Free Wavefunction Normalizer

Wavefunction Normalizer computes normalization constant and probability density for a 1D wave function from discrete points.

Integration Lower Limit (m):

Integration Upper Limit (m):

x1 (m):

ψ(x1):

x2 (m):

ψ(x2):

x3 (m):

ψ(x3):

x4 (m):

ψ(x4):

x5 (m):

ψ(x5):

Formulas Used in Wavefunction Normalizer

The calculator normalizes a 1D wave function using the following formulas:

Normalization Condition:

\\[ \int_{a}^{b} |\psi(x)|^2 \, dx = 1 \\]

Normalization Constant:

\\[ A = \sqrt{\frac{1}{\int_{a}^{b} |\psi(x)|^2 \, dx}} \\]

Normalized Wave Function:

\\[ \psi_{\text{norm}}(x) = A \cdot \psi(x) \\]

Probability Density Integral:

\\[ P = \int_{a}^{b} |\psi_{\text{norm}}(x)|^2 \, dx \\]

Where:

\\( \psi(x) \\): Unnormalized wave function
\\( a, b \\): Integration limits (m)
\\( A \\): Normalization constant (m⁻¹/²)
\\( \psi_{\text{norm}}(x) \\): Normalized wave function
\\( P \\): Probability density integral (should be 1)

Example Calculations

Example 1: Linear Wave Function (Triangular)

Input: Lower Limit = -1 m, Upper Limit = 1 m, Points: (-1, 0.5), (0, 1), (1, 0.5)

\\[ \int_{-1}^{1} |\psi(x)|^2 \, dx \approx \sum_{i=1}^{2} \frac{(|\psi(x_i)|^2 + |\psi(x_{i+1})|^2)(x_{i+1} – x_i)}{2} = \frac{(0.5^2 + 1^2)(0 – (-1)) + (1^2 + 0.5^2)(1 – 0)}{2} = 0.625 \\] \\[ A = \sqrt{\frac{1}{0.625}} \approx 1.265 \, \text{m}^{-1/2} \\] \\[ P = \int_{-1}^{1} |A \cdot \psi(x)|^2 \, dx \approx 1 \\]

Result: Normalization Constant: 1.265 m⁻¹/², ψ_norm(-1) = 0.632, ψ_norm(0) = 1.265, ψ_norm(1) = 0.632, Probability Density Integral: 1

Example 2: Quadratic Wave Function (Parabolic)

Input: Lower Limit = -1 m, Upper Limit = 1 m, Points: (-1, 0.5), (-0.5, 0.75), (0, 1), (0.5, 0.75), (1, 0.5)

\\[ \int_{-1}^{1} |\psi(x)|^2 \, dx \approx \sum_{i=1}^{4} \frac{(|\psi(x_i)|^2 + |\psi(x_{i+1})|^2)(x_{i+1} – x_i)}{2} \approx 0.6875 \\] \\[ A = \sqrt{\frac{1}{0.6875}} \approx 1.206 \, \text{m}^{-1/2} \\] \\[ P = \int_{-1}^{1} |A \cdot \psi(x)|^2 \, dx \approx 1 \\]

Result: Normalization Constant: 1.206 m⁻¹/², ψ_norm(-1) = 0.603, ψ_norm(-0.5) = 0.905, ψ_norm(0) = 1.206, ψ_norm(0.5) = 0.905, ψ_norm(1) = 0.603, Probability Density Integral: 1

Example 3: Sinusoidal Wave Function

Input: Lower Limit = -π m, Upper Limit = π m, Points: (-π, 0), (-π/2, 1), (0, 0), (π/2, -1), (π, 0)

\\[ \int_{-\pi}^{\pi} |\psi(x)|^2 \, dx \approx \sum_{i=1}^{4} \frac{(|\psi(x_i)|^2 + |\psi(x_{i+1})|^2)(x_{i+1} – x_i)}{2} \approx 1.571 \\] \\[ A = \sqrt{\frac{1}{1.571}} \approx 0.798 \, \text{m}^{-1/2} \\] \\[ P = \int_{-\pi}^{\pi} |A \cdot \psi(x)|^2 \, dx \approx 1 \\]

Result: Normalization Constant: 0.798 m⁻¹/², ψ_norm(-π) = 0, ψ_norm(-π/2) = 0.798, ψ_norm(0) = 0, ψ_norm(π/2) = -0.798, ψ_norm(π) = 0, Probability Density Integral: 1