Angular Momentum Homework: Calculating <Lz> & ΔLz

In summary, the conversation discusses the calculation of <\hat{L_{z}}> and \Delta L_{z} for a spherical solution of the Schrodinger equation in spherical coordinates given the superposition of spherical harmonic functions. The attempt at a solution uses a formula from the textbook, but the state in this case is a linear combination of spherical harmonics, so blindly using the formula is incorrect. The conversation also references another thread for further discussion.
  • #1
1,459
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
 
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  • #2
<L_z^2>=(+1+0+9)/14=5/7.
Thus, you should get Delta L_z=3/7.
 
  • #3
stunner5000pt said:

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
dextercioby said:
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
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
dextercioby said:
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...

... 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
 

1. What is Angular Momentum?

Angular momentum is a concept in physics that describes the rotational motion of an object. It is a measure of the object's resistance to changes in its rotation and is defined as the product of its moment of inertia and its angular velocity.

2. How is Angular Momentum calculated?

Angular momentum is calculated by multiplying the moment of inertia of an object by its angular velocity. The moment of inertia is a measure of an object's mass distribution around its axis of rotation, while angular velocity is the rate of change of its angular position.

3. What is and how is it related to Angular Momentum?

is the symbol used to represent the z-component of an object's angular momentum. It is related to angular momentum because it represents the angular momentum of an object in the z-direction, which is one component of its overall angular momentum.

4. What is ΔLz and how is it calculated?

ΔLz is the symbol used to represent the change in the z-component of an object's angular momentum. It is calculated by subtracting the initial value of from the final value of . This change in angular momentum can be caused by external torques acting on the object.

5. How can I use Angular Momentum to solve problems?

Angular momentum is a useful concept in solving problems related to rotational motion. It can be used to calculate the motion of objects in rotational systems, such as spinning tops or planets orbiting around a star. It can also be used to analyze the effects of external forces on rotating objects, such as balancing a spinning bike wheel.

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