Spin states of the addition of 2 spin 1/2 particles

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Discussion Overview

The discussion revolves around the addition of spin states for two spin-1/2 particles, specifically electrons. Participants explore the theoretical framework for determining the total spin states and their corresponding magnetic quantum numbers (ms). The conversation includes aspects of quantum mechanics and representation theory.

Discussion Character

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

Main Points Raised

  • Simon presents the problem of adding the spins of two electrons and seeks clarification on the derivation of the four spin states.
  • Some participants note that any linear combination of certain states can yield states with ms=0, but emphasize that the specific combinations mentioned are significant due to their relation to total spin (s).
  • One participant points out that the states aa and bb correspond to s=1 and suggests that they form an irreducible representation of the rotation group.
  • Another participant raises questions about how single spins transform under rotations, indicating a connection to the transformation properties of the spin states.

Areas of Agreement / Disagreement

There is no consensus on the derivation of the states or the implications of the transformations. Participants express differing views on the significance of the linear combinations and the representation theory involved.

Contextual Notes

Participants acknowledge limitations in the information provided, particularly regarding the value of total spin (s) and the mathematical steps necessary for a complete understanding of the problem.

Who May Find This Useful

This discussion may be useful for students and enthusiasts of quantum mechanics, particularly those interested in the addition of angular momentum and the mathematical representation of spin states.

SimonRoberts
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Hi everyone,

A QM question that I am stuck on:

Consider the addition of the spin of 2 electrons, s1=s2=½. Show
that there are one ms=1; two ms=0 and one ms=-1 states. Show the
4 spin states are given by:

a1a2
b1b2
(1/(sqrt(2)))(a1b2+b1a2)
(1/(sqrt(2)))(a1b2-b1a2)

I've used a and b in place of alpha and beta, which represent the spin up and spin down states.

I can see why there are one ms=1 and one ms=-1 states, and 2 ms=0 states, from adding the values of ms for individual particles, but am not sure on how to proceed to the second part.

Any nudge in the right direction would be great.

Cheers,

Simon.
 
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SimonRoberts said:
Hi everyone,

A QM question that I am stuck on:

Consider the addition of the spin of 2 electrons, s1=s2=½. Show
that there are one ms=1; two ms=0 and one ms=-1 states. Show the
4 spin states are given by:

a1a2
b1b2
(1/(sqrt(2)))(a1b2+b1a2)
(1/(sqrt(2)))(a1b2-b1a2)

I've used a and b in place of alpha and beta, which represent the spin up and spin down states.

I can see why there are one ms=1 and one ms=-1 states, and 2 ms=0 states, from adding the values of ms for individual particles, but am not sure on how to proceed to the second part.

Any nudge in the right direction would be great.

Cheers,

Simon.

Check out this thread: https://www.physicsforums.com/showthread.php?t=376142"

and the one referred to in it.
 
Last edited by a moderator:
SimonRoberts said:
I can see why there are one ms=1 and one ms=-1 states, and 2 ms=0 states, from adding the values of ms for individual particles, but am not sure on how to proceed to the second part.
That is because you are not given enough information. Any linear combination of the last two states is a state with ms=0.

Now, it happens that the first state has s=1 and the second state has s=0, and that is why these two particular linear combinations of ms=0 states are important. But the question does say anything about the value of s.
 
Cheers for the help.


The next part says:

Indicate which values of S and ms correspond to each of the 4 states.

So far what I've done is substituted the spin up (1 0 ) (<< vertically! don't know how to format that on here!) and spin down (0 1) into the states it says I am looking for .

e.g, (1/sqrt2)*(1 -1)

hmmm...
 
This area is very small to present these big issues. As you search on this topic you will go in deep. Online is best help for you. There you will find several e-books to solve your problems.
 
For the states aa and bb it should be clear that they can only belong to s=1. As they form a irreducible representation of the rotation group, the third state with s=1 should be obtainable from any of the two by applying a suitable rotation. How does a single spin transform under a rotation of 90 degrees? How does the product aa transform under the rotation?
 

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