Mathematical Principles of Theoretical Physics

102 CHAPTER 2. FUNDAMENTAL PRINCIPLES OF PHYSICS

In view of PHD for a quantum system, the conjugate fields are taken as real and imaginary
parts of the wave functions in (2.6.46):

(2.6.47) ψ 11 ,···,ψN^1 and ψ^21 ,···,ψN^2.

LetH=H(ψ)be the Hamilton energy. Then the Hamilton equations for the quantum system
are as follows:

(2.6.48)

α

∂ ψk^1

∂t

=

δ

δ ψk^2

H,

α

∂ ψk^2

∂t

=−

δ

δ ψk^1

H,

for 1≤k≤N,

whereαis a constant.
We now introduce quantum Hamiltonian systems for the Schr ̈odinger equation, the Weyl
equation, the Dirac equations, the Klein-Gordon equation,and the BEC equation.

1.Schrodinger equation: ̈

(2.6.49) i ̄h
∂ ψ
∂t

=−

h ̄^2

2 m

∆ψ+V(x)ψ,

whereψ=ψ^1 +iψ^2. The Hamilton energy of (2.6.49) is given by

H(ψ) =

1

2

∫

Ω

[

̄h^2

2 m

|∇ψ|^2 +V(x)|ψ|^2

]

dx.

It is easy to derive that

δ

δ ψ^1

H=−

h ̄^2

2 m

∆ψ^1 +Vψ^1 ,

δ

δ ψ^2

H=−

h ̄^2

2 m

∆ψ^2 +Vψ^2.

Hence, the Hamiltonian system is

(2.6.50)

̄h

∂ ψ^1

∂t

=−

̄h^2

2 m

∆ψ^2 +Vψ^2 ,

̄h

∂ ψ^2

∂t

=

̄h^2

2 m

∆ψ^2 −Vψ^2.

It is clear that (2.6.50) and (2.6.49) are equivalent.

2.Weyl equations:

(2.6.51)

1

c

∂ ψ

∂t

= (~σ·∇)ψ,