Gaussian Wave Function Calculator

Calculator

Enter Gaussian wave parameters. The layout uses three columns on large screens, two on smaller screens, and one on mobile.

Formula Used

The calculator uses a complex Gaussian wave packet:

ψ(x) = A × exp[-(x − μ)² / (2σ²)] × exp[i(kx + φ)]

Real part: Re(ψ) = A × exp[-(x − μ)² / (2σ²)] × cos(kx + φ)

Imaginary part: Im(ψ) = A × exp[-(x − μ)² / (2σ²)] × sin(kx + φ)

Probability density: |ψ|² = Re(ψ)² + Im(ψ)²

For normalized mode: A = 1 / (π^(1/4) × √σ)

How to Use This Calculator

1. Select normalized amplitude for probability work.
2. Select manual amplitude for scaled wave studies.
3. Enter the center value μ.
4. Enter a positive width σ.
5. Enter the wave number k and phase φ.
6. Add the selected x position for a point result.
7. Set range start, range end, and table points.
8. Press Calculate to view results above the form.
9. Use CSV or PDF download for saving results.

Example Data Table

What This Calculator Does

A Gaussian wave function is a compact model for a wave packet. It is useful in quantum study, signal work, and probability modeling. This calculator turns chosen parameters into real values, imaginary values, magnitude, and probability density. It also estimates normalization and basic uncertainty.

Why Gaussian Waves Matter

Gaussian forms are popular because they are smooth and localized. The center sets the main position. The width controls spread. A small width gives a sharp packet. A large width gives a broad packet. The wave number adds oscillation. The phase shifts the cycle without moving the envelope.

Reading the Output

The real part and imaginary part describe the complex wave. Their square sum gives probability density. The magnitude shows the envelope height at the selected point. When normalized mode is used, the total probability over all space is treated as one. Manual amplitude mode is useful for scaled studies and classroom checks.

Practical Uses

Use the range table to inspect many x values at once. It helps you see symmetry around the center. It also shows where the wave becomes very small. This is useful when choosing plotting limits. The CSV download supports spreadsheets. The PDF option helps keep a quick record.

Accuracy Notes

The calculator uses the common one dimensional Gaussian packet form. It assumes hbar equals one for the uncertainty estimate. It does not solve a full molecular orbital problem. It also does not run external quantum chemistry software. Instead, it evaluates the mathematical Gaussian expression directly. That makes it fast and transparent.

Better Input Choices

Start with center zero and width one. Then change one parameter at a time. Increase wave number to add faster oscillations. Change phase to shift real and imaginary parts. Use normalized mode when probability interpretation matters. Use manual mode when comparing relative amplitudes.

Final Tip

Gaussian functions are simple but powerful. They connect algebra, probability, and wave behavior. With careful parameters, this calculator becomes a useful conversion tool for turning abstract wave settings into readable numbers.

Use these results as guides. Verify critical research with trusted software. Document units, assumptions, and ranges before sharing any technical conclusion with care.

FAQs