Proving an eigenfunction (EF) from Lx as combination of EF of Lz

[sunrah]

Homework Statement

Show that the eigenfunction of L_x can be written as a combination of eigenfunctions from L_z with the same l but different m. Using the eigenfunction
$Y_{x} = \frac{1}{\sqrt{2}}(Y^{-1}_{1} - Y^{1}_{1})$ as the eigenfunction of L_x

Homework Equations

$Y^{m}_{l} = \frac{1}{\sqrt{2 \pi}}e^{im\varphi}$ eigenfunction of L_z
$L_{z}Y^{m}_{l}= l Y^{m}_{l}$

The Attempt at a Solution

$Y_{x} = \frac{1}{\sqrt{2}}(Y^{-1}_{1} - Y^{1}_{1}) = \frac{1}{\sqrt{4\pi}}(e^{-i\varphi} - e^{i\varphi})$

$\widehat{L}_{x} = i \frac{h}{2\pi}(sin\theta \frac{d}{d\varphi} + cot\theta cos\varphi\frac{d}{d\varphi})$

$\widehat{L}_{x}Y_{x} = i \frac{h}{2\pi}cot\theta cos\varphi\frac{d}{d\varphi}\frac{1}{\sqrt{4\pi}}(e^{-i\varphi} - e^{i\varphi})$

$\widehat{L}_{x}Y_{x} = \frac{h}{2\pi}cot\theta cos\varphi\frac{1}{\sqrt{4\pi}}(e^{-i\varphi} + e^{i\varphi}) ≠ lY_{x}$

can't see where I'm going wrong

 

[vela]

Homework Statement

Show that the eigenfunction of L_x can be written as a combination of eigenfunctions from L_z with the same l but different m. Using the eigenfunction
$Y_{x} = \frac{1}{\sqrt{2}}(Y^{-1}_{1} - Y^{1}_{1})$ as the eigenfunction of L_x

Homework Equations

$Y^{m}_{l} = \frac{1}{\sqrt{2 \pi}}e^{im\varphi}$ eigenfunction of L_z

The line above is your problem. $e^{i m \varphi}$ is an eigenfunction of L_z, but the spherical harmonic $Y^m_l$ is an eigenfunction of both L_z and L².

$L_{z}Y^{m}_{l}= l Y^{m}_{l}$

The Attempt at a Solution

$Y_{x} = \frac{1}{\sqrt{2}}(Y^{-1}_{1} - Y^{1}_{1}) = \frac{1}{\sqrt{4\pi}}(e^{-i\varphi} - e^{i\varphi})$

$\widehat{L}_{x} = i \frac{h}{2\pi}(sin\theta \frac{d}{d\varphi} + cot\theta cos\varphi\frac{d}{d\varphi})$

$\widehat{L}_{x}Y_{x} = i \frac{h}{2\pi}cot\theta cos\varphi\frac{d}{d\varphi}\frac{1}{\sqrt{4\pi}}(e^{-i\varphi} - e^{i\varphi})$

$\widehat{L}_{x}Y_{x} = \frac{h}{2\pi}cot\theta cos\varphi\frac{1}{\sqrt{4\pi}}(e^{-i\varphi} + e^{i\varphi}) ≠ lY_{x}$

can't see where I'm going wrong

 

[sunrah]

The line above is your problem. $e^{i m \varphi}$ is an eigenfunction of L_z, but the spherical harmonic $Y^m_l$ is an eigenfunction of both L_z and L².

Hi, should I be using

$Y^{m}_{l} \propto P^{m}_{l}(cos\theta)e^{i m \varphi}$ ?

 

[vela]

Yes.

 

[sunrah]

Yes.

just to confirm is

$P^{-1}_{1}(cos\theta) = cos\theta$

thanks!

 

[vela]

No, that's not correct. Don't you have a list of spherical harmonics in your textbook? If not, just google it.

 

[sunrah]

thank you no I wasn't aware of spherical harmonics, but i am now!