Mathematical Principles of Theoretical Physics

104 CHAPTER 2. FUNDAMENTAL PRINCIPLES OF PHYSICS

whereHis given by (2.6.55).

4.Klein-Gordon equation:

(2.6.56)

1

c^2

∂^2 ψ

∂t^2

−∆ψ+

(mc

h ̄

) 2

ψ= 0.

The conjugate fields are

ψ^1 =ψ, ψ^2 =

∂ ψ

∂t

.

The Hamilton energy is

(2.6.57) H=

1

2

∫

Ω

[

|ψ^2 |^2 +|∇ψ^1 |^2 +

(mc

h ̄

) 2

|ψ^1 |^2

]

dx.

Then (2.6.56) can be equivalently written as

1

c

∂ ψ^1

∂t

=

δ

δ ψ^2

H,

1

c

∂ ψ^2

∂t

=−

δ

δ ψ^1

H.

5.Bose-Einstein condensation (BEC) equation:

(2.6.58) ih ̄

∂ ψ

∂t

=−

h ̄^2

2 m

∆ψ+V(|ψ|^2 )ψ,

whereV(|ψ|^2 )is a function of|ψ|^2 , andψ=ψ^1 +iψ^2. The Hamilton energy of (2.6.58) is
given by

(2.6.59)

H=

∫

Ω

[

̄h^2

4 m

|∇ψ|^2 +G(|ψ|^2 )

]

dx,

G(z) =

1

2

∫z

0

V(s)ds.

For (2.6.59), the equation (2.6.58) can be rewritten as

̄h

∂ ψ^1

∂t

=

δ

δ ψ^2

H,

̄h

∂ ψ^2

∂t

=−

δ

δ ψ^1

H.

The examples given above show that PHD holds true in general in quantum physics.
In particular, we now show that the relations (2.6.16) and (2.6.17) also valid for quantum
Hamiltonian systems.
In fact, the Lagrange action of a quantum system with the Hamilton energyH(ψ)is given
by

(2.6.60)

L=

∫T

0

∫

R^3

L(ψ,ψ ̇)dxdt,

L=ih ̄ψ∗ψ ̇−H(ψ,Dψ),