At a glance
A mass on a spring and a quantum wavepacket sit side by side in the same harmonic trap. The sim shows what the classical and quantum descriptions share and what only the quantum one has, which is the clearest way to see what "quantum" adds to an oscillator.
What you'll learn
- How Newton's equation and the Schrödinger equation describe motion in the same potential V(x) = ½mω²x².
- How to read the density |ψ(x,t)|², and what the red bands and the orange region mean.
- How the packet's width Δx = σ depends on m, ω and A.
- How the classical energy Ecl compares with the quantum mean energy ⟨H⟩.
Both columns share V(x) and start at rest at +A. Amber: Ecl. Red dashed: the turning points ±A. Orange: |ψ|² beyond +A.
What's on screen
- Grey curve: the potential V(x) = ½mω²x², drawn in both columns.
- Amber dashed line: the classical energy Ecl = ½mω²A². It meets the curve at the red dashed lines x = ±A, the turning points.
- Red "Forbidden" bands: |x| > A, where V(x) > Ecl. A classical particle with this energy cannot be there.
- Blue mass on a spring, with a green arrow whose length is its speed.
- Purple outline: the probability density |ψ(x,t)|², filled blue inside ±A and orange beyond. A purple dashed line marks its centre, and the ±σ ticks mark its width.
- Live readouts: both positions, Δx = σ, k, ω, A, both energies and the turning points.
See itRun the opening settings, m = 1, ω = 1, A = 2. Watch for: the amber line meeting the grey curve exactly at the red ±A lines, and orange appearing on the density whenever it reaches past a red line.
The controls
Press ⌖ on a row to see where that control sits in the sim.
| Control | What it changes | Range | |
|---|---|---|---|
| Mass m | The mass of the block and of the quantum particle. It also sets the spring constant k = mω². | 0.25–4 opens at 1 | |
| Angular frequency ω | How fast the motion repeats, with period 2π/ω. It also sets k = mω². The motion carries on from its current phase. | 0.25–4 opens at 1 | |
| Amplitude A | How far the mass swings from x = 0, and where the packet's centre swings to. It sets the turning points ±A and Ecl. | 0.5–5 opens at 2 | |
| Legend | The key to the colours: the classically allowed region, the forbidden-region probability and the potential. | panel | |
| Live readouts | Classical x and Quantum ⟨x⟩, Δx = σ, k, ω, A, Classical E, Quantum ⟨H⟩ and the turning points. | panel | |
| Pause / Play | Freezes or runs both columns at the same moment, so you can compare them at one instant. | top bar | |
| Reset | Rewinds the clock to t = 0, with both objects at +A. The sliders keep their values, except in the Guided Inquiry, which restarts from m = 1, ω = 1, A = 2. | top bar |
Fixed: ħ = 1, so m, ω, A and the energies are in the sim's own units. The packet always has the shape of the oscillator's ground state, displaced to +A and released at rest.
The centre follows the classical path
The left column obeys Newton's equation. Released from rest at +A, the mass oscillates:
Force, equation of motion, solution.
The right column obeys the Schrödinger equation. For this potential the packet's mean position ⟨x⟩ obeys the same equation as the mass, because the force −mω²x is linear in x:
The mean position obeys the classical equation.
- Both columns start at +A at rest, so ⟨x⟩(t) = x(t) = A cos ωt. The readouts Classical x and Quantum ⟨x⟩ show the same number.
- ⟨x⟩ is the centre of the distribution, not a position the particle is known to have.
- The sim draws the packet from its known exact solution; it does not integrate the Schrödinger equation.
Key idea. For a linear force, the centre of the packet moves exactly as the classical mass does. The next sections cover what is not the same.
See itRelease both from +2 and let them swing. Watch for: Classical x and Quantum ⟨x⟩ showing the same number, and the packet's dashed centre line sitting at the mass's x.
The width is fixed by m and ω
The sim draws a Gaussian density whose centre is ⟨x⟩(t). Its standard deviation is the readout Δx = σ:
The density the sim plots, and its width.
- No t. The packet keeps the same width as it swings. A free packet would spread out; in this trap the packet does not.
- Only mω. A larger m and a larger ω narrow the packet in the same way, because σ depends on their product.
- No A. The amplitude moves the packet but does not change its width.
See itSet m = 4 with ω = 1 and A = 2. Watch for: Δx = σ reading 0.354 instead of 0.707 at the opening settings, and the packet staying that narrow as it swings.
The quantum energy has a floor
The classical energy is fixed by the swing. The turning points ±A are where V(x) = Ecl.
The quantities the readouts show for the classical side.
The packet's mean energy is larger by a constant, with n̄ = Ecl/ħω, the number of quanta the swing is worth:
The mean energy of the packet.
m
ω
A
These sliders move the sim too.
- ⟨H⟩ exceeds Ecl by ½ħω at every instant and every amplitude. This floor depends on ω only; it is the zero-point energy.
- Shrinking A lowers Ecl towards 0, but ⟨H⟩ levels off at ½ħω.
See itShrink the amplitude to A = 0.5 with m = ω = 1. Watch for: Classical E reading 0.125 while Quantum ⟨H⟩ reads 0.625.
Probability past the turning points
The orange part of |ψ|² lies where a classical particle with energy Ecl could not be. It is ordinary probability in a bound state: nothing leaves the trap, and the total area stays 1.
The orange share when the packet is centred on x = 0, and the product of the two spreads.
- At the opening settings the orange share at x = 0 is erfc(2) = 0.47%. At A = 0.5 it is 48%.
- When the centre reaches +A, half the packet lies beyond the +A line, because a Gaussian is symmetric about its own centre.
- The packet's momentum spread is Δp = √(mωħ/2), so Δx Δp = ħ/2: the smallest product the uncertainty relation allows. The sim shows Δx only.
See itUse A = 0.5 with m = ω = 1 and let the packet swing. Watch for: roughly half of the density drawn in orange at every moment, while the mass stays inside ±0.5.
Where it matters
The same comparison at n̄ = 2 and at n̄ ≈ 3×10²⁷.
- Molecular vibrations. Small vibrations of a bond are close to harmonic. For CO at 300 K the mean thermal occupation is about 3×10⁻⁵, so the molecule is almost always in its ground state, where all of its vibrational energy is the ½ħω floor.
- Trapped ions and laser light. An ion in a trap moves in a nearly harmonic potential and can be cooled to near its ground state. A single mode of laser light is described by a coherent state, the kind of state this sim draws.
- Everyday objects. A 1 g pendulum swinging 1 cm once a second has n̄ ≈ 3×10²⁷. The floor is about 2 parts in 10²⁸ of its energy, far too small to see.
Go deeper: what a coherent state is made of
- The packet is not an energy eigenstate. It is a superposition of oscillator levels n = 0, 1, 2, … whose mean is n̄, and ⟨H⟩ = ħω(n̄ + ½).
- With A = 0 it is the oscillator's ground state. The sim's slider stops at A = 0.5.
Summary
In one line. In a harmonic trap the packet's centre follows the classical motion exactly, while its width σ = √(ħ/2mω), its probability beyond ±A and its ½ħω energy floor are what the classical picture lacks.
Key points
- ⟨x⟩(t) = A cos ωt on the quantum side as well as the classical side.
- σ = √(ħ/2mω) depends on the product mω, and not on A or t.
- ⟨H⟩ = Ecl + ½ħω, so ⟨H⟩ never falls below ½ħω.
Common mix-ups
- "The orange tails mean the particle is leaking out of the trap." Nothing escapes: the state is bound and |ψ|² stays normalised.
- "A smaller amplitude makes the physics more classical." It makes it less classical: n̄ = mωA²/2ħ falls, and the classical limit is large n̄.
- "⟨x⟩ is where the particle is." ⟨x⟩ is the centre of a distribution of width σ, not the result of one measurement.
Things to try on your own
- Find two different pairs (m, ω) with the same product mω and compare their Δx = σ.
- With m = ω = 1, find the amplitude at which ⟨H⟩ is 10% above Ecl, then check it with ⟨H⟩/Ecl = 1 + 1/(2n̄).
- Set A = 0.5, pause at several moments, and estimate how much of the packet is orange at each.