#  Particle on a sphere

The Hamiltonial $H$ for a particle on a sphere is given by

\begin{align}
H &= \frac{p^2}{2m} = \frac{L^2}{2 I}
\end{align}




## Exercise 1
$\mathbf{L} = \mathbf{r} \times \mathbf{p}$ is the angular momentum operator
where $\mathbf{r} = (x, y, z)$ is the positional operator and $\mathbf{p} = (p_x, p_y, p_z)$ is the momentum operator

Show that:

\begin{align}
L_x &= - i \hbar \left(y \frac{\partial}{\partial z} - z \frac{\partial}{\partial y} \right) \\
L_y &= - i \hbar \left(z \frac{\partial}{\partial x} - x \frac{\partial}{\partial z} \right) \\
L_z &= - i \hbar \left(x \frac{\partial}{\partial y} - y \frac{\partial}{\partial x} \right)
\end{align}

Hint: $p_x = -i\frac{\partial}{\partial x}$ 

Hint 2: To do this exercise you need to do a cross product 

## Exercise 2

Find the commutator relations $[L_x, L_y]$ and $[L^2, L_z]$. 

Does $L_x$ and $L_y$ commute?

Does $L^2$ and $L_z$ commute?

Hint: $[a, b] = ab - ba$



The particle on a sphere is given by:

\begin{align}
L^2 Y_{m_l}^l (\varphi, \theta) = \hbar^2 l(l+1) Y_{m_l}^l (\varphi, \theta) \\
L_z Y_{m_l}^l (\varphi, \theta) = \hbar m_l Y_{m_l}^l (\varphi, \theta)
\end{align}

where $l = 0, 1, 2, ...$ and $m_l = -l, -l+1,...,0,..., l-1, l$



## Exercise 3

Make a surface plot of a sphere where the particle can be located

## Exercise 4

Find the analytic expressions for $Y_{m_l}^l (\varphi, \theta)$, for instance in the lecture notes on explain everything (code ATEHWGWY). 


 Make python functions for $L = 0$, $L = 1$ and $L = 2$ with all combination of $m_l$


```python
def Y_10(phi, theta):

def Y_11(phi, theta):

def Y_1m1(phi, theta): 

# or more generally using sph_harm from scipy.special

from scipy.special import sph_harm

 

```

## Exercise 5
The parametrization for the probability densities are:

\begin{align}
x &= \cos \theta \sin \phi |Y_{m_l}^l (\varphi, \theta)|^2 \\
y &= \sin \theta \sin \phi |Y_{m_l}^l (\varphi, \theta)|^2 \\
z &= \cos \phi |Y_{m_l}^l (\varphi, \theta)|^2
\end{align}

Give an explaination to why the parametrization looks like this.


Plot the probability density of $|Y_{m_l}^l (\varphi, \theta)|^2$ for $L = 0$, $L = 1$
and $L=2$ for $m_l = -l, -l+1,...,0,..., l-1, l$.

Try to plot them with the sphere from exercise 3, here it is a good idea to add the kword $alpha = 0.3$ for the sphere. 

## Exercise 6

Looking at the formulas for  $Y_{m_l}^l (\varphi, \theta)$

What is the probabilty densities $P_x$ and $P_y$, plot these for your favorite values of $l$ and $m_l$.

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ipynb
Jupyter Notebook
problem_candidates/Particle on a sphere exercises.ipynb
ilfreddy/seminars
c7e13874b41cc906a45b672e5b85c57d6880473e
MIT
problem_candidates/Particle on a sphere exercises.ipynb
ilfreddy/seminars
c7e13874b41cc906a45b672e5b85c57d6880473e
MIT
problem_candidates/Particle on a sphere exercises.ipynb
ilfreddy/seminars
c7e13874b41cc906a45b672e5b85c57d6880473e
MIT
Qwen/Qwen-72B
1. YES
2. YES