Quantum Mechanics Ladder Operator and Dirac Notation

[brooke1525]

Homework Statement

I'm given the eigenvalue equations
L$$^{2}$$|$$\ell$$,m> = h$$^2\ell(\ell + 1)|\ell$$,m>
$$L_z$$|$$\ell$$,m> = m|$$\ell$$
$$L_{\stackrel{+}{-}}$$|$$\ell$$,m> = h$$\sqrt{(\ell \stackrel{-}{+} m)(\ell \stackrel{+}{-} m + 1)}$$|$$\ell$$, m $$\stackrel{+}{-}$$ 1>

Compute <$$L_{x}$$>.

Homework Equations

Know that $$L_x = (1/2)(L_+ + L_-)$$.

The Attempt at a Solution

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

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

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

I don't think I have a good grasp on how to work with dirac notation, so this as far as I can get before I get stuck. Thanks in advance for any help!

 

[SporadicSmile]

You are on the right lines. Now you need to evaluate what the L₊ operator acting on the state |l,m> is, and the same for L_-. then you will have something which has the form < | > + < | >. From this point, the answer is straightforward.

Thats about as precise as i can be without doing it for you, i think, sorry if its a little abstract.

 

[brooke1525]

When I'm evaluating the ladder operators for the state |l,m>, how do I deal with the fact that the corresponding eigenvalue is for |l,m+-1>?

 

[malawi_glenn]

what is <l=1,m=1|l=1,m=0> for instance?

or what was your latest question reffering to?

 

[brooke1525]

That's what my question was about. I don't understand what it means to have <l_a, m_a | l_b, m_b>. If they were the same l's and m's, it should be 1, yes? But otherwise, I'm lost...

 

[malawi_glenn]

That's what my question was about. I don't understand what it means to have <l_a, m_a | l_b, m_b>. If they were the same l's and m's, it should be 1, yes? But otherwise, I'm lost...

If they are not same state, then it is ?... guess ;-)

 

[brooke1525]

Just 0?

 

[malawi_glenn]

YES! :-)

|L_a , M_b > is same state as |L_c, M_d> if and only if a=c and b=d

 

[brooke1525]

What is the physical interpretation of <quantum numbers "a" | quantum numbers "b">? Is it the probability that a system in the "b" state will simultaneously be in the "a" state?

 

[malawi_glenn]

$$<a|b> = \int _V\psi _a(\vec{r})^*\psi_b(\vec{r})d^3r$$

Where the $\psi _a(\vec{r})$ is the Wavefunction for the state a in position space.

Now you see why dirac formalism is superior, you don't need to know or write all the eigenfunction to the operators and so on, just label the state by the things that are important for you at the moment.

If you wanted to write your angular momentum states bra-ket in "old style":
$$<L_c , M_d |L_a , M_b > = \int Y_{L_c}^{M_d}^*(\theta,\phi)Y_{L_a}^{M_b}}(\theta,\phi)\sin \theta d\theta d\phi$$

where $Y_{L_c}^{M_d}(\theta,\phi)$ is the so called "Spherical harmonic" function.

 

[brooke1525]

Ohhh, okay. Thanks so much for all your help, I think things are becoming much clearer!

 

[malawi_glenn]

good luck and please ask questions here again.