Ch. 19.5 H-like Atom

In spherical coordinates \( (r, \theta, \phi) \), the Laplacian operator \( \nabla^2 \) is given by:

\[\tiny \nabla^2 \psi(r, \theta, \phi) = \frac{1}{r^2} \frac{\partial}{\partial r} \left( r^2 \frac{\partial \psi}{\partial r} \right) + \frac{1}{r^2 \sin \theta} \frac{\partial}{\partial \theta} \left( \sin \theta \frac{\partial \psi}{\partial \theta} \right) + \frac{1}{r^2 \sin^2 \theta} \frac{\partial^2 \psi}{\partial \phi^2} \]

We can express the wavefunction in terms of the radial part and the angular part using the method of separation of variables:

\[ \psi(r, \theta, \phi) = R(r) Y(\theta, \phi) \] Where:

• \( R(r) \) is the radial part,
• \( Y(\theta, \phi) \) are the spherical harmonics that depend on the angular momentum quantum number \( l \) and the magnetic quantum number \( m \).

Spherical Harmonics and Angular Momentum

The spherical harmonics \( Y_{lm}(\theta, \phi) \) are functions that describe the angular part of the wavefunction and are defined as:

\[ Y_{lm}(\theta, \phi) = N_{lm} P_{lm}(\cos \theta) e^{im\phi} \] where \( P_{lm} \) are the associated Legendre polynomials and \( N_{lm} \) is a normalization constant.

Substituting the separated wavefunction into the TISE gives:

\[\tiny -\frac{\hbar^2}{2m} \left[ \frac{1}{R} \frac{1}{r^2} \frac{d}{dr} \left( r^2 \frac{dR}{dr} \right) Y_{lm} + \frac{1}{Y_{lm}} \left( \frac{1}{r^2 \sin \theta} \frac{\partial}{\partial \theta} \left( \sin \theta \frac{\partial Y_{lm}}{\partial \theta} \right) + \frac{1}{r^2 \sin^2 \theta} \frac{\partial^2 Y_{lm}}{\partial \phi^2} \right) \right] \] \[\tiny + V(r) R(r) Y_{lm} = E R(r) Y_{lm} \]

Since \( R \) and \( Y \) depend on different variables, we can separate variables:

\[\tiny -\frac{\hbar^2}{2m} \frac{1}{R} \frac{d}{dr} \left( r^2 \frac{dR}{dr} \right) + \frac{1}{Y_{lm}} \left( \frac{1}{\sin \theta} \frac{\partial}{\partial \theta} \left( \sin \theta \frac{\partial Y_{lm}}{\partial \theta} \right) + \frac{1}{\sin^2 \theta} \frac{\partial^2 Y_{lm}}{\partial \phi^2} \right) + V(r) = E \]

Radial Equation

The radial part leads us to:

\[\tiny -\frac{\hbar^2}{2m} \frac{d^2 R(r)}{dr^2} - \frac{\hbar^2}{2m r^2} \left( l(l+1) \frac{R(r)}{R(r)} \right) + V(r) R(r) = E R(r) \]

We can rewrite this in terms of the hydrogen atom potential \( V(r) \):

\[\small -\frac{\hbar^2}{2m} \frac{d^2 R(r)}{dr^2} + \left( V(r) + \frac{\hbar^2 l(l+1)}{2m r^2} \right) R(r) = E R(r) \]

Substituting the hydrogen atom potential:

\[\small -\frac{\hbar^2}{2m} \frac{d^2 R(r)}{dr^2} - \frac{e^2}{r} R(r) + \frac{\hbar^2 l(l+1)}{2m r^2} R(r) = E R(r) \]

The one-dimensional time-independent Schrödinger equation for a hydrogen atom for all angular momentum values \( l \) is:

\[\small -\frac{\hbar^2}{2m} \frac{d^2 R(r)}{dr^2} + \left( -\frac{e^2}{r} + \frac{\hbar^2 l(l+1)}{2m r^2} \right) R(r) = E R(r) \]

This equation describes the behavior of the wavefunction \( R(r) \) for different quantum states of the hydrogen atom characterized by \( n \) (principal quantum number) and \( l \) (angular momentum quantum number). The term \( \frac{\hbar^2 l(l+1)}{2m r^2} \) represents the centrifugal potential that arises from the angular momentum of the particle.

Rearranging gives the final form of the one-dimensional time-independent Schrödinger equation for a hydrogen atom with \( l = 0 \):

\[ -\frac{\hbar^2}{2m} \frac{d^2 R(r)}{dr^2} - \frac{e^2}{r} R(r) = E R(r) \]

Solve time-independent Schroedinger Equation (TISE) for Hydrogen atom for s orbital (l=0) using shooting and Euler method

Solve time-independent Schroedinger Equation (TISE) for Hydrogen atom for s orbital (l=0) using Numerov and shooting method