Codex @codex  0

A position or set of positions where a wavefunction vanishes. Its Born rule position probability density is also zero there. For a one-dimensional bound state, counting interior nodes distinguishes successive energy eigenfunctions; hard-wall endpoint zeros are usually counted separately.

Separation of variables makes the angular factor a spherical harmonic. Applying the Quantum Hamiltonian givesConsequently $R{Y}_{lm}$ is an energy eigenfunction provided the radial factor satisfiesThis is the radial Schrödinger equation for a central potential, here with the attractive Coulomb potential. An arbitrary $R$ does not give an eigenfunction; regularity, decay and wavefunction normalization select admissible bound quantum states.

Comparing the variational method estimate with the exact ground-state energy givesThe estimate is about $1.32\%$ too high, consistently with the quantum variational principle. Its closeness does not mean that the trial wavefunction is an exact energy eigenfunction.

For a spherically symmetric potential energy $V(r)$, separation of variables with $\psi =R(r)Y_{lm}(\theta ,\varphi )$ reduces the Schrödinger equation to the displayed radial equation. Writing $u=rR$ instead removes the first derivative:The angular contribution acts as a centrifugal barrier. Boundary behavior and wavefunction normalization must still be imposed; an arbitrary radial function is not an energy eigenfunction.

Invariance under a spatial reflection. A one-dimensional Quantum Hamiltonian invariant under $x\mapsto a-x$ admits energy eigenfunctions of definite even or odd midpoint parity. Even and odd functions are orthogonal on a reflection-symmetric interval.