Two-Gaussian Superposition

Guided Inquiry

1 · Two packets, one particle

Here’s our cast. Two Gaussian packets sit on one axis: ψ₁ centred at X₁, ψ₂ centred at X₂, each localized by a. The particle’s state is the superposition Ψ = (αψ₁ + βψ₂)/√N′, with α² + β² = 1. The lower panel draws |Ψ(x)|², and over it you’ll see a dashed curve — the two packet densities added with those same weights.

2 · Two chances

Let’s start with the classical view. Classically, a particle that’s partly at X₁ and partly at X₂ simply has two chances of turning up. On that reckoning, what do you think its probability density is at any point on the axis?

3 · Bringing them together

Now let’s bring them together. The second centre X₂ comes in to 0, so the packets overlap appreciably. Midway between the two centres, how do you think |Ψ(x)|² will compare with the dashed curve?

4 · Paying for the rise

Here’s a little puzzle. Both curves are normalized: each encloses a total probability of exactly 1. Yet the solid curve now stands well above the dashed one between the centres. So what must follow elsewhere?

5 · Reading the amplitude

Now let’s peek at the amplitude itself. Read the same state as Re(ψ) instead of |ψ|², and the lower panel draws αψ₁ and βψ₂ faint and Ψ solid. Midway between the centres, where do you think the solid curve sits?

6 · Narrowing the packets

One last experiment. Back in |ψ|², a is raised so both packets become narrower, with X₁ and X₂ left where they are. What do you think happens to the gap between the solid and dashed curves?

Listen

Parameters

Physics

X₁ (center)-2.00

X₂ (center)2.00

a (localization)0.812

α (coefficient)0.707

β (coefficient)0.707

Coefficient choice: α² + β² = 1.000

View

The Physics

Two-Gaussian superposition · what this sim shows, the physics behind it, and every control

At a glance

A particle can be prepared in a superposition of two Gaussian wave packets, one centred at X₁ and one at X₂. This sim draws each packet alone, then the combined state's probability density, and sets that density beside a classical estimate so you can see how the two packets combine.

What you'll learn

How the state Ψ is built from two packets, and what the weights α and β do.
How |Ψ|² compares with the classical curve, and what sets the difference between them.
How the overlap S depends on the packet width a and the separation of the centres.
How a density that always encloses a probability of 1 can still change shape.

The three panels at first load. Packets 1 and 2 are drawn alone; the bottom panel is the combined state.

What's on screen

Top panel, |ψ₁(x)|² (blue): packet 1 alone, centred at X₁.
Middle panel, |ψ₂(x)|² (pink): packet 2 alone, centred at X₂.
Bottom panel, |Ψ(x)|² (green): the combined state.
Dashed grey curve: the "Classical (normalized)" comparison. The shading between the two curves is green where |Ψ|² is above it and pink where it is below; a ▲ or ▼ label marks the midpoint when the gap there is large.
Dotted lines X₁, X₂ mark the packet centres. The sidebar shows each slider's value and the check α² + β².

See itReturn to the sim's opening state: X₁ = −2, X₂ = 2, a = 0.812, α = β = 0.707. Watch for: a blue packet at X₁, a pink packet at X₂, and a two-humped green curve below them.

The controls

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

Control	What it changes	Range
X₁ (center)	Where packet 1 is centred.	−5 to 5 opens at −2
X₂ (center)	Where packet 2 is centred.	−5 to 5 opens at 2
a (localization)	How tightly both packets are squeezed. A larger a gives narrower packets.	0.1 to 3.5 opens at 0.812
α (coefficient)	The weight of packet 1 in Ψ. Moving it moves β so that α² + β² = 1.	0 to 1 opens at 0.707
β (coefficient)	The weight of packet 2 in Ψ. It is tied to α in the same way.	0 to 1 opens at 0.707
Coefficient choice	A check that prints α² + β², which stays at 1.000 (to rounding).	readout
View: \|ψ\|² / Re(ψ)	Switches all three panels between probability densities and amplitudes.	opens on \|ψ\|²
Maximize (panel corner)	Enlarges one panel to fill the plot area. Click again to restore.	per panel
↻ Reset	Returns every slider to its opening value and the view to \|ψ\|².	top bar

Fixed: the two packets are in phase, α and β are never negative, and the x-axis runs from −6 to 6, with every density normalised over that window.

How the state is built

Each packet is a normalised Gaussian amplitude. Both share the width parameter a:

One packet. Its width comes from a alone.

The state adds the two amplitudes, weighted by α and β, and divides by √N′ so that the total probability is 1:

The superposition and its normalisation. S is the overlap of the two packets.

Width. Δxᵢ is the standard deviation of |ψᵢ|². At a = 0.812 it is 0.555; across the slider range it runs from 1.58 (a = 0.1) down to 0.27 (a = 3.5).
Weights. α and β appear only in Ψ. The top two panels show |ψ₁|² and |ψ₂|² unweighted, so they do not change when you move α or β.
Phase. The sim keeps the packets in phase, so Ψ is real and never negative.

See itSet α to 0.9, which makes β = 0.436. Watch for: the top two panels keep their shape and height, while the bottom curve shifts weight towards X₁.

Adding amplitudes, not probabilities

Two classical alternatives contribute their probabilities, giving P_cl = (α²|ψ₁|² + β²|ψ₂|²)/(α² + β²). This is the sim's dashed curve, which encloses a total of 1. Quantum mechanics adds the amplitudes first and squares afterwards:

Squaring a sum is not summing the squares.

Cross term. The first two terms are the classical weights. The extra term 2αβψ₁ψ₂ is non-zero only where both packets are.
Sign. Because the packets are in phase and α, β ≥ 0, ψ₁ψ₂ is positive everywhere. The cross term adds to the density before it is normalised.
Amplitude view. In Re(ψ) the bottom panel is the pointwise sum of the two faint curves behind it. Nothing is squared there, so there is no cross term to see.

See itBring X₂ in to 0 and switch to Re(ψ). Watch for: the solid bottom curve is the sum of the two faint curves; midway between the centres it stands at twice either one.

How much the packets overlap

The overlap S sets the size of the cross term's effect:

S depends on a and the separation only through a(X₁ − X₂)².

S against separation, at the default a and at a = 2.053 (the narrowed setting used below).

X₁ X₂ a α

These sliders move the sim too. β follows α.

Scaling. Halving the separation has the same effect on S as cutting a to a quarter.
Reach. S falls to ½ when the centres are one packet FWHM apart, of |ψᵢ|², which is 1.31 at the default a. It is 0.197 at separation 2 and 0.0015 at separation 4.
Visibility. At first load the midpoint of |Ψ|² is already 2.0 times the dashed curve, but it stands at only 0.6% of the peak height, so you see nothing.

See itPut the centres 2 apart, then narrow the packets. Watch for: the green bump between the centres, then the solid and dashed curves merging after a rises to 2.053.

Where the probability goes

Both curves enclose a total probability of 1, so interference only moves probability from one place to another. The sim divides |Ψ|² by N′ and the dashed curve by α² + β².

With α² + β² = 1, the gap between the curves is

|Ψ|² − P_cl = (2αβ/N′)(ψ₁ψ₂ − S·P_cl)

The two areas cancel, because ∫ψ₁ψ₂ dx = S and ∫P_cl dx = 1.

Gain. |Ψ|² is above the dashed curve around the midpoint (green), where ψ₁ψ₂ is large.
Loss. It is below the dashed curve (pink) on the outer flanks. With the centres 2 apart at the default a, the two peaks sit in this region too and fall to 0.90 of the dashed curve. At a = 0.1 the packets are so wide that the peaks lie in the green region instead.
Legend. The legend names only the green shading. The pink is drawn but not labelled.

See itBring X₂ in to 0 at the default a. Watch for: green shading between the centres, and pink on the outer flanks and at both peaks.

Where it matters

Chemical bonds. The bonding orbital of H₂⁺ is (φ_A + φ_B)/√(2(1 + S)), which is this sim's Ψ at α = β. The extra density between the nuclei is the textbook picture of the bond. H₂⁺ is bound by about 2.8 eV at a separation of 0.106 nm (literature values; the sim has no potential and computes neither).
Gaussian basis sets. Quantum-chemistry codes build orbitals from Gaussians because overlaps such as S, and other integrals over Gaussians, have closed forms.
Any two-route state. Whenever a particle can reach a detector by two routes, as in a double slit, the amplitudes add before the density is found.

Go deeper: the antibonding state

Two 1s-like Gaussians on two nuclei. The bonding curve is this sim at α = β.

The same normalisation, written the way a chemist writes it.

Flipping the sign of β reverses the cross term and puts a node between the centres. The sim's sliders stop at β = 0, so this case is out of its range.

Summary

In one line. A superposition adds amplitudes before squaring, so |Ψ|² differs from the classical curve by a cross term set by S = e^{−a(X₁−X₂)²/2}, and normalisation means it only moves probability around.

Key points

Ψ = (αψ₁ + βψ₂)/√N′; the weights act on Ψ only.
The classical curve adds probabilities. |Ψ|² adds amplitudes first, leaving the cross term 2αβψ₁ψ₂.
S falls as the centres move apart or the packets narrow.
Both curves enclose 1, so a gain between the centres is paid for elsewhere.

Common mix-ups

"The particle is in two places at the same time." A position measurement returns one x. The sim plots only the density |Ψ|².
"A superposition just means we don't know which packet it is in." Not knowing which packet would give the dashed curve. The solid curve differs from it wherever the packets overlap.
"More separation means more interference." The reverse: S, and with it the effect, falls as the centres move apart.

Things to try on your own

Set X₁ = X₂ and drag α. Check that the solid curve stays on the dashed one.
With X₂ = 0, use the live panel in section 05 to find the a at which S = ½, then compare it with 2 ln 2/d² for d = 2.
Set α = 1 so that β = 0. Check that |Ψ|² matches the top panel.