mirror of
https://asciireactor.com/otho/phy-4600.git
synced 2025-01-18 19:15:05 +00:00
44 lines
1.7 KiB
Plaintext
44 lines
1.7 KiB
Plaintext
|
Exam 2 Problem 2
|
|||
|
|
━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━
|
|||
|
|
|
|||
|
|
(pic) Finished the problem using the boundary conditions
|
|||
|
|
|
|||
|
|
boundary conditions limit the number of possible spherical harmonics.
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
Two Similar Particles
|
|||
|
|
━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━
|
|||
|
|
|
|||
|
|
❙a,b❭ = ❙a❭⊗❙b❭
|
|||
|
|
|
|||
|
|
Exchange Operator
|
|||
|
|
──────────────────────────────────────────────────────────────────────────
|
|||
|
|
|
|||
|
|
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
|
|||
|
|
──────────────────────────────────────────────────────────────────────────
|
|||
|
|
(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
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|