Welcome to the hydrogen atom: the hydrogen of atoms. The starter pack of quantum mechanics. The MVP of Schrödinger’s early access demo.
🧠 Why do we care about hydrogen?
Because it’s the only atom we can solve exactly using the Schrödinger equation.
It’s just:
- One electron
- One proton
- One huge existential crisis
📜 Schrödinger says:
We solve the time-independent Schrödinger equation in spherical coordinates:
$$ -\frac{\hbar^2}{2\mu} \nabla^2 \psi(r, \theta, \phi) - \frac{e^2}{4\pi \varepsilon_0 r} \psi = E \psi $$
Where:
- ( \mu ) is the reduced mass (basically just the electron’s mass, but fancy)
- ( r, \theta, \phi ) are spherical coordinates
- ( \psi ) is the wavefunction, our favorite quantum spaghetti
🔮 The result?
Energy levels are quantized:
$$ E_n = -\frac{13.6 , \text{eV}}{n^2} $$
where ( n = 1, 2, 3, \dots )
🎨 And the orbitals?
We get beautiful shapes from ( \psi_{n\ell m} ) — the quantum numbers give us the “address” of the electron.
- ( n ): principal quantum number (shell)
- ( \ell ): angular momentum (shape)
- ( m ): magnetic (orientation)
🧘♂️ Fun fact:
An electron in the ground state of hydrogen isn’t “orbiting” — it’s just existing as a probability cloud.
It’s chill. It’s centered. It’s peak capybara energy.
Stay tuned for: “Why the 2p orbital is a diva and the s orbital is a minimalist monk.”