Formalism and Angular Momentum Expectation Values

[brooke1525]

I seem to be having a rather difficult time understand all the details of the notation used in this quantum material. If I'm given the eigenvalue equations for L$$^{2}$$ and $$L_z$$ and $$L_{\stackrel{+}{-}}$$ for the state |$$\ell$$,m>, how do I compute <$$L_{x}$$> using bra-ket formalism? I know that $$L_x$$ = (1/2)($$L_+$$ + $$L_-$$).

What I've got so far:

Need to compute <$$\ell$$,m|$$L_x$$|$$\ell$$,m>.

= (1/2)(<$$\ell$$,m)($$L_+$$ + $$L_-$$)($$\ell$$,m>)

=?

Algebraically, what's the next step?

Thanks in advance,
AB

 

[malawi_glenn]

$$(1/2)\cdot (<l,m|L_+|l,m> +<l,m|L_-|l,m> ) =$$

the result should be pretty obious ;-)

Anyway, this is general:

Suppose we have, two operators A and B:

<psi_i|(A+B)|psi_j> = <psi_i|A|psi_j> + <psi_i|B|psi_j>

Next time you have homework related questions, ask them in the homework section.

 

[Entropia]

I understand your struggle with the notation and concepts used in quantum mechanics. Let me try to provide a response that will help clarify the steps needed to compute <L_{x}> using bra-ket formalism.

First, we need to understand that the notation <\ell,m| represents the bra vector, while |\ell,m> represents the ket vector. These are simply different representations of the same vector in a bra-ket notation. In this case, the state |\ell,m> represents the state with angular momentum L_{z} and its eigenvalue m.

Next, we need to use the fact that L_{x} = (1/2)(L_{+} + L_{-}). This means that we can rewrite the expression <L_{x}> as:

<L_{x}> = (1/2)(<\ell,m|L_{+}|\ell,m> + <\ell,m|L_{-}|\ell,m>)

Now, we can use the eigenvalue equations for L^{2} and L_{z} to simplify the expressions for L_{+} and L_{-}. Recall that L_{+} = L_{x} + iL_{y} and L_{-} = L_{x} - iL_{y}. Using these equations, we can rewrite the expression for <L_{x}> as:

<L_{x}> = (1/2)(<\ell,m|L_{x} + iL_{y}|\ell,m> + <\ell,m|L_{x} - iL_{y}|\ell,m>)

From here, we can use the properties of the bra-ket notation to simplify the expressions further. Specifically, we can use the fact that <\ell,m|L_{x}|\ell,m> = \ell(\ell+1)<\ell,m|\ell,m> and <\ell,m|L_{y}|\ell,m> = -i\ell(\ell+1)<\ell,m|\ell,m>. This allows us to rewrite the expression for <L_{x}> as:

<L_{x}> = (1/2)(\ell(\ell+1)<\ell,m|\ell,m> + i\ell(\ell+1)<\ell,m|\ell,m> - i\ell(\ell+1)<\ell,m|\ell,m> + \ell(\ell+1)<\ell