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23 lines
805 B
Plaintext
23 lines
805 B
Plaintext
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A(x) is a real-space vector
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Ψ(x) is a wave function
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A acts on Ψ
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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,
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A Ψₐ(x) = a Ψₐ(x), where a is complex.
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Ψₐ is called an eigenstate of Α corresponding to the eigenvalue a.
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If A is a Hermitian operator corresponding to some physical dynamical variable:
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∞ ∞
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〈A〉 = ∫ Ψₐ⃰ A Ψₐ dx = a ∫ Ψₐ˟ Ψₐ dx = a
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-∞ -∞
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ₐ
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∞ ∞ ∞
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〈A²〉 = ∫ Ψₐ⃰ A² Ψₐ dx = a ∫ Ψ⃰ₐ A Ψₐ dx = a² ∫ Ψₐ˟ Ψₐ dx = a²
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-∞ -∞ -∞
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σ² = 〈A²〉 - 〈A〉² = a² - a² = 0
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ᴬ
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