Question

when is angular momentum maximum and when is it minimum​

Answer 1

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PLZ write full question bro

Answer 2

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I want to work out the maximum and minimum values for mℓ. I know that λ≥mℓ, therefore mℓ is bounded. In the lectures notes there is the following assumption:

L+^|λ,mmax⟩=|0⟩L−^|λ,mmin⟩=|0⟩

I think I understand this. Since the action of the ladder operators is the keep the value of λ and raise (or lower) mℓ, you cannot "go up" from mmax or down from mmin. However, I do not understand why the result of the operation should be |0⟩.

It turns out we can write the produt of L−^L+^ as:

L−^L+^=L2^−L2z^−ℏLz^

Then we we evalute the following expression:

L2^|λ,mmax⟩=(L−^L+^+L2z^+ℏLz^)|λ,mmax⟩

Since L+^|λ,mmax⟩=|0⟩, then L−^L+^|λ,mmax⟩=L−^|0⟩=|0⟩. And Lz^|λ,mmax⟩=ℏmℓ|λ,mmax⟩. These two relations imply:

L2^|λ,mmax⟩=ℏ2mmax(mmax+1)|λ,mmax⟩

Now I want to know how to compute L2^|λ,mmin⟩ since my lecture notes only state the result. My problem is that I will have L−^L+^|λ,mmin⟩, but I can no longer say that L+^|λ,mmin⟩=|0⟩. I tried to compute L+^L−^ and try to plug in the expression, but I had no success. How can I solve this?

PS. This is not homework, I'm just trying to derive the expression stated in the lecture notes.

quantum-mechanics angular-momentum operators representation-theory textbook-erratum

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edited Aug 2 '14 at 16:08

rob♦

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asked Jun 1 '14 at 3:26

Thiago

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L+^|λ,mmin⟩=|0⟩ is not 0, it should be sth like L+^|λ,mmin⟩=|1⟩, the next level (of ang. momentum) L+ and L- operators raise or lower the levels of ang. momentum, also cannot raise above max and cannot lower below min. – Nikos M. Jun 1 '14 at 3:57  

I have already mentioned this in my question. – Thiago Jun 1 '14 at 5:05

The result is zero because this level has no states of the system that are compatible, i.e zero number of states of the system. Does this answer your question? Since L+ raised mmin to one (from zero), applying L- will lower the state back to zero – Nikos M. Jun 1 '14 at 5:08  

No, it does not. The result is a ket vector |0⟩ not the number 0. And my question was on how to compute L2^|λ,mmin⟩ – Thiago Jun 1 '14 at 5:11

Ket vector representing the number of states in this level, correct? It is a quantized harmonic oscillator-type creation-annihilation model, if i am not mistaken – Nikos M. Jun 1 '14 at 5:13  

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