# Angular Momentum Homework: Calculating <Lz> & ΔLz

• stunner5000pt
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.

## 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 $<\hat{L_{z}}>$ 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 $<L_{z}> [/tex]? thanks in advance for any input <L_z^2>=(+1+0+9)/14=5/7. Thus, you should get Delta L_z=3/7. stunner5000pt said: ## 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 [itex] <\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 $<L_{z}> [/tex]? thanks in advance for any input How did you get that $$\langle L_{z}^2\rangle = (\hbar m_{l})^2$$ ? dextercioby said: How did you get that $$\langle L_{z}^2\rangle = (\hbar m_{l})^2$$ ? my textbook says so...also $$<L_{z}^2>=\int \Psi^{*}_{nlm}\left(\frac{\hbar}{i}\frac{\partial}{\partial\phi}\right)\left(\frac{\hbar}{i}\frac{\partial}{\partial\phi}\right)\Psi_{nlm} d\tau = (\hbar m)^2$$ Last edited: Yes, but in your case the state is no longer [itex] \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...

dextercioby said:
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 ...

## 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.

• Quantum Physics
Replies
3
Views
190
Replies
21
Views
1K
Replies
6
Views
2K
Replies
2
Views
1K
Replies
4
Views
3K
Replies
2
Views
841
Replies
5
Views
1K
Replies
3
Views
1K
• Mechanics
Replies
2
Views
793
• Introductory Physics Homework Help
Replies
45
Views
2K