6.2. FOUNDATIONS OF QUANTUM PHYSICS 343
LetHbe the Hamiltonian energy of a conservative quantum system,which can be de-
scribed by the following Hamiltonian equations:
(6.2.43)
∂Ψ
∂t=HˆΦ(Ψ,Φ),
∂Φ
∂t=−HˆΨ(Ψ,Φ),
where
HˆΦ=δH
δΦ, HˆΨ=
δH
δΨ.
LetLbe an observable physical quantity with the corresponding Hermitian operatorLˆfor
the conjugate fields(Ψ,Φ)Tof (6.2.43), andLˆis expressed as
Lˆ=
(
Lˆ 11 Lˆ 12
Lˆ 21 Lˆ 22
)
, LˆT 12 =Lˆ∗ 21.
Then the physical quantityLof system (6.2.43) is given by
(6.2.44) L=
∫
(Ψ†,Φ†)Lˆ(
Ψ
Φ
)
dx=∫[
Ψ†Lˆ 11 Ψ+Φ†Lˆ 22 Φ+2Re(Ψ†Lˆ 12 Φ)]
dx.It is clear that the quantityLof (6.2.44) is conserved if for the solution(Ψ,Φ)Tof (6.2.43)
we have
dL
dt
= 0 ,
which is equivalent to
∫[
(6.2.45) HˆΦ†Lˆ 11 Ψ+Ψ†Lˆ 11 HˆΦ−HˆΨ†Lˆ 22 Φ−Φ†Lˆ 22 HˆΨ
+2Re(HˆΦ†Lˆ 12 Φ−Ψ†Lˆ 12 HˆΨ)]dx= 0.We remark here that if the QHD is described by a complex valuedwave function:ψ=Ψ+iΦ,and its dynamic equation is linear, then (6.2.43) can be written as
(6.2.46) ih ̄
∂ ψ
∂t
=Hˆψ, H=∫
ψ†Hˆψdx.In this case, the physical quantityLin (6.2.44) is in the form
(6.2.47) L=
∫
ψ†Lˆψdx,and the conservation law (6.2.45) ofLis equivalent to
(6.2.48) LˆHˆ−HˆLˆ= 0.