Runge Lenz Vector Commutation With Hamiltonian

In classical and quantum mechanics, the Runge-Lenz vector plays a crucial role in describing the conservation of motion in central force problems, particularly in the Kepler problem. It provides an additional conserved quantity beyond energy and angular momentum, revealing the hidden symmetry of the system.

In quantum mechanics, understanding the commutation of the Runge-Lenz vector with the Hamiltonian is essential to analyzing the exact solvability of the hydrogen atom and the SO(4) symmetry of the Coulomb potential. This topic explores the Runge-Lenz vector, its commutation relations, and its role in quantum mechanics.

Table of Contents

1. Understanding the Runge-Lenz Vector

Definition of the Runge-Lenz Vector

The Runge-Lenz vector (sometimes called the Laplace-Runge-Lenz vector) is defined in a central force system with an inverse-square law potential, such as the Coulomb potential:

mathbf{A} = mathbf{p} times mathbf{L} – m k hat{mathbf{r}}

Where:

mathbf{p} is the linear momentum,
mathbf{L} = mathbf{r} times mathbf{p} is the angular momentum,
m is the mass of the ptopic,
k is a constant related to the central force,
hat{mathbf{r}} is the unit radial vector.

This vector remains conserved in an inverse-square force, such as the gravitational and Coulomb potentials.

Physical Meaning of the Runge-Lenz Vector

It points along the major axis of the elliptical orbit.
Its magnitude is proportional to the eccentricity of the orbit.
In quantum mechanics, it provides an additional conserved quantity that helps explain the degeneracy of energy levels in the hydrogen atom.

2. The Hamiltonian of a Central Force System

The Hamiltonian of a ptopic moving in a central inverse-square force field is given by:

H = frac{p^2}{2m} – frac{k}{r}

Where:

p^2 = mathbf{p} cdot mathbf{p} is the squared momentum,
r = |mathbf{r}| is the radial coordinate,
k = Ze^2 for the Coulomb interaction in hydrogen-like atoms.

The conservation of energy follows directly from the Hamiltonian, while the conservation of angular momentum follows from its commutation with H . However, an additional conserved quantity, the Runge-Lenz vector, leads to deeper symmetries in the system.

3. Commutation of the Runge-Lenz Vector with the Hamiltonian

**Commutator $[H, mathbf

To determine whether the Runge-Lenz vector is a conserved quantity in quantum mechanics, we calculate its commutator with the Hamiltonian. If:

[H, mathbf{A}] = 0

then mathbf{A} is conserved, meaning that its expectation value remains constant over time.

To compute the commutator, we use the quantum versions of momentum, angular momentum, and position operators:

mathbf{A} = mathbf{p} times mathbf{L} – m k frac{mathbf{r}}{r}

Using the standard commutators:

[H, mathbf{r}] = -frac{i hbar}{m} mathbf{p}
[H, mathbf{p}] = -nabla V
[mathbf{p}, mathbf{L}] = i hbar (mathbf{p} times mathbf{r})

After a detailed calculation (often found in textbooks on quantum mechanics), it follows that:

[H, mathbf{A}] = 0

This confirms that the Runge-Lenz vector is a conserved quantity in the Coulomb potential.

Interpretation of the Commutation Relation

Since the Hamiltonian commutes with both mathbf{L} and mathbf{A} , the system exhibits a higher symmetry than just the usual rotational invariance. This hidden symmetry is described by the SO(4) group, which is responsible for the degeneracy of the hydrogen atom energy levels.

4. The Role of the Runge-Lenz Vector in Hydrogen Atom Symmetry

SO(3) vs. SO(4) Symmetry in Quantum Mechanics

The usual symmetry in central potentials is SO(3), corresponding to conservation of angular momentum.
The additional conservation of the Runge-Lenz vector extends this to SO(4) symmetry in bound states.
This extra symmetry explains why the energy levels of hydrogen-like atoms depend only on the principal quantum number n , not on l (orbital angular momentum).

Energy Degeneracy and the Hidden Symmetry

In quantum mechanics, energy levels of the hydrogen atom are given by:

E_n = -frac{mk^2}{2 hbar^2 n^2}

where n = 1, 2, 3, dots .

Without the Runge-Lenz vector, one would expect the energy to depend on both n and l .
The SO(4) symmetry, arising from **the conservation of both mathbf{L} and mathbf{A} , ensures that energy depends only on n **.

5. Applications of the Runge-Lenz Vector in Physics

1. Classical Mechanics and Orbital Dynamics

The Runge-Lenz vector helps describe planetary orbits, ensuring that Kepler’s first law (elliptical orbits) holds.
It explains the precession of orbits in relativistic corrections.

2. Quantum Mechanics and Atomic Physics

The SO(4) symmetry simplifies the solution of the hydrogen atom using algebraic methods.
It provides insights into degenerate energy levels and selection rules in spectroscopy.

3. Relativistic and Quantum Field Theory Extensions

The Runge-Lenz vector concept extends to Dirac’s equation for relativistic hydrogen.
It plays a role in advanced field theories that generalize the hidden symmetries of nature.

6. Summary and Conclusion

The Runge-Lenz vector is a fundamental conserved quantity in systems governed by an inverse-square law, such as the Coulomb and gravitational potentials. Its commutation with the Hamiltonian confirms that it is a conserved operator in quantum mechanics, leading to the SO(4) symmetry of the hydrogen atom.

The commutation relation [H, mathbf{A}] = 0 shows that the Runge-Lenz vector is a quantum conserved quantity.
This symmetry explains the degeneracy of hydrogen energy levels, ensuring that energy depends only on n .
Applications range from classical orbital dynamics to modern quantum field theory, highlighting the importance of this vector in physics.

Understanding the Runge-Lenz vector and its commutation with the Hamiltonian provides deep insights into hidden symmetries, quantum mechanics, and celestial mechanics, making it a crucial concept in theoretical physics.