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.