Consider the Coulomb potential V(𝐱)=-Z e^2 / r and define the (quantum-mechanical operator analo

step 1 So let's start the explanation of this system. We know that. So first of all we discuss part a.

Consider the Coulomb potential V(𝐱)=-Z e^2 / r and define...

Consider the Coulomb potential $V(\mathbf{x})=-Z e^{2} / r$ and define the (quantum-mechanical operator analogue of the) Runge-Lenz vector $$ \mathbf{M}=\frac{1}{2 m}(\mathbf{p} \times \mathbf{L}-\mathbf{L} \times \mathbf{p})-\frac{Z e^{2}}{r} \mathbf{x} . $$ Prove that $\mathbf{M}$ is Hermitian and that it commutes with the Hamiltonian. We will return to $\mathbf{M}$ when we go through Pauli's algebraic solution for this Hamiltonian in Section 4.1.4.

Consider the Coulomb potential $V(\mathbf{x})=-Z e^{2} / r$ and define the (quantum-mechanical operator analogue of the) Runge-Lenz vector
$$
\mathbf{M}=\frac{1}{2 m}(\mathbf{p} \times \mathbf{L}-\mathbf{L} \times \mathbf{p})-\frac{Z e^{2}}{r} \mathbf{x} .
$$
Prove that $\mathbf{M}$ is Hermitian and that it commutes with the Hamiltonian. We will return to $\mathbf{M}$ when we go through Pauli's algebraic solution for this Hamiltonian in Section 4.1.4.

Key Concepts

Hermitian Operators

In quantum mechanics, an operator that is equal to its own adjoint is known as Hermitian, which guarantees that its eigenvalues are real and that it corresponds to an observable quantity. Proving that an operator like the Runge-Lenz vector is Hermitian is essential for ensuring that its measurable predictions are physically meaningful.

Coulomb Potential

This is the potential energy function corresponding to the electrostatic interaction between charged particles. In quantum mechanics, it is fundamental in the analysis of systems like the hydrogen atom, where it produces a central potential that allows for specific analytical solutions with high symmetry properties. It underlies many of the symmetry and conservation law discussions in the problem.

Runge-Lenz Vector

Originally found in classical mechanics for the Kepler problem, the Runge-Lenz vector is extended to quantum mechanics as an operator that, together with the angular momentum, reflects a hidden symmetry of the Coulomb problem. Its conservation, when appropriate commutation relations are satisfied, is what allows one to obtain degenerate energy levels and provides an alternative method to solving for the spectrum of the hydrogen atom.

Commutation with the Hamiltonian

When an operator commutes with the Hamiltonian of a system, it represents a conserved quantity due to the time invariance of the associated physical observable. Demonstrating that the Runge-Lenz vector commutes with the Hamiltonian confirms that it is a constant of motion and plays a central role in analyzing the system's symmetries.

Operator Algebra and Commutation Relations

This concept involves the algebraic manipulations of the quantum mechanical operators using commutators to derive properties such as conservation laws and symmetries. In the context of the Runge-Lenz vector, understanding the underlying operator algebra is crucial to establishing its Hermiticity and its commutation with the Hamiltonian, thus uncovering the hidden dynamical symmetry of the system.

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