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Angular momentum

  • #1
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2

Homework Statement


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

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

calculate [itex] <\hat{L_{z}}> [/itex] and [itex] \Delta L_{z} [/itex]


2. The attempt at a solution

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

while [tex] <L_{z}^2> = (\hbar m_{l})^2 [/tex]

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

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

am i missing something

is there a difference between [itex] <\hat{L_{z}}> [/itex] and [itex] <L_{z}> [/tex]?

thanks in advance for any input
 

Answers and Replies

  • #2
Meir Achuz
Science Advisor
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<L_z^2>=(+1+0+9)/14=5/7.
Thus, you should get Delta L_z=3/7.
 
  • #3
dextercioby
Science Advisor
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540

Homework Statement


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

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

calculate [itex] <\hat{L_{z}}> [/itex] and [itex] \Delta L_{z} [/itex]


2. The attempt at a solution

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

while [tex] <L_{z}^2> = (\hbar m_{l})^2 [/tex]

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

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

am i missing something

is there a difference between [itex] <\hat{L_{z}}> [/itex] and [itex] <L_{z}> [/tex]?

thanks in advance for any input

How did you get that

[tex] \langle L_{z}^2\rangle = (\hbar m_{l})^2 [/tex] ?
 
  • #4
1,444
2
How did you get that

[tex] \langle L_{z}^2\rangle = (\hbar m_{l})^2 [/tex] ?
my textbook says so....


also

[tex] <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 [/tex]
 
Last edited:
  • #5
dextercioby
Science Advisor
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Yes, but in your case the state is no longer [itex] \langle r, \theta, \varphi|n, l, m \rangle [/itex] , but a linear combination of spherical harmonics. So blindly using a fomula in the book is a wrong decision...
 
  • #6
1,444
2
Yes, but in your case the state is no longer [itex] \langle r, \theta, \varphi|n, l, m \rangle [/itex] , 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 ...
https://www.physicsforums.com/showthread.php?t=157392
 

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