Schrödinger Equation Solver

Schrödinger Equation Solver finds energy levels, wave functions, and normalization for a particle in a 1D finite well.

Particle Mass (kg):

Well Half-Width (nm):

Potential Barrier Height (eV):

Formulas Used in Schrödinger Equation Solver

The calculator solves the time-independent 1D Schrödinger equation for a finite square well:

Schrödinger Equation:

\\[ -\frac{\hbar^2}{2m} \frac{d^2\psi(x)}{dx^2} + V(x)\psi(x) = E\psi(x) \\]

Finite Square Well Potential:

\\[ V(x) = \begin{cases} 0 & \text{if } -a \leq x \leq a \\ V_0 & \text{if } |x| > a \end{cases} \\]

Normalization Constant:

\\[ A = \sqrt{\int_{-\infty}^{\infty} |\psi(x)|^2 \, dx} \\]

Where:

\\( \hbar \\): Reduced Planck constant (\\( 1.0545718 \times 10^{-34} \, \text{J·s} \\))
\\( m \\): Particle mass (kg)
\\( a \\): Well half-width (nm)
\\( V_0 \\): Potential barrier height (eV)
\\( \psi(x) \\): Wave function
\\( E \\): Energy eigenvalue (eV)
\\( A \\): Normalization constant

Example Calculations

Example 1: Electron in a Shallow Well

Input: Particle Mass = \\( 9.11 \times 10^{-31} \\) kg, Well Half-Width = 1 nm, Potential Barrier Height = 5 eV

Result: \\( E_1 \approx 0.94 \, \text{eV}, E_2 \approx 3.76 \, \text{eV}, E_3 \approx 8.46 \, \text{eV}, \psi(-1) \approx 0.71, \psi(0) \approx 1.00, \psi(1) \approx 0.71, A \approx 0.50 \, \text{nm}^{-1/2} \\)

Example 2: Electron in a Standard Well

Input: Particle Mass = \\( 9.11 \times 10^{-31} \\) kg, Well Half-Width = 1 nm, Potential Barrier Height = 10 eV

Example 3: Heavier Particle in a Deep Well

Input: Particle Mass = \\( 1.67 \times 10^{-27} \\) kg, Well Half-Width = 0.5 nm, Potential Barrier Height = 50 eV

Result: \\( E_1 \approx 0.03 \, \text{eV}, E_2 \approx 0.12 \, \text{eV}, E_3 \approx 0.27 \, \text{eV}, \psi(-0.5) \approx 0.71, \psi(0) \approx 1.00, \psi(0.5) \approx 0.71, A \approx 0.71 \, \text{nm}^{-1/2} \\)