Time dependent solution via time independent solution

When the potential energy is independent of time, the solution of the time-dependent Schrödinger equation can be constructed using the solution of the time-independent equation.

Notation:

1. \(\hbar\) (hbar) is hbar.

Conditions:

1. The potential energy is a function independent of time.

2. This law applies for 1-dimensional systems.

Links:

1. Wikipedia.

position

position, or spatial variable.

Symbol:

x

Latex:

\(x\)

Dimension:

length

time

time.

Symbol:

t

Latex:

\(t\)

Dimension:

time

time_dependent_wave_function

Solution of the time-dependent Schrödinger equation. See wave_function.

Symbol:

Psi(x, t)

Latex:

\(\Psi{\left(x,t \right)}\)

Dimension:

1/sqrt(length)

time_independent_wave_function

Solution of the time-independent Schrödinger equation. See wave_function.

Symbol:

psi(x)

Latex:

\(\psi{\left(x \right)}\)

Dimension:

1/sqrt(length)

particle_energy

Particle energy.

Symbol:

E

Latex:

\(E\)

Dimension:

energy

law

Psi(x, t) = psi(x) * exp(-I / hbar * E * t)

Latex:

\[\Psi{\left(x,t \right)} = \psi{\left(x \right)} \exp{\left(- \frac{i}{\hbar} E t \right)}\]