Time independent solution in one dimension

The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system.

Notation:

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

Condition:

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.

2. The wave function is independent of time.

Links:

1. Wikipedia.

position

position.

Symbol:

x

Latex:

\(x\)

Dimension:

length

wave_function

wave_function as a function of position.

Symbol:

psi(x)

Latex:

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

Dimension:

1/sqrt(length)

potential_energy

potential_energy as a function of position.

Symbol:

U(x)

Latex:

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

Dimension:

energy

particle_mass

mass.

Symbol:

m

Latex:

\(m\)

Dimension:

mass

particle_energy

energy of the particle.

Symbol:

E

Latex:

\(E\)

Dimension:

energy

law

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

Latex:

\[- \frac{\hbar^{2}}{2 m} \frac{d^{2}}{d x^{2}} \psi{\left(x \right)} + U{\left(x \right)} \psi{\left(x \right)} = E \psi{\left(x \right)}\]