Angular Momentum Homework: Calculating <Lz> & ΔLz

[stunner5000pt]

Homework Statement

For the spherical solution of the Schrödinger 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 $<\hat{L_{z}}>$ and $\Delta L_{z}$2. The attempt at a solution

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

while $$<L_{z}^2> = (\hbar m_{l})^2$$

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 $<\hat{L_{z}}>$ and [itex]<L_{z}> [/tex]?<br /> <br /> thanks in advance for any input[/itex]

 

[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 Schrödinger 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 $<\hat{L_{z}}>$ and $\Delta L_{z}$


2. The attempt at a solution

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

while $$<L_{z}^2> = (\hbar m_{l})^2$$

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 $<\hat{L_{z}}>$ and [itex]<L_{z}> [/tex]?<br /> <br /> thanks in advance for any input[/itex]

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

 

[stunner5000pt]

How did you get that

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

my textbook says so...also

$$<L_{z}^2>=\int \Psi^{*}_{nlm}\left(\frac{\hbar}{i}\frac{\partial}{\partial\phi}\right)\left(\frac{\hbar}{i}\frac{\partial}{\partial\phi}\right)\Psi_{nlm} d\tau = (\hbar m)^2$$

 

[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