Calculator Inputs
The page uses a single stacked layout, while the calculator fields switch to three columns on large screens, two on smaller screens, and one on mobile.
Example Data Table
| Example wavefunction | Input values | Normalization constant A |
|---|---|---|
| Gaussian State | α = 0.50 | 0.751126 |
| Decaying Exponential State | λ = 1.20 | 1.095445 |
| Uniform Box State | L = 2.50 | 0.632456 |
| Infinite Well Sine State | L = 3.00, n = 2 | 0.816497 |
| Odd Gaussian State | α = 0.75 | 1.439779 |
Formula Used
General normalization condition:
1 = ∫ |ψ(x)|2 dx
If ψ(x) = Aφ(x), then A = 1 / √(∫ |φ(x)|2 dx).
Gaussian State
φ(x) = e-αx2
A = (2α/π)1/4
Decaying Exponential State
φ(x) = e-λ|x|
A = √λ
Uniform Box State
φ(x) = 1 for 0 ≤ x ≤ L
A = 1 / √L
Infinite Well Sine State
φ(x) = sin(nπx/L) for 0 ≤ x ≤ L
A = √(2/L)
Odd Gaussian State
φ(x) = x e-αx2
A = 25/4α3/4/π1/4
Interval probability:
P(a,b) = ∫ab |ψ(x)|2 dx
How to Use This Calculator
- Select the wavefunction family that matches the state you want to normalize.
- Enter the needed parameters such as α, λ, box length L, or quantum number n.
- Set the probability interval you want to evaluate.
- Choose the graph range and sample count for the visualization.
- Click Normalize Wave Function to show the result above the form.
- Review the normalization constant, interval probability, numerical check, and plotted curves.
- Use the CSV button for spreadsheet data and the PDF button for a saved report.
FAQs
1) What does normalization mean in quantum mechanics?
Normalization makes the total probability equal to one. A valid wave function must give a full probability of finding the particle somewhere in its allowed region.
2) Why is the normalization constant important?
The constant scales the raw wavefunction into a physically meaningful state. Without that scaling, probability values and expectation calculations become inconsistent or misleading.
3) Why does the numerical check sometimes differ slightly from one?
Small differences come from finite graph ranges and numerical integration steps. Increasing the graph range or sample density usually moves the result closer to one.
4) Can I use this for bound and unbound style examples?
Yes. The calculator handles bounded box and sine states, plus unbounded Gaussian and exponential style states through exact constants and numerical verification.
5) What is the interval probability output?
It is the probability of finding the particle between your chosen start and end values. The calculator integrates the normalized probability density over that interval.
6) Why do sine and box states ignore values outside the box?
Those states are defined only inside the well or box. Outside that region, the wavefunction is zero, so the probability density also becomes zero.
7) What does the plotted density curve represent?
The density curve shows |ψ(x)|². That curve, not ψ(x) alone, determines measurable position probabilities across the plotted range.
8) Can this help with coursework and derivation checks?
Yes. It is useful for homework review, derivation checks, and quick comparisons between common wavefunction families when studying introductory or intermediate quantum mechanics.