Mathematical Principles of Theoretical Physics

(Rick Simeone) #1

2.6. PRINCIPLE OF HAMILTONIAN DYNAMICS (PHD) 105


whereHreads
H=



R^3

H(ψ,Dψ)dx.

When taking the variation, the functional


̃L=


QT

iψ∗ψ ̇dxdt

=



QT

i(ψ^1 −iψ^2 )(ψ ̇^1 +iψ ̇^2 )dxdt

=



QT

[


i
2


∂t

|ψ|^2 −


∂t

(ψ^1 ψ^2 )+ 2 ψ^2 ψ ̇ 1

]


dxdt

=


∫T

0


R^3

2 ψ^2 ψ ̇^1 dxdt,

whereQT=R^3 ×( 0 ,T). Hence, the densityLin (2.6.60) is equivalent to


(2.6.61) L= 2 ̄hψ^2 ψ ̇^1 −H.


It follows from (2.6.61) that


(2.6.62)


ψ^2 =

1


2 h ̄

∂L/∂ψ ̇^1 ,

ψ^2 ψ ̇^1 −

1


2 h ̄

L=


1


(^2) ̄h


H.


The relations (2.6.62) are what we expected.

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