Time independent Schrodinger equation

The Time-Independent Schrödinger Equation is a fundamental equation in quantum mechanics that describes the behavior of a quantum system in a stationary state, where the system’s properties do not change with time. It is derived from the more general time-dependent Schrödinger equation by assuming that the wave function can be separated into time-dependent and time-independent parts.

Formulation of the Time-Independent Schrödinger Equation

The time-independent Schrödinger equation is given by:

$[\hat{H} \psi(\mathbf{r}) = E \psi(\mathbf{r})]$

where:

$( \hat{H} )$ is the Hamiltonian operator (which represents the total energy of the system).
$( \psi(\mathbf{r}) )$ is the wave function of the system, which depends on the position $( \mathbf{r} )$ (and possibly other variables).
( E ) is the energy eigenvalue, representing the energy associated with the state $( \psi(\mathbf{r}) )$ .

The Hamiltonian Operator

The Hamiltonian operator $( \hat{H} )$ typically consists of two parts: the kinetic energy operator $( \hat{T} )$ and the potential energy operator $( \hat{V} )$ :

$[\hat{H} = \hat{T} + \hat{V}]$

For a single particle of mass ( m ) moving in a potential $( V(\mathbf{r}) )$ , the Hamiltonian in three dimensions is:

$[\hat{H} = -\frac{\hbar^2}{2m} \nabla^2 + V(\mathbf{r})]$

where:

$( \hbar )$ is the reduced Planck’s constant.
$( \nabla^2 )$ is the Laplacian operator, which represents the kinetic energy term in this context.

Thus, the time-independent Schrödinger equation can be written as:

$[-\frac{\hbar^2}{2m} \nabla^2 \psi(\mathbf{r}) + V(\mathbf{r}) \psi(\mathbf{r}) = E \psi(\mathbf{r})]$

Interpretation

$( \psi(\mathbf{r}) )$ is the wave function, which contains all the information about the quantum state of the system. The square of the wave function’s magnitude, $( |\psi(\mathbf{r})|^2 )$ , gives the probability density of finding the particle at position $( \mathbf{r} )$ .
The equation describes how the wave function $( \psi(\mathbf{r}) )$ behaves in a given potential $( V(\mathbf{r}) )$ and how this behavior is related to the energy ( E ) of the system.

Solving the Schrödinger Equation

The process of solving the time-independent Schrödinger equation involves finding the wave functions $( \psi(\mathbf{r}) )$ that satisfy the equation for a given potential $( V(\mathbf{r}) )$ . These solutions are called eigenfunctions, and the corresponding energies ( E ) are called eigenvalues.

Examples of the Schrödinger Equation

Particle in a One-Dimensional Box (Infinite Potential Well):

For a particle confined to a box of length ( L ) with infinite potential barriers at the edges, the potential ( V(x) ) is zero inside the box and infinite outside.
The solutions are sinusoidal wave functions $( \psi_n(x) = \sqrt{\frac{2}{L}} \sin\left(\frac{n\pi x}{L}\right) )$ , with energies $( E_n = \frac{n^2 \pi^2 \hbar^2}{2mL^2} )$ , where ( n ) is a positive integer.

Harmonic Oscillator:

For a particle in a harmonic oscillator potential $( V(x) = \frac{1}{2} m \omega^2 x^2 )$ , the solutions are Hermite polynomials multiplied by a Gaussian function, with quantized energy levels $( E_n = \hbar \omega \left(n + \frac{1}{2}\right) )$ .

Hydrogen Atom:

In the case of the hydrogen atom, the potential is the Coulomb potential $( V(r) = -\frac{e^2}{4\pi \epsilon_0 r} )$ , and the Schrödinger equation describes the electron’s wave function around the nucleus. The solutions lead to discrete energy levels $( E_n = -\frac{13.6 \, \text{eV}}{n^2} )$ , where ( n ) is the principal quantum number.

Significance

The time-independent Schrödinger equation is crucial for understanding quantum systems that are in stationary states. It allows physicists to determine the allowed energy levels of a quantum system, the shape of the wave functions, and the behavior of particles at the quantum level. The equation plays a central role in fields such as quantum chemistry, solid-state physics, and atomic and molecular physics.

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Time independent Schrodinger equation