Angular Momentum Homework: Calculating <Lz> & ΔLz

[stunner5000pt]

Homework Statement

For the spherical solution of the Schrodinger equation in spherical coordinates given the superposition of spherical harmonic functions

\frac{1}{\sqrt{14}} (Y_{1,-1}+ 2Y_{1,0}+3Y_{1,1})

calculate &lt;\hat{L_{z}}&gt; and \Delta L_{z}2. The attempt at a solution

now from my textbook (brehm and mullin)
&lt;\hat{L_{z}}&gt; = \hbar m_{l}
&lt;\hat{L_{z}}&gt; = \frac{\hbar}{14} (-1 + 4(0) +9(1)) = \frac{8}{14} \hbar = \frac{4}{7} \hbar

while &lt;L_{z}^2&gt; = (\hbar m_{l})^2

this implies that the uncertainty in the Z component of the angular momentum \Delta L_{z} =0

but i was marked wrong in my assignment for this...

am i missing something

is there a difference between &lt;\hat{L_{z}}&gt; and &lt;L_{z}&gt; [/tex]?<br /> <br /> thanks in advance for any input

 

[Meir Achuz]

<L_z^2>=(+1+0+9)/14=5/7.
Thus, you should get Delta L_z=3/7.

 

[dextercioby]

Homework Statement

For the spherical solution of the Schrodinger equation in spherical coordinates given the superposition of spherical harmonic functions

\frac{1}{\sqrt{14}} (Y_{1,-1}+ 2Y_{1,0}+3Y_{1,1})

calculate &lt;\hat{L_{z}}&gt; and \Delta L_{z}


2. The attempt at a solution

now from my textbook (brehm and mullin)
&lt;\hat{L_{z}}&gt; = \hbar m_{l}
&lt;\hat{L_{z}}&gt; = \frac{\hbar}{14} (-1 + 4(0) +9(1)) = \frac{8}{14} \hbar = \frac{4}{7} \hbar

while &lt;L_{z}^2&gt; = (\hbar m_{l})^2

this implies that the uncertainty in the Z component of the angular momentum \Delta L_{z} =0

but i was marked wrong in my assignment for this...

am i missing something

is there a difference between &lt;\hat{L_{z}}&gt; and &lt;L_{z}&gt; [/tex]?<br /> <br /> thanks in advance for any input

<br /> <br /> <br /> How did you get that<br /> <br /> \langle L_{z}^2\rangle = (\hbar m_{l})^2 ?

 

[stunner5000pt]

How did you get that

\langle L_{z}^2\rangle = (\hbar m_{l})^2 ?

my textbook says so...also

&lt;L_{z}^2&gt;=\int \Psi^{*}_{nlm}\left(\frac{\hbar}{i}\frac{\partial}{\partial\phi}\right)\left(\frac{\hbar}{i}\frac{\partial}{\partial\phi}\right)\Psi_{nlm} d\tau = (\hbar m)^2

 

[dextercioby]

Yes, but in your case the state is no longer \langle r, \theta, \varphi|n, l, m \rangle , but a linear combination of spherical harmonics. So blindly using a fomula in the book is a wrong decision...

 

[stunner5000pt]

Yes, but in your case the state is no longer \langle r, \theta, \varphi|n, l, m \rangle , but a linear combination of spherical harmonics. So blindly using a fomula in the book is a wrong decision...

... I am not sure how to proceed then...

do i 'prove' it?

thanks for the help so far...

but could you look at this thread of mine... its in more of ugent need ...
https://www.physicsforums.com/showthread.php?t=157392