Calculator
Formula used
This tool reports the probability density ρ(x)=|ψ(x)|², where ψ(x) is a normalized wavefunction. Units follow normalization: in one dimension, ρ has units of 1/length.
- Gaussian wave packet: ψ(x)=(1/(πσ²))^(1/4) · exp(-(x−x₀)²/(2σ²)), so ρ(x)= (1/(√π σ)) · exp(-(x−x₀)²/σ²).
- Infinite square well (0 to L): ψₙ(x)=√(2/L) · sin(nπx/L) for 0≤x≤L, and 0 otherwise. Then ρ(x)= (2/L) · sin²(nπx/L).
- Harmonic oscillator ground state: with α=mω/ħ, ψ₀(x)=(α/π)^(1/4)·exp(−αx²/2), so ρ₀(x)=√(α/π)·exp(−αx²).
How to use this calculator
- Select a model that matches your physical system.
- Enter the position x and choose its unit.
- Fill the model parameters (σ and x₀, or L and n, or m and ω).
- Optional: enter an interval [a,b] to compute probability in that region.
- Press Calculate to display results above the form.
- Use Download CSV or Download PDF for exports.
Example data table
| Model | Inputs | x (m) | ρ(x) (1/m) |
|---|---|---|---|
| Gaussian | x₀=0 m, σ=0.01 m | 0.000 | 56.41896 |
| Gaussian | x₀=0 m, σ=0.01 m | 0.010 | 20.75537 |
| Square well | L=1 m, n=1 | 0.500 | 2.00000 |
| Square well | L=1 m, n=2 | 0.250 | 2.00000 |
| Oscillator | m=9.11e−31 kg, ω=1e13 | 0.000 | 5230078.2 |
Understanding probability density in one dimension
In quantum physics, ψ(x) carries amplitude, while ρ(x)=|ψ(x)|² gives the likelihood per unit length near x. In one dimension, ρ has units of 1/length so that the normalization ∫ρ(x)dx equals 1. This calculator evaluates ψ(x), ρ(x), and an optional interval probability P[a,b] for three widely used wavefunctions.
Model 1: Gaussian wave packet
Gaussian packets describe localized particles and short pulses. The width σ controls spread: larger σ lowers the peak density and increases coverage, without changing total probability. Shifting x₀ moves the distribution along x, which is useful for modeling displaced initial states.
Model 2: Infinite square well
The infinite well confines motion to 0≤x≤L and creates standing waves. ψₙ(x)=√(2/L)sin(nπx/L) produces nodes where ψ=0 and a density ρ(x) that oscillates across the well. Increasing n adds more lobes and sharper spatial variation.
Model 3: Harmonic oscillator ground state
The harmonic oscillator approximates motion near a stable equilibrium. The ground state is Gaussian with α=mω/ħ, so heavier masses or larger ω make the distribution narrower. This model underpins vibrational physics and near-minimum approximations in many potentials.
Interval probability and quick validation
P[a,b]=∫abρ(x)dx is dimensionless and must lie between 0 and 1. If you choose a broad interval for the Gaussian or oscillator models, the probability should approach 1. For a Gaussian, the region x₀±2σ contains about 95% of the total probability. For the well model, the interval is effectively limited to the physical region [0,L].
Units, scaling, and reporting
Because ρ scales as 1/length, changing the length unit changes the numeric density. A value reported as 10 1/m corresponds to 0.10 1/cm. For consistent comparisons, keep units fixed across runs and report both the parameters and the chosen unit system. As a guide, teaching examples often use L from 0.1 to 2 m and σ from 0.001 to 0.1 m.
Interpreting peaks, nodes, and tails
A higher ρ(x) means more probability density at that location, not certainty at a single point. Nodes in the well force ρ=0 at specific x values, while Gaussian tails never become exactly zero. For decisions, compare integrated probability over regions instead of single-point values.
Using exports for analysis workflows
Use CSV export to collect multiple evaluations into spreadsheets and plots. Use PDF export to capture a clean snapshot for lab notes or sharing. For model checks, sample ρ(x) at many x values and verify that numerical integration approximates 1 within your tolerance.
FAQs
1) Why does probability density have units of 1/length?
Because total probability is an integral: ∫ρ(x)dx must equal 1. In one dimension, dx has units of length, so ρ must carry 1/length to keep the probability dimensionless.
2) Is ψ(x) always real in this calculator?
These three models are implemented in their common real-valued forms. Many quantum states are complex, but the probability density still uses ρ=|ψ|², which the calculator conceptually follows.
3) What does a larger σ mean for the Gaussian model?
A larger σ spreads the wave packet over a wider region. The peak probability density decreases while the distribution becomes broader, keeping total probability normalized to 1.
4) Why is ρ(x)=0 outside the square well?
The infinite well assumes impenetrable walls, so the wavefunction is constrained to be zero outside 0≤x≤L. Squaring it keeps the probability density zero there.
5) How should I choose interval limits [a,b]?
Pick a region that matches your question, such as “within one σ of the center” or “the left half of the well.” The output P[a,b] summarizes probability mass in that region.
6) What happens if I swap a and b?
The calculator automatically orders the interval so that a≤b. This prevents negative probabilities and makes it easier to enter bounds without worrying about direction.
7) Why can ρ(x) be very large for the oscillator example?
If α is large, the distribution becomes very narrow, concentrating probability near x=0. A narrow distribution can have a high peak density while still integrating to a total probability of 1.