Raising and lowering operators / spherical harmonics

[nowits]

This isn't exactly a part of any problem, but a part of a generic principle. I don't understand the use of raising and lowering operators.

$$L_{^+_-}=\hbar e^{^+_- i l \phi}({^+_-}\frac{\partial}{\partial \theta}+ i cot \theta \frac{\partial}{\partial \phi})$$

So how does one use $$L_{^+_-}Y_l^m$$ to gain $$Y_l^{m{^+_-}1}$$

 

[malawi_glenn]

you can use this definition to apply it to the general expression for the spherical harmonics. The general expression for an arbitrarty spherical harmonics can be found either in your textbook, or google it.

 

[nowits]

Do you mean that I simply:
$$\hbar e^{^+_- i l \phi}({^+_-}\frac{\partial Y}{\partial \theta}+ i cot \theta \frac{\partial Y}{\partial \phi})$$

But what happens to hbar? There isn't supposed to be any hbars in Y's?

 

[malawi_glenn]

it is just a constant..

 

[malawi_glenn]

Did you get it? $$\hbar$$ is only a multiplcative constant, same as $$e^{^+_- i l \phi}$$. So you find out what the derivative operators does on the general spherical harmonic.

 

[nrqed]

Do you mean that I simply:
$$\hbar e^{^+_- i l \phi}({^+_-}\frac{\partial Y}{\partial \theta}+ i cot \theta \frac{\partial Y}{\partial \phi})$$

But what happens to hbar? There isn't supposed to be any hbars in Y's?

applying, say $L_+ Y^l_m$ does not give $Y^l_{m+1}$, it gives a constant times $Y^l_{m+1}$. You may find the general constant in eq 4.121 of Griffiths, for example.

 

[nowits]

Yes, I think I understand it now.

Thank you both.