Hydrogen atom

The Hydrogen atom is the simplest real case of a spherically symmetric potential. We examine the behavior of a single electron in Hydrogen to understand the more general behavior of a particle in a spherical potential. Starting with the Schrödinger equation in spherical coordinates:

\begin{align*} -\frac{\hbar^2}{2m} \left[ \frac{\partial^2}{\partial r^2} + \frac{2}{r}\frac{\partial}{\partial r} - \frac{\LL^2}{\hbar^2r^2} \right] \psi(\rr) + V(r) \psi(\rr) = E\psi(\rr). \end{align*}

We assume that our wave function factorizes into a radial part and an angular part $\psi(r,\theta,\phi) = R(r) Y(\theta,\phi).$ This is the same assumption as we make when considering angular momentum. With this assumption, our Schrödinger equation becomes (using the fact $\LL^2 Y_l^m = \hbar^2 l(l+1)$ )

\begin{align*} \biggl[ -\frac{\hbar^2}{2m} \left( \frac{\partial^2}{\partial r^2} + \frac{2}{r}\frac{\partial}{\partial r} \right) + \underbrace{\frac{\hbar^2 l(l+1)}{2mr^2} + V(r)}_{V_\text{eff}} \biggr] R(r) = ER(r). \end{align*}

$V_\text{eff}$ is the effective potential, where the first term is called the centrifugal barrier, and accounts for the centrifugal force. For Hydrogen, $V(r)$ is the Coulomb potential, which gives us

\begin{align*} V_\eff = \frac{\hbar^2 l(l+1)}{2mr^2} - \frac{Z e^2}{4\pi\epsilon_0 r}. \end{align*}

Where $Ze$ is the charge of the nucleus ( $Z=1$ for H).

We see that our radial wave function will depend on the radial quantum number $l$ as well as another quantum number which we will call $n$ . The magnetic quantum number $m$ does not appear. We can write the SE in a simpler form if we introduce $u(r) = rR(r)$ :

\begin{align*} -\frac{\hbar^2}{2m} \frac{\partial^2 u}{\partial r^2} + V_\text{eff}(r) u(r) = Eu(r). \end{align*}

We introduce two dimensionless variables

\begin{align*} \rho = \sqrt{\frac{8m|E|}{\hbar^2}}, \quad \lambda = \frac{Ze^2}{4\pi\epsilon_0 \hbar} \sqrt{\frac{m}{2|E|}}. \end{align*}

Which we use to write the Schrödinger equation as

\begin{align*} u'' + \left(\frac{\lambda}{\rho} - \frac14 - \frac{l(l+1)}{\rho^2} \right) u = 0. \end{align*}

We consider the asymptotic behavior of $u$ . For small $\rho$ we have

\begin{align*} \frac{\d^2}{\d \rho^2} u &= \frac14 u \\ u(\rho) &\propto e^{-\rho/2}. \end{align*}

For small $\rho$ we have

\begin{align*} \frac{\d^2}{\d \rho^2} u &= \frac{l(l+1)}{\rho^2} u \\ u(\rho) &\propto \rho^{l+1} \end{align*}

Incorporating the asymptotic behavior, we write $u(\rho) = s(\rho) \rho^{l+1} e^{-\rho/2}$ . We can perform a Taylor series expansion of the differential equation to solve for $s$ . We find that

\begin{align*} s(\rho) = \sum a_k \rho^k \quad \underset{\text{from SE}}{\implies} \quad \frac{a_{k+1}}{a_k} = \frac{k+l+1-\lambda}{(k+1)(k+2(l+1))}. \end{align*}

The series must terminate to match the asymptotic behavior: the numerator of the ratio must be zero for some $k$ , so $\lambda = k + l + 1$ . Let $n_r$ be the integer such that $n_r = \lambda - l - 1$ (here the series terminates when $k = n_r$ ). We define the principle quantum number

\begin{align*} n := n_r + l + 1 = \lambda. \end{align*}

From the definition of $\lambda$ above, we see that

\begin{align*} n &= \frac{Ze^2}{4\pi\epsilon_0\hbar} \sqrt{\frac{m}{2|E_n|}} \\ &= Z\alpha \sqrt{\frac{mc^2}{2|E_n|}} \\ E_n &= -\frac12 mc^2 \left(\frac{Z\alpha}{n}\right)^2 = \frac{E_1}{n^2}. \end{align*}

Where $\alpha$ is the fine structures constant. This is the same energy equation we get from the Bohr model! There are a few things to note about this result:

The principle quantum number $n = n_r + l + 1$ is the sum of the radial and angular quantum numbers. Since all are non-negative, $n \ge l + 1$ .
From $R(r)$ and $Y(\theta, \phi)$ we have a full spacial representation of the electron, which is more general than a classical (Bohr) orbit.
This formula is not relativistic. The $mc^2$ appears because of the fine structure constant $\alpha$ .

Back to the actual wavefunction: $s(\rho)$ is the associated (generalized) Laguerre polynomial:

\begin{align*} s_{nl}(\rho) &= \mathcal L_{n-l-1}^{2l+1}(\rho) \\ \mathcal L_n^\alpha (\rho) &= \sum_{m=0}^n {{n+\alpha} \choose {n-m}} \frac{(-\rho)^m}{m!} \end{align*}

Putting everything together, we get the wave function

\begin{align*} \psi_{nlm}(r,\theta,\phi) &= R_{nl}(r) Y_l^m (\theta,\phi) = \frac{u_{nl}(r)}{r} Y_l^m(\theta,\phi) \\ u_{nl}(\rho) &= s_{nl}(\rho) \rho^{l+1} e^{-\rho/2}. \end{align*}

Degeneracy

For some $n$ , we know $l \in [0, n-1]$ . Similarly, for some $l$ , $m \in [-l, l]$ , so there are $2l+1$ states for each $l$ . Since $n$ fully determines the energy $N_l$ , for each energy eigenvalue we have

\begin{align*} \underset{l=0}{1} + \underset{l=1}{3} + \cdots + \underbrace{2(n-1)+ 1}_{l=n-1} = n^2 \end{align*}

linearly independent eigenvectors (degeneracies). There are also two spin states each, although these don’t have a spacial representation. The figure below shows the first few Hydrogen states and their degeneracies, including the orbital names.

The names correspond to the principle quantum number and the orbital quantum number

"sharp" | S | $l=0$ ||
"principle" | P | $l=1$ ||
"diffuse" | D | $l=2$ ||
| F | $l=3$ || 
| G | $l=4$

Wave functions

The first few Hydrogen wave functions are given below

\begin{align*} \psi_{100} &= \frac{2}{a_0^{3/2}} e^{-r/a_0} Y_0^0(\theta,\phi) \\ \psi_{200} &= \frac{2}{(2a_0)^{3/2}} \left(1-\frac{r}{2a_0} \right) e^{-r/2a_0} Y_0^0(\theta,\phi) \\ \mat{\psi_{211} \\ \psi_{210} \\ \psi_{21-1}} &= \frac{1}{\sqrt3 (2a_0)^{3/2}} \frac{r}{a_0} e^{-r/2a_0} \mat{Y_1^1(\theta,\phi) \\ Y_1^0(\theta,\phi) \\ Y_1^{-1}(\theta,\phi)}. \end{align*}

$u(r)$ and the probability amplitude $r^2|u(r)|^2$ are plotted for these first few states below. Notice that the number of nodes in the radial wave function equals $n_r$ , the radial quantum mumber.