# The Hydrogenic atom

By separating the angular and radial components of the Wavefunction

\begin{align}
\Psi(x, y, z) \rightarrow R(r) Y(\phi, \theta)
\end{align}

the radial wavefunction for a Hydrogenic atom takes the form

\begin{align}
-\frac{1}{2 \mu} \frac{d^2 u(r)}{dr^2} + V_{eff}u(r) &= E u(r) \\
u(r) &= rR(r) \\
V_{eff}(r) &= -\frac{1}{r} + \frac{\hbar^2 l (l + 1)}{2 \mu r^2} 
\end{align}

where $\mu$ is the reduced mass.

## Exercise 1

Plot the effective potential $V_{eff}(r)$ for different values of $l$. Note how different the potential is for $l = 0$ and for $l \neq 0$. Can you explain what happens?

## Exercise 2

Find the analytic expression for $R_{nl}(r)$ at [Explain Everything](https://drive.explaineverything.com/thecode/FNPFRHMN) and make Python Functions for
n = 1, l = 0, n = 2, l = 0, 1 and n = 3, l = 0, 1, 2.

Note that on Expliain Everything the functions are given as a function of $\rho$. The relation between $\rho$ and $r$ is: $\rho = \frac{2 Z}{n a}r$ where $a = 4\pi \epsilon_0 \frac{ \hbar^2 }{\mu e^2}$

```python
def R10(r):
    # n = 1, l = 0
```
Bonus points to those who can make a General R(n, l, r).
```python
def R(n, l, r):
 
```
I was not able to do this so please tell me if you come up with a solution.


## Exercise 3

Now let's study the radial probablilty density $P(r)$, the total wavefunction is
$\Psi_{nlm_l}$. To study the radial probability density we take the probability density $|\Psi_{nlm_l}|$ and integrate the non-radial parts ($\theta, \phi$).

\begin{align}
P(r) = \int_{surface} |\Psi_{nlm_l}(\theta, \phi, r)| d\tau = \int_0^\pi \int_0^{2 \pi} R_{nl}(r)^2 |Y_{lm_l}(\theta, \phi)| r^2\sin (\theta) d\theta d \phi
\end{align}

we then find $P(r) = R_{nl}(r)^2 r^2$

Plot the Quantity $P(r) = R_{nl}(r)^2 r^2 $ for various values of Z.

What happens when Z increases, why do you think this happens?

## Exercise 4

The total wavefunciton $Y_{n l m_l} = R_{nl}(r) Y_{lm_l}(\theta, \phi)$ where $R_{nl}$ is the solution to the Radial wavefunciton and 

Make a function that returns the probability density of finding the particle within a given volume $\theta \in (\theta_0, \theta_1)$ $r \in (r_0, r_1)$$\phi \in (\phi_0, \phi_1)$ 

```python 
def two_pz_probability(theta_0, theta_1, phi_0, phi_1, r_0, r_1):
```


```python

```

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ipynb
Jupyter Notebook
radial_wavefunction.ipynb
KJE2001/seminars
b42c5c0a945795f6f625db35cafefa6662e5ce7e
MIT
2017-02-04T01:34:33.000Z
2021-06-12T12:27:37.000Z
radial_wavefunction.ipynb
KJE2001/seminars
b42c5c0a945795f6f625db35cafefa6662e5ce7e
MIT
2020-03-30T11:00:35.000Z
2020-05-12T05:42:24.000Z
radial_wavefunction.ipynb
KJE2001/seminars
b42c5c0a945795f6f625db35cafefa6662e5ce7e
MIT
2016-04-26T20:42:43.000Z
2022-02-06T11:12:57.000Z
Qwen/Qwen-72B
1. YES
2. YES