21.5. Schrodinger model#

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Schrodinger equation. Schrodinger equation in space basis for the electron in a \(\text{H}\) atom reads

\[i \hbar \partial_t \Psi(\mathbf{r},t) = \left[ -\frac{\hbar^2}{2m} \nabla^2 - \frac{q^2}{4 \pi \varepsilon}\frac{1}{|\mathbf{r}|} \right] \Psi(\mathbf{r},t) \ \]

where the Hamiltonian is the sum of the kinetic energy and a Coulomb potential contribution, \(\hat{V}\), so that \(\langle \mathbf{r} | \hat{V} | \Psi \rangle = - \frac{q^2}{4 \pi \varepsilon |\mathbf{r}|} \Psi(\mathbf{r},t)\)

Spectral decomposition of the Hamiltonian operator. The stationary states, and the corresponding energy levels - i.e. the eigenfunctions, and the eigenvalues of the Hamiltonian operator \(\hat{H}\) - are the result of the eigenvalue problem, \(\hat{H} | \Psi \rangle = E | \Psi \rangle\), or using space basis,

\[\left[ -\frac{\hbar^2}{2m} \nabla^2 - \frac{q^2}{4 \pi \varepsilon}\frac{1}{|\mathbf{r}|} \right] \Psi(\mathbf{r}) = E \Psi(\mathbf{r}) \ .\]

Quantum numbers. The stationary states depend on three - in this model, no spin exists - quantum numbers:

  • principal quantum number \(n \in \{ 1, 2, \dots \}\)

  • azimuthal quantum number \(\ell \in \{ 0, 1,\dots, n-1 \} \)

  • magnetic quantum number \(m_{\ell} \in \{ -\ell, -\ell+1, \dots, \ell \}\)

Energy levels only depend on the principal quantum number, \(E(n)\). Thus, Schrodinger model of the atom has degenerate stationary states, i.e. stationary states with the same energy value.

Commutation of \(\hat{H}\), \(\hat{L}^2\), \(\hat{L}_z\). As angular momentum operators \(\hat{L}^2\), \(\hat{L}_z\) commute with the Hamiltonian operator \(\hat{H}\) of the \(\text{H}\) atom, these three operators share common eigenvectors. These relations are proved in the note section.