A(x) is a real-space vector
Ψ(x) is a wave function
A acts on Ψ

Some Ψ, when acted upon by A, result in a multiple of the original Ψ. These Ψ are called eigenstates and may be denoted Ψₐ. In algebraic terms,

    A Ψₐ(x) = a Ψₐ(x), where a is complex.

Ψₐ is called an eigenstate of Α corresponding to the eigenvalue a.

If A is a Hermitian operator corresponding to some physical dynamical variable:
      ∞                 ∞
〈A〉 = ∫ Ψₐ⃰ A Ψₐ dx = a ∫ Ψₐ˟ Ψₐ dx = a
     -∞                -∞
ₐ
       ∞                  ∞                  ∞
〈A²〉 = ∫ Ψₐ⃰ A² Ψₐ dx = a ∫ Ψ⃰ₐ A Ψₐ dx = a² ∫ Ψₐ˟ Ψₐ dx = a² 
      -∞                 -∞                 -∞

σ² = 〈A²〉 - 〈A〉² = a² - a² = 0
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