Observables Are Operators, Averages Are Expectation Values
In quantum mechanics every observable, position, momentum, energy, is an operator acting on the wavefunction. Position is plain multiplication x̂, momentum reads a slope p̂ = −iℏ d/dx, and energy is the Hamiltonian H. One measurement returns a single eigenvalue of that operator, while the average over many runs is the expectation value ⟨A⟩ = ∫ψ*Âψ dx. How much the operators fail to commute, namely [x̂, p̂] = iℏ, is the structural root of uncertainty. We begin by seeing what ⟨x⟩ even means in the first figure.
Averaging an observable means taking a weighted average with the probability density |ψ|². The expectation of position is ⟨x⟩ = ∫ x|ψ|² dx, the center of mass of the bump. Move the bump with the slider or tilt it to one side and the ⟨x⟩ marker follows, while the width-σ band shows how widely outcomes spread around that average. Here σ = √(⟨x²⟩ − ⟨x⟩²).
Momentum lives not in the magnitude of the wavefunction but in the slope of its phase. Slide the k0 control that multiplies the plane-wave phase eik0x: |ψ|² (the filled curve) does not change at all, yet the expectation of momentum swings as ⟨p⟩ = ℏk0. That is because the momentum operator p̂ = −iℏ d/dx reads exactly this phase slope. Remember: position lives in the magnitude, momentum in the phase.
In an eigenstate of an operator, the measured result is definite. With a single energy eigenstate there is just one bar, the same eigenvalue every time, and the spread σE = 0. Mix in a second eigenstate with the slider and two bars |c₁|², |c₂|² split by Born probability, so a distribution appears. Eigenstate means definite, a mix means a distribution.
Why can some pairs never both be sharp? Applying x̂ first then p̂ to the same test function differs from applying p̂ first then x̂. Step through the two orderings and the difference is left over as exactly [x̂, p̂]ψ = iℏ ψ. This failure of the operators to commute is precisely the structural root of uncertainty.
How do averages connect to classical mechanics? Move a wave packet in a parabolic potential V(x) with the time slider and the average position ⟨x⟩ oscillates left and right like a classical ball. This is Ehrenfest's theorem: d⟨x⟩/dt = ⟨p⟩/m and d⟨p⟩/dt = −⟨dV/dx⟩, so the expectation values follow the classical equations of motion on average.