Calculator Inputs
Use the responsive input grid below. It displays three columns on large screens, two on smaller screens, and one on mobile.
Formula Used
The core requirement is one-dimensional normalization:
∫ |ψ(x)|² dx = 1
If the trial wave function is written as ψ(x) = A f(x), then the normalization constant is:
A = 1 / √(∫ |f(x)|² dx)
Gaussian
f(x) = exp(-(x - μ)² / (2σ²))
A = (σ√π)-1/2
Symmetric Exponential
f(x) = exp(-|x - x₀| / a)
A = 1 / √a
Particle in a Box
f(x) = sin(nπx/L), for 0 ≤ x ≤ L
A = √(2 / L)
Polynomial Window
f(x) = xm(L - x)k, for 0 ≤ x ≤ L
A is found numerically from the integral.
The numerical engine uses Simpson’s rule for stable integration, plus a probability recheck after normalization.
How to Use This Calculator
- Select the wave profile that best matches your physical model.
- Enter the profile parameters such as σ, a, L, n, m, or k.
- Choose automatic or custom domain selection.
- Set the integration intervals and plot points for numerical quality.
- Press the calculate button to generate the normalization constant.
- Review the graph, table, probability check, and export options.
Example Data Table
| Profile | Example Inputs | Integral of |f(x)|² | Normalization Constant A | Comment |
|---|---|---|---|---|
| Gaussian | σ = 1, μ = 0 | 1.77245385 | 0.75112554 | Centered bell-shaped wave function. |
| Exponential | a = 2, x₀ = 0 | 2.00000000 | 0.70710678 | Symmetric decaying state with wider tails. |
| Particle in a Box | L = 1, n = 1 | 0.50000000 | 1.41421356 | Ground-state standing wave in a finite interval. |
| Polynomial Window | L = 1, m = 1, k = 1 | 0.03333333 | 5.47722558 | Peaks inside the interval and vanishes at boundaries. |
Why Normalization Matters
Normalization makes the total probability equal to one. Without it, probability density loses physical meaning and comparisons between states become misleading. This calculator helps verify both the normalization constant and the resulting probability check over the chosen interval.
FAQs
1) What is a normalization constant in physics?
It is the factor that scales a trial wave function so its total probability becomes exactly one over the chosen domain.
2) Why does the calculator show a probability check?
The probability check confirms that the normalized function integrates to about one after numerical processing and domain selection.
3) Why can custom x limits change my result?
If you truncate the tails of a wave function, the integral changes. That directly changes the normalization constant and the probability check.
4) Which model should I choose?
Use Gaussian for bell-shaped packets, exponential for decays, box for standing waves in a finite region, and polynomial for custom finite-window shapes.
5) Does the box-state constant depend on n?
For the standard sine basis on 0 to L, the normalization constant stays √(2/L). The shape changes with n, but the constant does not.
6) Why use Simpson’s rule here?
Simpson’s rule gives accurate numerical integration for smooth functions and is a strong choice for normalization and expectation-value calculations.
7) What does the spread σx represent?
It is the standard deviation of position, showing how widely the normalized probability density is distributed around the mean position.
8) Can I export my results?
Yes. The page includes CSV export for data rows and PDF export for the visible result panel and graph.