Quantum Mechanics · Lesson 09

Chapter practice

Bring the chapter together: atomic units, the termination of the radial series, the spectrum and its series, the $n^{2}$ degeneracy and its hidden symmetry, radial functions and densities, orbitals and their nodes, averages and the virial theorem, and one-electron ions. No worked example sits above the answers.

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V (r) = - \frac{e ^{2}}{4 π ε _{0} r}

E_{n} = - \frac{13.6 eV}{n ^{2}}

a_{0} = \frac{4 π ε _{0} ℏ ^{2}}{m _{e} e ^{2}}

ψ_{n ℓ m} = R_{n ℓ} Y_{ℓ m}

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Intuition

Bring the chapter together: atomic units, the termination of the radial series, the spectrum and its series, the $n^{2}$ degeneracy and its hidden symmetry, radial functions and densities, orbitals and their nodes, averages and the virial theorem, and one-electron ions. No worked example sits above the answers.

Hydrogen is the tuning fork of atomic physics: every more complicated atom is heard against it.

Before you practise

$E_{n}=-13.6\ \text{eV}\,Z^{2}/n^{2}$ ; $\ell=0,\dots,n-1$ ; $n^{2}$ states per level; radial nodes $n-\ell-1$ ; $\langle T\rangle=-E$ , $\langle V\rangle=2E$ .

\begin{aligned}a_{0}&\approx0.0529\ \text{nm},\qquad E_{h}\approx27.2\ \text{eV}\\ \frac{hc}{\lambda}&=13.6\ \text{eV}\Bigl(\frac{1}{n_{1}^{2}}-\frac{1}{n_{2}^{2}}\Bigr)\end{aligned}

Reminders

$\langle r\rangle=\frac{a_{0}}{2}(3n^{2}-\ell(\ell+1))$ and $\langle1/r\rangle=1/n^{2}a_{0}$ .
Only $s$ states reach the nucleus; $R_{n\ell}\propto r^{\ell}$ near it.
Lyman ends on $n=1$ (ultraviolet), Balmer on $n=2$ (visible), Paschen on $n=3$ (infrared).
Sizes scale as $1/Z$ and $1/\mu$ ; energies as $Z^{2}$ and $\mu$ .

Applications