Expectation Value Calculator

An expectation value ⟨A⟩ is the probability-weighted average of the possible outcomes of measuring an observable A, computed as ⟨ψ|A|ψ⟩ for a normalized state ψ. It is not the result of any single measurement (which always yields one specific eigenvalue), but the average you would obtain repeating the same measurement on many identically-prepared systems.

For the infinite square well centered symmetrically, the probability density is symmetric about the well's midpoint, so ⟨x⟩ = L/2 (the midpoint, not zero, since the well spans 0 to L) and ⟨p⟩ = 0 by symmetry of the momentum-space wavefunction. For the quantum harmonic oscillator, the potential is symmetric about x=0, so both ⟨x⟩ = 0 and ⟨p⟩ = 0 for every stationary state.

Δx and Δp are the standard deviations (uncertainties) of position and momentum in a given quantum state, computed as Δx = √(⟨x²⟩ − ⟨x⟩²) and Δp = √(⟨p²⟩ − ⟨p⟩²). They quantify how spread out a repeated measurement of position or momentum would be, not any single measurement's precision.

The n=0 quantum harmonic oscillator state is the unique 'minimum uncertainty' state for this system: its ΔxΔp product equals exactly ħ/2, the smallest value the Heisenberg uncertainty principle allows. Every other state, in either the oscillator or the box, has a strictly larger ΔxΔp.

No. Even its ground state (n=1) gives ΔxΔp ≈ 1.14 × (ħ/2), noticeably above the absolute minimum, because the box's hard walls force a wavefunction shape different from the perfect Gaussian that uniquely minimizes the uncertainty product. Higher box states push the product up further, roughly proportional to n for large n.

For the harmonic oscillator, ΔxΔp = (n + 1/2)ħ exactly, growing linearly with n. For the infinite square well, ΔxΔp also grows with n (though not by quite as simple a formula), since higher-energy states have more oscillations packed into the same box, spreading out the momentum distribution.

The infinite square well and quantum harmonic oscillator are the two textbook systems with fully exact, closed-form energy eigenstates and expectation values, no numerical approximation is needed for either. This makes them the standard first examples for learning to compute expectation values and verify the uncertainty principle explicitly.

For a well of width L with the stationary state ψ_n(x) = √(2/L) sin(nπx/L) on [0,L], ⟨x²⟩ = L²(1/3 − 1/(2n²π²)). Combined with ⟨x⟩ = L/2, this gives the exact position uncertainty Δx for any quantum number n.

For a stationary state n with angular frequency ω = 2πf, ⟨x²⟩ = (ħ/mω)(n + 1/2), and correspondingly ⟨p²⟩ = mħω(n + 1/2). These come directly from the ladder-operator algebra of the harmonic oscillator and are exact at every n.

No, it is deliberately scoped to the two exactly-solvable systems with the simplest, cleanest closed-form expectation values. Hydrogen atom expectation values (like ⟨r⟩ or ⟨r²⟩ for a given n, l) follow similar exact formulas but involve more complex expressions specific to the Coulomb potential, outside this calculator's scope.