Angular momentum

[stunner5000pt]

1. The problem statement, all variables and given/known data
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 <\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]?

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]

1. The problem statement, all variables and given/known data
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 <\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]?

thanks in advance for any input


How did you get that

\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

<L_{z}^2>=\int \Psi^{*}_{nlm}\left(\frac{\hbar}{i}\frac{\partial} {\partial\phi}\right)\left(\frac{\hbar}{i}\frac{\p artial}{\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...

... im 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 ...
http://physicsforums.com/showthread.php?t=157392