Schrodinger's Equation

Time-dependent #

The time-dependent Schrodinger equation is a partial differential equation that describes the behavior of a quantum system evolving with time. In one spacial dimension it is given by:

$$ \scriptsize i\hbar\frac{\partial}{\partial t}\Psi(x,t) = \left[-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2} + V(x, t)\right]\Psi(x,t) $$

Where

• \(\small \Psi(x,t)\) is the wave function of the particle,
• \(\small V \) is the potential energy function,
• \(\small m \) is the mass of the particle, and
• \(\small \hbar \) is the reduced Planck constant.

In three spacial dimensions, the partial derivative in the second term is replaced with the Laplacian operator:

$$ \scriptsize i\hbar\frac{\partial}{\partial t}\Psi(r,t) = \left[-\frac{\hbar^2}{2m}\nabla^2 + V(r,t)\right]\Psi(r,t) $$

Where \(\small r \) is the position vector of the particle.

Time-independent #

If \(\small V \) is independent of time, the wave function can form stationary states that are described by the time-independent Schrodinger equation. In one spacial dimension it is given by:

$$ \scriptsize \left[-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2} + V(x)\right]\Psi(x) = E\Psi(x) $$

Where \(\small E \) is the energy of the stationary state.

In three spacial dimensions, the partial derivative is again replaced with the Laplacian operator:

$$ \scriptsize \left[-\frac{\hbar^2}{2m}\nabla^2 + V(r)\right]\Psi(r) = E\Psi(r) $$

Sources #

• Wikipedia
• Griffiths’ Introduction to Quantum Mechanics, P. 171