Mathematical Principles of Theoretical Physics

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.