Single Gaussian

Guided Inquiry

1 · The packet

Let’s meet the packet. A single quantum particle is described here by a real Gaussian centred at X₁: ψ₁(x) = e^{−a(x−X₁)²} / √N. The prefactor 1/√N normalises it, so the area under |ψ₁|² is always 1. The number a sets how tightly the packet is localised, and the plot can draw either |ψ₁(x)|² or ψ₁(x) itself.

2 · Squeezing the packet

Now let’s squeeze it. The localisation a is increased, so the packet occupies a narrower stretch of x. What do you think happens to the peak of |ψ₁|²?

3 · The wave and its square

Here’s a subtler one. The same packet can be drawn as ψ₁ or as |ψ₁|², on the same axes and at the same a. How do you think their two widths compare?

4 · How the peak scales

One last step: let’s put a number on it. Starting from a = 0.40, the localisation is quadrupled to a = 1.60. By what factor do you expect the peak of |ψ₁|² to grow?

Listen

Controls

Physics

X₁ (center)-2.00

-5 0+5

a (localization)1.000

0.1 σ=0.50 3.5

View

The Physics

The single Gaussian packet · what this sim shows, the physics behind it, and every control

At a glance

A quantum particle in a Gaussian state is not a small ball sitting at X₁: it is a probability spread over x. This sim lets you set the packet's centre X₁ and its localisation a, and draw either the amplitude ψ₁ or the probability density |ψ₁|².

What you'll learn

What ψ₁ and |ψ₁|² each mean, and which one is a probability density.
How the localisation a sets the packet's width and its height, and why the two are tied together.
How the width of ψ₁ compares with the width of |ψ₁|² on the same axes.
How narrowing the packet in position changes its spread in momentum.

The opening packet (X₁ = −2, a = 0.812) as the sim draws it.

What's on screen

Blue curve: the packet. It is |ψ₁(x)|² at first and Re(ψ₁(x)) after you flip the View toggle; the plot title says which.
Dashed line: the centre X₁.
Slider readouts: X₁ and a beside their sliders, and σ under the a slider, the width of the packet as the standard deviation of |ψ₁|².
Axes: x from −6 to 6, with no unit; the vertical axis refits to the curves, so read heights off its numbers.

See itReturn to the opening packet. Watch for: the blue bump centred on the dashed X₁ line at −2, with σ reading 0.555.

The controls

Press ⌖ on a row to see where that control sits in the sim.

Control	What it changes	Range
X₁ (center)	Where the packet is centred along x. Only the position of the packet changes.	−5 to +5 opens at −2
a (localization)	How tightly the packet is localised. The σ readout under the slider is the packet's width. The slider is logarithmic: a = 0.1 × 35^v/1000.	0.1–3.5 opens at 0.812
View	Draws \|ψ\|² (the probability density) or Re(ψ) (the amplitude itself).	\|ψ\|² / Re(ψ) opens on \|ψ\|²
The plot	The packet against x. Both views share one vertical scale that refits as the curves change.	x: −6 to 6
Maximize (plot corner)	Expands the plot to fill the panel; press again to bring the sliders back.	on / off
Reset	Puts X₁, a and the view back to their opening values.	top bar

Fixed: one real packet at one instant. The sim has no time evolution, no play button and no units.

What the curve represents

The sim's state is one Gaussian amplitude, with a normalising constant N:

ψ₁ is an amplitude; |ψ₁|² is the probability density. The probability of finding the particle between x and x + dx is |ψ₁(x)|² dx, and over all x it adds up to 1.
ψ₁ is real and never negative: the sim puts no phase factor in it. So the Re(ψ₁) view shows the whole wavefunction, not a slice of it.
Away from the plot edges, X₁ only slides the packet. Its shape, width and area are unchanged, which is why X₁ is the only readout that moves when you drag it.
The x axis has no unit. a is a plain number here; with a length unit attached it would have units of 1/length².

See itSlide the centre X₁ to the other side of the plot. Watch for: the whole bump moving with its shape unchanged, while a and σ keep the same readings.

Height and width are tied together

Setting the area to 1 gives N = √(π/2a). That fixes the density's width and peak in terms of a alone:

a is a localisation, not a width. The width of |ψ₁|² is σ = 1/(2√a), also written Δx, the spread in position. It falls only as 1/√a.
The area stays 1, so a narrower packet must be a taller one: the peak grows as √a. Quadrupling a (0.40 to 1.60) halves σ and doubles the peak, from 0.50 to 1.01.

|ψ₁|² at the slider's two ends and at its opening value. All three have area 1.

X₁ a

These sliders move the sim too.

The sim normalises over the plotted window x ∈ [−6, 6] only. If a wide packet is pushed against an edge, part of it is cut off, and the peak drawn is higher than √(2a/π).

See itQuadruple a, from 0.40 to 1.60. Watch for: the packet's width halving, with the card listing σ and the peak at both settings.

Squaring narrows the curve

The density is the amplitude squared, and squaring a Gaussian doubles its exponent coefficient from a to 2a:

On the same axes, |ψ₁|² is √2 narrower than ψ₁. The width of a Gaussian goes as 1/√(exponent coefficient).
σ = 1/(2√a) is the width of |ψ₁|². Taken as a curve, ψ₁ has width 1/√(2a) = √2 σ.
The peaks are linked by a square root: peak of ψ₁ = √(peak of |ψ₁|²). They are equal, both 1, at a = π/2 ≈ 1.57. Below that ψ₁ is the taller curve; above it |ψ₁|² is.
The two views are one state drawn two ways: they differ in how it is drawn, not in the state.

See itFlip the View toggle at the opening a = 0.812. Watch for: the curve turning wider and a little taller on the same axes when it switches to Re(ψ).

Squeezing in x spreads momentum

The same ψ₁ also fixes the packet's spread in momentum, though the sim never draws it. The Fourier transform of a Gaussian is a Gaussian:

Δx falls as 1/√a while Δp = ħ√a rises. Here a is in 1/length² and ħ in matching units.
The product stays at ħ/2 for every a. The uncertainty relation allows Δx Δp ≥ ħ/2 for any state, and a Gaussian sits exactly on that limit.
⟨p⟩ = 0, because ψ₁ is real: the packet has no net momentum. The sim draws one instant, so nothing moves.

Δp against Δx for the slider's whole range.

See itSweep a from 0.10 to 3.50. Watch for: the packet narrowing in the sim, while the card's Δp grows and Δx Δp stays at 0.500 ħ.

Where it matters

Harmonic traps. The ground state of a particle of mass m in a harmonic potential of angular frequency ω is exactly this Gaussian, with a = mω/2ħ and σ = √(ħ/2mω).
Trapped ions. A ⁴⁰Ca⁺ ion in the ground state of a trap with ω/2π = 1 MHz has σ ≈ 11 nm.
An electron, as an estimate. If you read the sim's x axis as nanometres, the opening packet has σ = 0.555 nm, and Δp = ħ√a gives an electron a speed spread of about 104 km/s. This is our own estimate; the sim carries no units.

Go deeper: a free packet spreads in time

A free Gaussian does not stay put. Its width grows as σ(t) = σ √(1 + (2ħat/m)²). That is real physics, and it is not in this sim, which has no time.
A packet with ⟨p⟩ ≠ 0 would carry a phase e^ikx and drift. The sim's ψ₁ is real, so its ⟨p⟩ is zero.

Summary

In one line. A Gaussian packet's localisation a fixes its width σ = 1/(2√a), its peak √(2a/π), and a momentum spread ħ√a, so that Δx Δp is always ħ/2.

Key points

|ψ₁|² is the probability density; ψ₁ is the amplitude.
Normalisation ties the packet's height to its width.
|ψ₁|² is √2 narrower than ψ₁ at the same a.
Squeezing in x spreads the momentum by exactly the amount that keeps Δx Δp at ħ/2.

Common mix-ups

"a is the width of the packet." a is the localisation. The width is σ = 1/(2√a), so a larger a gives a narrower packet.
"The Re(ψ) curve is the probability." It is the amplitude. The probability density is its square.
"Leave it alone and the packet will spread out." A free packet would, but this sim has no time in it. It draws ψ₁ at one instant.

Things to try on your own

Set a to about 1.57 and flip the View toggle. Compare the two peaks against the vertical axis.
Set X₁ = 5 and a = 0.10, then compare the peak you read off the axis with √(2a/π) = 0.252, and explain the difference.