Categories: Physics, Quantum mechanics.

# Hydrogen atom

The quantum-mechanical calculation of the **hydrogen atom** is,
in my opinion, the single most important model in all of physics:
miraculously, it is possible to find closed-form solutions
for the wave function of an electron in a proton’s potential well.
The results are highly educational, and also qualitatively
tell us a lot about all other chemical elements.

We start from the time-independent Schrödinger equation, where $\mu$ is the reduced mass of the electron-proton system, and $V$ is the proton’s Coulomb potential:

$\begin{aligned} E \psi = - \frac{\hbar^2}{2 \mu} \nabla^2 \psi + V \psi \end{aligned}$In spherical coordinates $(r, \theta, \varphi)$ it becomes as follows, where $V$ only depends on $r$:

$\begin{aligned} E \psi = - \frac{\hbar^2}{2 \mu} \bigg( \pdvn{2}{\psi}{r} + \frac{2}{r} \pdv{\psi}{r} + \frac{1}{r^2} \pdvn{2}{\psi}{\theta} + \frac{1}{r^2 \tan{\theta}} \pdv{\psi}{\theta} + \frac{1}{r^2 \sin^2{\theta}} \pdvn{2}{\psi}{\varphi} \bigg) + V \psi \end{aligned}$We will use the method of *separation of variables*
by making the following ansatz,
such that the Schrödinger equation takes the form below:

After multiplying by $- 2 \mu r^2 / (\hbar^2 R Y)$, each term depends on $r$ or $(\theta, \varphi)$, but not both:

$\begin{aligned} 0 &= \bigg( r^2 \frac{R''}{R} + 2 r \frac{R'}{R} - \frac{2 \mu}{\hbar^2} r^2 V + \frac{2 \mu}{\hbar^2} r^2 E \bigg) + \bigg( \frac{Y_{\theta\theta}}{Y} + \frac{1}{\tan{\theta}} \frac{Y_\theta}{Y} + \frac{1}{\sin^2{\theta}} \frac{Y_{\varphi\varphi}}{Y} \bigg) \end{aligned}$Since these two groups are independent,
this equation can only hold if there exists
a *separation constant* $C$ such that:

Now we have two simpler equations than the one we started with.
We multiply them by $R$ and $Y$ respectively,
and define $C \equiv \ell (\ell + 1)$ to help us later
($\ell$ is unknown for now).
The results are the **radial equation**
and the **angular equation**:

Note that this calculation has not really been specific to hydrogen so far: it is applicable to all spherically symmetric quantum systems.

## Angular equation

Let us keep this generality, by keeping $V$ unspecified for now, In that case, the radial equation cannot be solved yet, but the angular one can. We separate the variables again:

$\begin{aligned} Y(\theta, \varphi) = \Theta(\theta) \: \Phi(\varphi) \end{aligned}$Insert this into the equation and multiply by $\sin^2\theta / (\Theta \Phi)$ to get a clean separation:

$\begin{aligned} \sin^2{\theta} \frac{\Theta''}{\Theta} + \sin{\theta} \cos{\theta} \frac{\Theta'}{\Theta} + \ell (\ell + 1) \sin^2{\theta} = - \frac{\Phi''}{\Phi} \end{aligned}$Each term depends on $\theta$ or $\varphi$ but not both, so there exists a constant $m^2$ such that:

$\begin{aligned} m^2 &= \sin^2{\theta} \frac{\Theta''}{\Theta} + \sin{\theta} \cos{\theta} \frac{\Theta'}{\Theta} + \ell (\ell + 1) \sin^2{\theta} \\ &= - \frac{\Phi''}{\Phi} \end{aligned}$These are two distinct equations; multiplying by $\Theta$ and $\Phi$ respectively yields:

$\begin{aligned} \boxed{ \begin{aligned} m^2 \Theta &= \sin^2{\theta} \:\Theta'' + \sin{\theta} \cos{\theta} \:\Theta' + \ell (\ell + 1) \sin^2{\theta} \:\Theta \\ - m^2 \Phi &= \Phi'' \end{aligned} } \end{aligned}$Clearly the latter is the simplest, so we start there. It is an eigenvalue problem for $m^2$, but it looks like a harmonic oscillator equation, so the solutions are easily found to be:

$\begin{aligned} \boxed{ \Phi(\varphi) = e^{i m \varphi} } \end{aligned}$Because the coordinate $\varphi$ is only defined in the interval $[0, 2\pi]$, we demand periodic boundary conditions $\Phi(0) = \Phi(2 m \pi)$, which tells us that $m$ is an integer.

The other equation, for $\Theta$, needs a bit more work. We write it out like so:

$\begin{aligned} 0 &= \dvn{2}{\Theta}{\theta} + \frac{\cos{\theta}}{\sin{\theta}} \dv{\Theta}{\theta} + \Big( \ell (\ell + 1) - \frac{m^2}{\sin^2{\theta}} \Big) \Theta \end{aligned}$And then perform a change of variables $\theta \to \xi$ where $\xi \equiv \cos{\theta}$, leading to:

$\begin{aligned} 0 &= - \dv{}{\theta} \bigg( \sin{\theta} \dv{\Theta}{\xi} \bigg) - \cos{\theta} \dv{\Theta}{\xi} + \bigg( \ell (\ell + 1) - \frac{m^2}{\sin^2{\theta}} \bigg) \Theta \\ &= \sin^2{\theta} \dvn{2}{\Theta}{\xi} - 2 \cos{\theta} \dv{\Theta}{\xi} + \bigg( \ell (\ell + 1) - \frac{m^2}{\sin^2{\theta}} \bigg) \Theta \\ &= (1 - \xi^2) \dvn{2}{\Theta}{\xi} - 2 \xi \dv{\Theta}{\xi} + \bigg( \ell (\ell + 1) - \frac{m^2}{1 - \xi^2} \bigg) \Theta \end{aligned}$This result can be recognized as
Legendre’s generalized equation,
a known eigenvalue problem for $\ell (\ell + 1)$,
which has solutions when $\ell$ is a non-negative integer.
Those solutions are called the *associated Legendre polynomials*
$P_\ell^m(x)$ of degree $\ell$ and order $m$.
For a given $\ell$, there exist $2 \ell + 1$
such “polynomials” (they actually contain square roots too)
indexed by the integer $m$ in the range $[-\ell, \ell]$,
so e.g. for $\ell = 2$ there is $m = -2, -1, 0, 1, 2$.
We now have:

We are still missing a constant factor, found by imposing the normalization condition:

$\begin{aligned} \int_0^{2 \pi} \int_0^\pi |Y_\ell^m|^2 \sin{\theta} \dd{\theta} \dd{\varphi} = 1 \end{aligned}$Calculating the normalization constant (not shown here) leads to the
following definition of the so-called **spherical harmonics**
$Y_\ell^m$ of degree $\ell$ and order $m$:

These are important functions: the wave function of any spherically symmetric quantum system is a superposition of $Y_\ell^m$ with $r$-dependent coefficients. And, as befits a (component of a) wave function, they form an orthonormal basis, specifically:

$\begin{aligned} \int_0^{2 \pi} \int_0^\pi Y_\ell^m \:Y_{\ell'}^{m'} \:\sin\theta \:d\theta \:d\varphi = \delta_{\ell\ell'} \delta_{mm'} \end{aligned}$## Radial equation

With the angular part solved, we now turn to the radial part. Introducing $u(r) = r R(r)$, such that the derivatives of $R(r)$ become:

$\begin{aligned} R' = \frac{r u' - u}{r^2} \qquad\qquad R'' = \frac{r^2 u'' - 2 r u' + 2 u}{r^3} \end{aligned}$Inserting this into the radial equation and cancelling some of the terms:

$\begin{aligned} \ell (\ell + 1) \frac{u}{r} &= \frac{r^2 u'' - 2 r u' + 2 u}{r} + \frac{2 r u' - 2 u}{r} - \frac{2 \mu}{\hbar^2} (V - E) r u \\ &= r u'' - \frac{2 \mu}{\hbar^2} (V - E) r u \end{aligned}$After multiplying by $\hbar^2 / (2 \mu r)$ and rearranging, this turns into:

$\begin{aligned} E u = - \frac{\hbar^2}{2 \mu} u'' + \bigg( V + \frac{\hbar^2}{2 \mu} \frac{\ell (\ell + 1)}{r^2} \bigg) u \end{aligned}$Here it is useful to define an **effective potential** $V_{\mathrm{eff}}(r)$ as below.
Keep in mind that $\ell$ is known after solving the angular equation:

This yields a relation of the same form as the
time-independent Schrödinger equation, just with $V$ replaced by
$V_{\mathrm{eff}}$. This is the “true” **radial equation**,
an eigenvalue problem for $E$:

Now, finally, we specialize for the hydrogen atom. Coulomb’s law tells us the attractive force $F(r)$ between the electron and the proton, which we integrate to find the potential energy $V(r)$:

$\begin{aligned} F(r) = \frac{q^2}{4 \pi \varepsilon_0 r^2} \qquad \implies \qquad V(r) = - \frac{q^2}{4 \pi \varepsilon_0 r} \end{aligned}$Where $q < 0$ is the electron’s charge,
and $\varepsilon_0$ is the permittivity of free space.
Note that $V < 0$, so there is a natural distinction
between **bound states** $E < 0$
(where the electron is trapped in the proton’s well),
and **scattering states** $E > 0$
(where the electron is free).
The true radial equation, after dividing by $E$, is now given by:

For brevity, let us introduce new constants $\kappa$ and $\rho_0$, defined as follows:

$\begin{aligned} \kappa \equiv \frac{\sqrt{-2 \mu E}}{\hbar} \qquad\qquad \rho_0 \equiv \frac{\mu q^2}{2 \pi \varepsilon_0 \hbar^2 \kappa} \end{aligned}$Where $E < 0$, as we are interested in bound states. Now the radial equation has become:

$\begin{aligned} 0 = \frac{1}{\kappa^2} u'' + \Big( \frac{\rho_0}{\kappa r} - \frac{\ell (\ell + 1)}{\kappa^2 r^2} - 1 \Big) u \end{aligned}$To clean this up further, we switch to the dimensionless variable $\rho \equiv \kappa r$, yielding:

$\begin{aligned} 0 = u'' + \Big( \frac{\rho_0}{\rho} - \frac{\ell (\ell + 1)}{\rho^2} - 1 \Big) u \end{aligned}$We then choose the following ansatz for $u(\rho)$, where $v(2 \rho)$ is unknown:

$\begin{aligned} u(\rho) &= w(2 \rho) \: \rho^{\ell + 1} \: e^{- \rho} \end{aligned}$For reference, we also calculate its first and second derivatives:

$\begin{aligned} u'(\rho) &= \Big( 2 \rho w' + (\ell + 1 - \rho) w \Big) \rho^\ell \: e^{- \rho} \\ u''(\rho) &= \bigg( 4 \rho^2 w'' + 4 (\ell + 1 - \rho) \rho w' + \big( \rho^2 - 2 \rho (\ell + 1) + \ell (\ell + 1) \big) w \bigg) \rho^{\ell-1} \: e^{- \rho} \end{aligned}$Inserting this into the radial equation and dividing out all common factors gives:

$\begin{aligned} 0 &= 4 \rho w'' + 4 (\ell + 1 - \rho) w' + \big( \rho_0 - 2 (\ell + 1) \big) w \end{aligned}$Let us rearrange this to put it in a more suggestive form. Keep in mind that $w = w(2 \rho)$:

$\begin{aligned} 0 &= (2 \rho) w'' + \Big( (2 \ell + 1) + 1 - (2 \rho) \Big) w' + \Big( \frac{\rho_0}{2} - \ell - 1 \Big) w \end{aligned}$This can be recognized as Laguerre’s generalized equation,
a well-known eigenvalue problem for $\lambda \equiv (\rho_0 / 2 \!-\! \ell \!-\! 1)$.
It has solutions when $\lambda$ is a non-negative integer,
in other words for $\rho_0 = 2n$ with $n = 1, 2, 3,...$,
which also tells us that $\ell$ cannot be larger than $n - 1$.
Then the solutions are the so-called *associated Laguerre polynomials*
$L_{n - \ell - 1}^{2 \ell + 1}(2 \rho)$, therefore:

We are still missing a constant factor, found by imposing the normalization condition:

$\begin{aligned} \int_0^\infty R^2 \: r^2 \dd{r} = 1 \end{aligned}$Calculating the normalization constant (not shown here) leads to this radial solution $R_{n\ell}(r)$:

$\begin{aligned} R_{n \ell}(r) = \sqrt{\frac{(n - \ell - 1)!}{2 n (n + \ell)!}} \: (2 \kappa)^{3/2} \: (2 \kappa r)^\ell \: e^{-\kappa r} \: L_{n - \ell - 1}^{2 \ell + 1}(2 \kappa r) \end{aligned}$Meanwhile, by isolating the definitions of $\kappa$ and $\rho_0$ for $E$, we find the eigenenergies to be:

$\begin{aligned} E = - \frac{\hbar^2}{2 \mu} \kappa^2 = - \frac{\hbar^2}{2 \mu} \bigg( \frac{\mu q^2}{2 \pi \varepsilon_0 \hbar^2 \rho_0} \bigg)^2 \end{aligned}$Since $\rho_0 = 2 n$, these allowed **Bohr energies** $E_n$
of the electron are as follows:

At this point, it is customary to also define
the **reduced Bohr radius** $a_0^*$, given by:

The non-reduced **Bohr radius** $a_0$ simply uses
the electron’s raw mass $m_e$ instead of $\mu$.
Roughly speaking, $a_0^*$ is the most probable electron-proton distance
after a measurement of the electron’s position while it is in its ground state.
This is often to used to write $R_{n \ell}(r)$ as:

## Quantum numbers

Multiplying the angular and radial parts together, we thus arrive at the following expression for the full wave function $\psi_{n \ell m}$:

$\begin{aligned} \boxed{ \psi_{n \ell m}(r, \theta, \varphi) = R_{n \ell}(r) \: Y_\ell^m(\theta, \varphi) } \end{aligned}$The indices $n$, $\ell$, and $m$ are the **quantum numbers**,
which describe the state of the electron.
There is also a fourth not shown here, the **spin quantum number**,
which is $+1/2$ or $-1/2$ for spin-up or spin-down electrons respectively.

The **principal quantum number** $n$, often called the **shell number**,
gives the energy level (shell) of the electron,
because the other numbers do not appear in $E_n$’s formula.
Since $E_n = E_1 / n^2$,
the energy differences decrease with increasing $n$,
so electrons in higher shells can be excited more easily
(i.e. they need less energy to get excited).

The **azimuthal quantum number** $\ell$ gives the **subshell**
of shell $n$ in which the electron is located.
It takes integer values from $0$ to $n - 1$ inclusive,
with $0$, $1$, $2$, and $3$ respectively
also called the $s$, $p$, $d$, and $f$ subshells.
The electron’s total angular momentum is given by $\hbar \sqrt{\ell (\ell + 1)}$.

The **magnetic quantum number** $m$ splits the electrons in each subshell
into **orbitals**, and takes integer values from $-\ell$ to $\ell$.
The $z$-component of the electron’s angular momentum is $\hbar m$.

The total degeneracy of each energy level $n$ can be calculated as the sum of an arithmetic series, and is found to be $n^2$ excluding spin (or $2 n^2$ with spin).

Unsurprisingly, all these wave functions form an orthonormal basis
(although not a *complete* one unless the scattering states with $E > 0$ are included):

When an excited electron drops from a state with energy $E_i$ to a lower level $E_f$, it emits a photon with energy $\hbar \omega$, where $\omega$ is the angular frequency of the resulting electromagnetic wave:

$\begin{aligned} \hbar \omega = E_i - E_f = E_1 \bigg( \frac{1}{n_i^2} - \frac{1}{n_f^2} \bigg) \end{aligned}$The corresponding vacuum wavelength is $\lambda_0 = 2 \pi c / \omega$,
leading to the **Rydberg formula**,
which was discovered empirically before the hydrogen atom had been solved:

Quantum mechanics then successfully gave a theoretical value
to the experimentally determined **Rydberg constant**
$R_\mathrm{H}$ (or $R_\infty$ if the raw electron mass $m_e$ is used):

The transitions from excited states to the ground state $n_f = 1$
correspond to ultraviolet spectral lines known as the **Lyman series**.
Similarly, transitions to $n_f = 2$ give visible lines known as the **Balmer series**,
and transitions to $n_f = 3$ explain the **Paschen series** of infrared lines.

The Rydberg constant is not to be confused
with the **Rydberg energy** $\mathrm{Ry}$,
which is the ionization energy of ground-state hydrogen,
and is sometimes used as a unit in calculations:

The point is that the hydrogen atom’s solution gave clear
explanations for known experimental data,
and settled the mystery of what an atom *actually looks like*.
While other elements’ atoms generally do not have such closed-form solutions
(because they have more than one electron),
their orbitals are qualitatively very similar.
In short, this model is the foundation of our modern understanding of atoms.

## References

- D.J. Griffiths, D.F. Schroeter,
*Introduction to quantum mechanics*, 3rd edition, Cambridge. - R. Shankar,
*Principles of quantum mechanics*, 2nd edition, Springer.