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44 lines
1.7 KiB
Plaintext
44 lines
1.7 KiB
Plaintext
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Exam 2 Problem 2
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━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━
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(pic) Finished the problem using the boundary conditions
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boundary conditions limit the number of possible spherical harmonics.
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Two Similar Particles
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━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━
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❙a,b❭ = ❙a❭⊗❙b❭
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Exchange Operator
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──────────────────────────────────────────────────────────────────────────
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𝓟₁₂❙a,b❭ = ❙a,b❭
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𝓟₁₂(❙a❭⊗❙b❭) = ❙b❭⊗❙a❭
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If particles are indistinguishable
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𝓟₁₂❙a,b❭ = exp(iδ) ❙a,b❭ = λ ❙a,b❭
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𝓟²₁₂❙a,b❭ = λ² ❙a,b❭ = ❙a,b❭ ⇒ λ=±1
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Symmetry States
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──────────────────────────────────────────────────────────────────────────
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(pic) Two cases: symmetric, antisymmetric
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Symmetric States, λ=1
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𝓟₁₂❙a,b❭
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(pic) Constructed the exchange operator in matrix form, then found eigen states
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(pic) Showed that the exchange operator leads to the Pauli Exclusion Principle
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