Laplace–Runge–Lenz Vector - Quantum Mechanics of The Hydrogen Atom

Quantum Mechanics of The Hydrogen Atom

Poisson brackets provide a simple guide for quantizing most classical systems: the commutation relation of two quantum mechanical operators is specified by the Poisson bracket of the corresponding classical variables, multiplied by iħ.

By carrying out this quantization and calculating the eigenvalues of the Casimir operator for the Kepler problem, Wolfgang Pauli was able to derive the energy levels of hydrogen-like atoms (Figure 6) and, thus, their atomic emission spectrum. This elegant derivation was obtained before the development of the Schrödinger equation.

A subtlety of the quantum mechanical operator for the LRL vector A is that the momentum and angular momentum operators do not commute; hence, the cross product of p and L must be defined carefully. Typically, the operators for the Cartesian components As are defined using a symmetrized product,

A_{s} = - m k \hat{r}_{s} + \frac{1}{2} \sum_{i=1}^{3} \sum_{j=1}^{3} \epsilon_{sij} \left( p_{i} l_{j} + l_{j} p_{i} \right) ,

from which the corresponding additional ladder operators for L can be defined,

J_{0} = A_{3} \,
J_{\pm 1} = \mp \frac{1}{\sqrt{2}} \left( A_{1} \pm i A_{2} \right) ~.

These further connect different eigenstates of L2.

A normalized first Casimir invariant operator, quantum analog of the above, can likewise be defined,

C_{1} = - \frac{m k^{2}}{2 \hbar^{2}} H^{-1} - I ~,

where H−1 is the inverse of the Hamiltonian energy operator, and I is the identity operator.

Applying these ladder operators to the eigenstates of the total angular momentum, azimuthal angular momentum and energy operators, the eigenvalues of the first Casimir operator C1 are n2 − 1. Importantly, by dint of the vanishing of C2, they are independent of the l and m quantum numbers, making the energy levels degenerate.

Hence, the energy levels are given by

E_{n} = - \frac{m k^{2}}{2\hbar^{2} n^{2}} ,

which coincides with the Rydberg formula for hydrogen-like atoms (Figure 6). The additional symmetry operators A have implicitly connected the different l multiplets among themselves, for a given energy (and C1). In effect, they have enlarged the group SO(3) to SO(4).

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