusc]

Close, there should be a constant multiplying the state on the rhs of each equation above. But anyway, that isn't what you posted originally .. your expression from your OP has two applications of the lowering operator ...

And just to clarify, S_-*S_+ = 1 without constants?

 

[Nusc]

ALso:

S_+S_- + S_-S_+ = S^2-S_z^2 but this book I found says its also equal to 1 for a spin 1/2 particle?? Why?

They also have

S_+S_z = -1/2 S_+

why is that one true?

 

[SpectraCat]

And just to clarify, S_-*S_+ = 1 without constants?

No, ... you gave the correct expression for this *operator* earlier ... I showed you how to apply it to an eigenstate. It most certainly does not equal one.

EDIT: However, since I now know you are talking about spin 1/2 eigenstates, I think you will find that the result of applying this to the eigenstate will give you 1, (well, actually hbar², as I have said).

Did you check out the website I linked?

 

[SpectraCat]

ALso:

S_+S_- + S_-S_+ = S^2-S_z^2 but this book I found says its also equal to 1 for a spin 1/2 particle?? Why?

Again, those are operators on the right hand side .. a spin 1/2 particle is an eigenstate of both of those operators ... plug in the state and evaluate the expression. However, it does not evaluate to one when applied to a spin 1/2 eigenstate .. work it out and see what you get. You are missing some factors of hbar in there as well.

They also have

S_+S_z = -1/2 S_+


why is that one true?

again, missing some factors of hbar ... and that is a possible result, but not the only one.

 

[Nusc]

Again, those are operators on the right hand side .. a spin 1/2 particle is an eigenstate of both of those operators ... plug in the state and evaluate the expression. /QUOTE]

Is the same true for a spin 3/2?