Exam 2 Problem 2
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(pic) Finished the problem using the boundary conditions

    boundary conditions limit the number of possible spherical harmonics.



                        Two Similar Particles
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❙a,b❭ = ❙a❭⊗❙b❭

                            Exchange Operator
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𝓟₁₂❙a,b❭ = ❙a,b❭

𝓟₁₂(❙a❭⊗❙b❭) = ❙b❭⊗❙a❭

If particles are indistinguishable 

𝓟₁₂❙a,b❭ = exp(iδ) ❙a,b❭ = λ ❙a,b❭

𝓟²₁₂❙a,b❭ = λ² ❙a,b❭ = ❙a,b❭ ⇒ λ=±1


                            Symmetry States
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(pic) Two cases: symmetric, antisymmetric

Symmetric States, λ=1

    𝓟₁₂❙a,b❭

(pic) Constructed the exchange operator in matrix form, then found eigen states

(pic) Showed that the exchange operator leads to the Pauli Exclusion Principle