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Lecture 2

# Lecture 2 - Clas sical Mecha nics c onjugated coordinates p...

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st conjugated coordinates p & q are used to obtain a full description of the system. 6N 1 order differential equations DETERMINE the state of the system at a Clas ny f sical Mecha uture time nics: Q . uantu Ψ Ψ Ψ Ψ g : p & q cannot be simultaneously specified. No trajectory = wavefunction. Knowledge of ( , ) provides ALL neccessaryinformation m Mech ancis pro ( , ) ( , ) ( , ) q t q t q t q t dq + bability of finding particle between and at time q q dq t

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α β κ α β κ α β κ α α α α ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ε ε ε ε - = + + + = = Ψ = = h K K K g g g For a time independent potential, where ˆ Many times can be written as and wher ˆ ˆ ˆ ˆ ˆ ( ) e E= + + + For a ( ) is the solution to ( , ) ( ) ( ) ( ) iEt H h h h q h H q e q t q H q E q µ ψ ψ + + K given , there might be multiple solutions For several the energy might be the , same DEGENERA , CY 1 ( ) ( ) j j j j j H n q q
ε ε = = + + + + 2 2 2 2 2 2 2 2 2 2 2 Example: Translational energy of a particle in a box For 1D box For 3D box ( ) 8 8 Degeneracy occurs when different combination of , ,

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Lecture 2 - Clas sical Mecha nics c onjugated coordinates p...

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