Time dependent Schrödinger equation in one dimension

The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. This law describes the general case of a time-dependent potential and a time-dependent wave function.

Notation:

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

Notes:

1. This law works in the case of a single spatial dimension. To use it for the \(3\)-dimensional space, replace the spatial second derivative with the Laplace operator.

Links:

1. Wikipedia <https://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation#Separation_of_variables>

position

position.

Symbol:

x

Latex:

\(x\)

Dimension:

length

time

time

Symbol:

t

Latex:

\(t\)

Dimension:

time

wave_function

Time-dependent wave_function as a function of position and time.

Symbol:

psi(x, t)

Latex:

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

Dimension:

1/sqrt(length)

potential_energy

Time-independent potential_energy as a function of position.

Symbol:

U(x)

Latex:

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

Dimension:

energy

particle_mass

mass of the quantum particle.

Symbol:

m

Latex:

\(m\)

Dimension:

mass

law

-hbar^2 / (2 * m) * Derivative(psi(x, t), (x, 2)) + U(x) * psi(x, t) = I * hbar * Derivative(psi(x, t), t)

Latex:

\[- \frac{\hbar^{2}}{2 m} \frac{\partial^{2}}{\partial x^{2}} \psi{\left(x,t \right)} + U{\left(x \right)} \psi{\left(x,t \right)} = i \hbar \frac{\partial}{\partial t} \psi{\left(x,t \right)}\]