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37 lines
940 B
Plaintext
37 lines
940 B
Plaintext
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A particle in an infinite square well has an initial state vector
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❙Ψ(t=0)❭ = A(❙φ₁❭ - ❙φ₂❭ + ι❙φ₃❭).
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where ❙φₙ❭ are the energy eigenstates. This also means
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❬Ψ(t=0)❙ = A⃰(❬φ₁❙ - ❬φ₂❙ + ι❬φ₃❙)
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❙Ψ(t=0)❭ = _͟A͟ (αβ❙φ₁❭ - βγ❙φ₂❭ + αγι❙φ₃❭)
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αβγ
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In the energy basis,
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❙φ₁❭ ≐ ⎛1⎞ ❙φ₂❭ ≐ ⎛0⎞ and ❙φ₃❭ ≐ ⎛0⎞
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⎜0⎟ ⎜1⎟ ⎜0⎟
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⎝0⎠, ⎝0⎠, ⎝1⎠.
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So,
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❙Ψ(t=0)❭ ≐ ⎛ A ⎞
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⎜-A ⎟
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⎝ιA ⎠.
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(𝐚) Multiplying the state vector by its magnitude normalizes it.
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❙Ψ′(t=0)❭ ≐ __͟A͟__ ⎛ 1 ⎞ = _͟1͟ ⎛ 1 ⎞
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√(3A²) ⎜-1 ⎟ √3 ⎜-1 ⎟.
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⎝ ι ⎠ ⎝ ι ⎠
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