Raising and lowering operators on a ket?

[philip041]

How do you use the S(+) and S(-) operators on integer kets, |1>, |-1>, |0>?

I'm told the outcome of the ones which aren't zero will be something like h(bar)/sqrt(2) * |ket>

Confused!? I thought operators are 2 x 2 matrices...

Any help much appreciated,

Philip

 

[malawi_glenn]

Operating with S+ on |+1> Will give you Zero if you are operating on S = 1 states.. you are not so clear in your question here, but I assume you have S = 1 states here..

Operating with S- on |-1> = 0 as well.

Operating with S- on |0> you use formulas:

http://www.mt.luth.se/~nikle/Education/MTF067/Lectures/PL13.pdf

(just replace L with S , these operators work the same no matter what kind of angular momentum you have)

 

[malawi_glenn]

And since you have S = 1 states here, you should have a 3x3 matrix

 

[philip041]

Cheers for the answer, I have a question though:

In the third slide you posted, what does the l = 3 correspond to? The question I am looking at says for spin 1. What is l and what is m. Should my l in the examples always be 1, as they are spin 1, and my m be either -1, 0 or 1?

 

[malawi_glenn]

yes :-)

 

[philip041]

cool, so if it were for spin 1/2, would l = 1/2 and m = 1/2, 0 or 1/2?

 

[malawi_glenn]

No, think! For a spin 1/2 particle, which possible z-projections exists? What is the general relation for an Angularmomentum J and its z-projections m_J ?

 

[philip041]

none?

 

[malawi_glenn]

Given an angular momentum $j$, which can have values: 0,1/2 ,1, 3/2 ,2 ,5/2, ...

the z-component, $m_j$, can obtain values: $m_j = -j, -j +1, \ldots , j-1, j$

So if j = 1/2, what possible values can you have of the z-component? ;-)

 

[philip041]

-5/2 to 5/2, why didn't i say that the first time? sorry for being dim

 

[malawi_glenn]

-5/2 to 5/2, why didn't i say that the first time? sorry for being dim

No you must have missunderstood. In your case, j = 1/2, what values of m_j can you have?

 

[philip041]

no that's wrong

 

[philip041]

-3/2 3/2

miss print

 

[malawi_glenn]

-3/2 3/2

miss print

now you are REALLY making a fool out of yourself ;-)

If j = 1/2 and the general "formula" for possible m_j's for a given j are -j, -j+1, .. ,j -1, j

What values of m_j can you have for j = 1/2 ?

 

[philip041]

3/2 to 3/2?

 

[philip041]

missed out the minus by mistake.

 

[malawi_glenn]

3/2 to 3/2?

are you guessing or using the formula?

if you are guessing, stop doing that and apply formula.

if you use formula, tell me how you do it, you are doing something REALLY wrong.

According to formula, for a given j, the maximum m_j is j, and the minumum is -j.

 

[philip041]

1/2 and -1/2

 

[malawi_glenn]

1/2 and -1/2

good :-)