The physical question
Which quantum states move without changing their probability profile?
Individual energy eigenstates acquire only phase factors: their position probabilities are stationary. A coherent state superposes number states with precisely coordinated phases. Its wave-packet center follows the classical oscillator trajectory while the spatial variance stays constant. A squeezed vacuum makes the quantum fluctuations themselves evolve.
Position-space probability
Wave-packet density |ψ(x,t)|²
The x coordinate is in oscillator units √(ℏ/(mω)); the plotted area under the probability density equals one. Number states have Hermite-polynomial nodes; coherent and squeezed vacua have Gaussian position densities.
Phase space
Wigner quasiprobability W(x,p)
Axes show canonical dimensionless x and p. Blue indicates positive Wigner values, orange indicates negative values; white is near zero. Wigner negativity for excited number states is physical quasiprobability structure, not an error in normalization.
Number statistics
How much of each energy eigenstate is present?
The oscillator energies are En = ℏω(n+½). Changing time alone rotates relative phases but leaves the probability of measuring each energy unchanged. The distribution below distinguishes Poisson statistics, definite excitation number and the even-parity structure of a squeezed vacuum.
Probabilities P(n), with the last displayed bin n=18. Any residual weight above the displayed range is reported below.
Mathematical derivation
Why does the coherent packet retain its shape?
With H = ℏω(a†a+½) and |α〉 satisfying a|α〉=α|α〉, write the initial state in the number basis:
Each |n〉 gains a phase e−iωt(n+½). The resulting relative phases are exactly those of a new coherent state with α(t)=αe−iωt. Its center is a classical oscillator orbit.
Derive the number-state position density
In oscillator units, ψn(x) = π−1/4Hn(x)e−x²/2/√(2nn!). Thus |ψn|² = e−x²Hn(x)²/(√π·2nn!). Hn obeys the recurrence Hn+1=2xHn−2nHn−1. Time changes only the global phase of each number state.
Why does the squeezed Gaussian rotate?
For ζ=reiφ, the initial covariance matrix has Vxx = ½(cosh2r−cosφ sinh2r), Vpp = ½(cosh2r+cosφ sinh2r), Vxp = −½sinφ sinh2r. Harmonic evolution applies a phase-space rotation R(t) to this matrix: V(t)=R(t)V(0)R(t)T. Its determinant remains ¼, while a single quadrature's variance oscillates.
Interpret the Wigner function
Gaussian states have positive Wigner functions W(z)=[2π√det V]−1exp[−½(z−μ)TV−1(z−μ)]. For |n〉, Wn(x,p)=(−1)nπ−1e−(x²+p²)Ln[2(x²+p²)], which can be negative. In all cases, integrating W over p yields the actual position density.
Independent exploration
Three informative comparisons
- Coherent oscillation. Set |α|=2, track the peak through quarter periods, then inspect 〈x〉 and the constant Δx². Which features agree with a classical phase-space orbit?
- Quantum nodes and negativity. Switch to |n=1〉 and |n=2〉. Locate spatial nodes and negative Wigner regions. These are differences between energy eigenstates and positive Gaussian states.
- Squeezing and its trade-off. Set squeeze r=1, vary the angle and time. Find where Δx² is minimal. Verify that the covariance determinant remains ¼ even when Δx²Δp² exceeds ¼ due to x–p covariance.
Self-check: For a coherent state in a harmonic potential, what happens to Δx² over one period?
Assumptions and further study
The plots describe an ideal one-dimensional harmonic oscillator with no environmental decoherence or anharmonicity. The coherent state is an infinite Fock superposition evaluated analytically; the occupation histogram truncates only the displayed bins and reports the omitted probability. Squeezed vacuum has strictly even number parity. For number states, the Wigner function is a signed quasiprobability, so it should never be interpreted as an ordinary joint probability over x and p.
Lecture-note sources: Oscillator ladder operators, Coherent states and Squeezed states.