Expectation Value Calculator
Calculate expectation values of quantum observables for various systems
Expectation Values
Position ⟨x⟩
0.0000e+0 m
Position² ⟨x²⟩
1.7366e-19 m²
Momentum ⟨p⟩
0.0000e+0 kg m/s
Momentum² ⟨p²⟩
1.4409e-49
Energy ⟨E⟩
1.5819e-19 J
Position Uncertainty Δx
4.1672e-10 m
Momentum Uncertainty Δp
3.7959e-25
ΔxΔp Product
1.5819e-34
About Expectation Values
The expectation value ⟨A⟩ = ∫ψ*Âψ dx gives the average result of measuring observable A on a quantum system in state ψ. The uncertainty ΔA = √(⟨A²⟩ - ⟨A⟩²) measures the spread of measurement outcomes. The product ΔxΔp must satisfy the Heisenberg uncertainty principle: ΔxΔp ≥ ℏ/2.