quantum operators expectation value
In quantum mechanics every observable, position, momentum, and energy, is an operator that acts on the wave function. Position is multiplication by x̂, momentum reads the slope as p̂ = −iℏ d/dx, and energy is the Hamiltonian H. One measurement returns one eigenvalue of that operator, and the average of many measurements is the expectation ⟨A⟩ = ∫ψ*Âψ dx. How far the order of these operators refuses to swap, namely [x̂, p̂] = iℏ, is the structural root of uncertainty. The first picture shows what ⟨x⟩ is.
Averaging an observable means 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 hump. Move the hump or tilt it and the ⟨x⟩ mark follows, and a band of width shows how far the results spread around that average. That width is √(⟨x²⟩ − ⟨x⟩²).
Momentum lives in the slope of the phase, not in the size of the wave function. Move the slider that multiplies by the plane-wave phase eik0x, with wavenumber k0, and |ψ|² stays put while only the expectation of momentum shifts as ⟨p⟩ = ℏk0. The momentum operator p̂ = −iℏ d/dx is what reads that slope of the phase. Position lives in the size, and momentum lives in the phase.
In an eigenstate of an operator the measurement result is fixed. In a single energy eigenstate there is only one bar, so the same eigenvalue always comes out and the spread is σE = 0. Mix in a second eigenstate and the two bars |c₁|², |c₂|² split by the Born probabilities and a distribution appears. An eigenstate is certainty, and a mix is a distribution.
Why can some pairs not be sharp at the same time. Acting with x̂ first and then p̂ on the same function is not the same as acting with p̂ first and then x̂. Follow the two orders and the difference that remains is exactly [x̂, p̂]ψ = iℏ ψ. That the operators do not commute is the structural root of uncertainty.
How does the average connect to classical mechanics. Move a wave packet inside the parabolic potential V(x) and the average position ⟨x⟩ oscillates left and right like a classical ball. That is Ehrenfest's theorem. d⟨x⟩/dt = ⟨p⟩/m and d⟨p⟩/dt = −⟨dV/dx⟩, so the expectations follow the classical equation of motion on average.