Why does |s=1,m=1> equal |++> ?

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Discussion Overview

The discussion revolves around the relationship between the quantum states |s=1, m=1> and |++> in the context of total spin angular momentum for a system of two spin-1/2 particles. Participants explore the definitions and calculations related to spin, including the addition of angular momenta and the implications of different spin states.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • Some participants express confusion about the relationship between the states |s=1, m=1> and |++>, noting that |++> represents both particles being spin-up.
  • One participant explains that the total spin is defined based on the projections of individual spins, where spin-up corresponds to +1/2 and spin-down to -1/2.
  • Another participant deduces that m=1 from the state |++>, but questions how to determine that s=1.
  • It is mentioned that the addition of angular momenta follows specific rules, leading to possible total spin values of 0 and 1 for two spin-1/2 particles.
  • Participants discuss the concept of treating angular momenta as vectors in quantum mechanics, while emphasizing that they are not vectors in the classical sense.

Areas of Agreement / Disagreement

Participants generally agree on the definitions and calculations related to spin states, but there remains uncertainty about how to conclusively determine the total spin value s from the given states.

Contextual Notes

There are limitations in the discussion regarding the assumptions made about angular momentum addition and the specific definitions of the quantum states involved. Some mathematical steps and definitions may not be fully resolved.

Ethan0718
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Sorry...I know it's a very stupid problem .. =.=

But i just don't know why...

ihttp://imageshack.us/photo/my-images/594/imagewyt.jpg/

it's mentioned at p205 of Modern Quantum Mechanics by J.J.Sakurai...=======

sorry... i found i post at wrong place after i received the PM from PF mentor...

please forgive me @@... i won't do this again!..
 
Last edited:
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Ethan0718 said:
Sorry...I know it's a very stupid problem .. =.=

But i just don't know why...

ihttp://imageshack.us/photo/my-images/594/imagewyt.jpg/

it's mentioned at p205 of Modern Quantum Mechanics by J.J.Sakurai...

Because it is defined that way. You are talking about the total spin angular momentum and its projection for a system consisting of two spin-1/2 particles. Spin-up or "+" denotes a projection of +1/2 and spin-down "-" denotes a projection of -1/2. If you consider the case where both particles are spin-up (i.e. |++>), then the total spin is 1, and the projection m=(1/2)+(1/2)=1. If you consider the case where both are spin-down (i.e. |-->), then the total spin is still 1, but the projection is now m=(-1/2)+(-1/2)=-1.

Does that help?
 
Thank you very much...but... I still have some questions >"<

i know the following information

S2 = (S1+S2)2 : s(s+1) * h'
SZ = S1Z + S2Z : m * h'
S1Z : m1 * h'
S2Z : m2 * h'

h' means h bar

I can deduce m = 1 from the fact that |++> means two spin up ; 1/2 + 1/2 = 1

but i don't know how to know s = 1... @@
 
Last edited:
Ethan0718 said:
Thank you very much...but... I still have some questions >"<

i know the following information

S2 = (S1+S2)2 : s(s+1) * h'
SZ = S1Z + S2Z : m * h'
S1Z : m1 * h'
S2Z : m2 * h'

h' means h bar

I can deduce m = 1 from the fact that |++> means two spin up ; 1/2 + 1/2 = 1

but i don't know how to know s = 1... @@

Spins are angular momenta, so you have to follow the rules for their addition. Specifically, any two angular momenta j1 and j2, can add together to give a resultant total angular momentum in the set, {(j1+j2, j1+j2 - 1, j1+j2 - 2, ... , j1- j2+1, j1 - j2}.

In QM, angular momenta can often be treated "as if" they are vectors (although they are *not* vectors), particularly with respect to coupling (addition) of multiple angular momenta. Note that the above expression is just the usual triangle inequality for vector addition, with the additional stipulation that angular momentum is quantized such that the smallest allowable step size is hbar.

For this case, we have s1=s2=1/2, so possible values for s are 0 and 1.
 

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