Raising and lowering operators on a ket?

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Discussion Overview

The discussion revolves around the application of raising and lowering operators, specifically S(+) and S(-), on integer kets such as |1>, |0>, and |-1>. Participants explore the outcomes of these operations in the context of angular momentum states, particularly focusing on spin-1 and spin-1/2 systems.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant questions how to use the S(+) and S(-) operators on integer kets and expresses confusion about the expected outcomes.
  • Another participant states that operating with S(+) on |+1> results in zero for S = 1 states and provides a link to a resource for further clarification.
  • It is mentioned that for S = 1 states, a 3x3 matrix representation is appropriate.
  • A participant inquires about the meaning of l in the context of spin and whether it should always be 1 for spin-1 states.
  • There is a discussion about the possible values of m for spin-1/2 particles, with participants debating the correct application of the formula for angular momentum projections.
  • Multiple participants express uncertainty and confusion regarding the values of m_j for different j values, leading to corrections and clarifications about the range of possible projections.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the application of the operators or the values of m_j for different spin states, with ongoing debate and corrections indicating multiple competing views.

Contextual Notes

Some participants reference specific formulas and resources, but there are unresolved questions about the definitions of l and m in the context of angular momentum, as well as the correct interpretation of the operators' effects on the kets.

philip041
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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
 
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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)
 
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And since you have S = 1 states here, you should have a 3x3 matrix
 
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?
 
yes :-)
 
cool, so if it were for spin 1/2, would l = 1/2 and m = 1/2, 0 or 1/2?
 
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 ?
 
none?
 
Given an angular momentum [itex]j[/itex], which can have values: 0,1/2 ,1, 3/2 ,2 ,5/2, ...

the z-component, [itex]m_j[/itex], can obtain values: [itex]m_j = -j, -j +1, \ldots , j-1, j[/itex]

So if j = 1/2, what possible values can you have of the z-component? ;-)
 
  • #10
-5/2 to 5/2, why didn't i say that the first time? sorry for being dim
 
  • #11
philip041 said:
-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?
 
  • #12
no that's wrong
 
  • #13
-3/2 3/2

miss print
 
  • #14
philip041 said:
-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 ?
 
  • #15
3/2 to 3/2?
 
  • #16
missed out the minus by mistake.
 
  • #17
philip041 said:
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.
 
  • #18
malawi_glenn;2001489 If j = 1/2 and the general "formula" for possible m_j's for a given j are -j said:
1/2 and -1/2
 
  • #19
philip041 said:
1/2 and -1/2

good :-)
 

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