Calculator
Select a one-dimensional trial state, define the interval, choose a region, and calculate the normalization constant numerically.
Example Data Table
These benchmark rows help validate the calculator with common trial states.
| State | Interval | Inputs | Raw Integral ∫|ψ₀|²dx | Normalization Constant A | Interpretation |
|---|---|---|---|---|---|
| Sinusoidal | [0, 1] | n = 1 | 0.500000 | 1.414214 | Ground-state style well function. |
| Sinusoidal | [0, 1] | n = 2 | 0.500000 | 1.414214 | Higher node count, same raw norm. |
| Exponential | [-6, 6] | α = 1, x₀ = 0 | 0.999994 | 1.000003 | Nearly full support across the interval. |
| Polynomial | [0, 1] | m = 1, k = 1 | 0.033333 | 5.477226 | Bounded shape with edge suppression. |
Formula Used
Normalization condition
A valid state must carry unit probability.
∫[a to b] |ψ(x)|² dx = 1
Normalization constant
Scale the raw trial state ψ₀(x).
A = 1 / √(∫[a to b] |ψ₀(x)|² dx)
Normalized wavefunction
The calculator reports the rescaled amplitude.
ψ(x) = A · ψ₀(x)
Region probability
Measure likelihood inside a user-defined subrange.
P(x₁ to x₂) = ∫[x₁ to x₂] |ψ(x)|² dx
Expectation value
This gives the position average for the normalized state.
<x> = ∫[a to b] x |ψ(x)|² dx
Numerical integration
Simpson’s rule estimates smooth integrals accurately.
I ≈ h/3 · [f₀ + fN + 4Σfodd + 2Σfeven]
Built-in raw states
- Gaussian: ψ₀(x) = exp(-α(x - x₀)²)
- Exponential: ψ₀(x) = exp(-α|x - x₀|)
- Sinusoidal: ψ₀(x) = sin(nπ(x - a)/(b - a))
- Polynomial: ψ₀(x) = ((x-a)/(b-a))m · ((b-x)/(b-a))k
How to Use This Calculator
- Choose the trial state that best matches your physics problem.
- Enter the normalization interval from a to b.
- Set sample points for numerical accuracy.
- Provide the state parameters such as α, x₀, n, m, or k.
- Define a subregion if you want a local probability estimate.
- Press the calculate button to display the result above the form.
- Review the graph, summary metrics, and sample output table.
- Use the export buttons to save the computed data as CSV or PDF.
FAQs
1. What does normalization mean in quantum mechanics?
Normalization makes the total probability equal one over the chosen domain. After scaling, the area under |ψ(x)|² becomes one, so probability statements remain consistent.
2. Why does this calculator use a finite interval?
A finite interval can represent a box, a measurement window, or a numerical truncation. Gaussian and exponential states extend indefinitely, so the calculator normalizes them only across the interval you enter.
3. Why is Simpson’s rule used here?
Simpson’s rule is accurate for smooth one-dimensional functions and is efficient for classroom and engineering use. More sample points usually improve precision for narrow peaks or strongly oscillating states.
4. Does this page support complex wavefunctions?
This version focuses on real trial states. For complex states, the normalization density should use ψ*ψ. You can extend the raw-state function in the code if your problem requires complex amplitudes.
5. Why can the normalization constant become very large?
A large constant means the raw integral is very small. That happens when the state is sharply localized, heavily truncated, or nearly zero throughout most of the selected interval.
6. What does the region probability tell me?
It estimates the chance of finding the particle between your chosen region bounds. The calculator integrates only the normalized density inside that subrange.
7. How many sample points should I choose?
Start with 401 or 801 points. Increase the count for narrow Gaussians, steep exponentials, or larger quantum numbers. The code automatically keeps the final count odd for Simpson’s rule.
8. What does <x> represent in the results?
It is the expectation value of position on the selected interval. This value gives the density-weighted average location of the particle for the normalized state.