Mathematical Principles of Theoretical Physics

6.2. FOUNDATIONS OF QUANTUM PHYSICS 339

Here the conjugate fields are the real and imaginary parts ofψ=ψ^1 +iψ^2.
By applying the PHD, from the Hamiltonian energies in (6.2.30) we can derive the dy-
namic equations of the three systems, which are equivalent to the form (6.2.27); see (2.6.49)-
(2.6.55).
However the Klein-Gordon system is different. In fact, we can write the equation (6.2.11)
in the form

(6.2.31) i ̄h

∂

∂t

(

ψ

φ

)

=Hˆ

(

ψ

φ

)

,

whereHˆis given by

(6.2.32) Hˆ=

(

0 1

−h ̄^2 c^2 ∆+m^2 c^40

)

.

However, it is clear that the operatorHˆof (6.2.32) is not Hermitian, and thereforeHˆis not a
Hamiltonian. Consequently the quantity

〈Φ|Hˆ|Φ〉=

∫

R^3

[

ψ∗φ+φ∗(− ̄h^2 c^2 ∆ψ+m^2 c^4 ψ)

]

dx

is also not a physical quantity because it is not a real numberin general. In other words, under
the theoretic frame based on Postulate6.5and PLD, the Klein-Gordon equation can not be
regarded as a model to describe a conservation quantum system.
With PHD, we can, however, show that the Klein-Gordon equation is a model for a con-
served system. As seen in (2.6.57), we take a pair conjugate fields(ψ,φ)and the Hamiltonian
energy

(6.2.33) H=

1

2

∫

R^3

[

φ^2 +c^2 |∇ψ|^2 +

m^2 c^4

h ̄^2

|ψ|^2

]

dx.

Then the Klein-Gordon equation (6.2.11) can be rewritten as

(6.2.34)

∂

∂t

(

ψ

φ

)

=J

(

δ

δ ψH^0

(^0) δ φδH

)(

ψ

φ

)

,

whereHis as in (6.2.33), and

(6.2.35) J=

(

0 1

−1 0

)

.

The Hamiltonian operatorHˆfor the Klein-Gordon system reads

(6.2.36) Hˆ=δH=

(

−c^2 ∆+m

(^2) c 4
̄h^20
0 1

)

.

The model of (6.2.33)-(6.2.36) is in a standard form of PHD. In fact, all conservation
quantum systems, including the four classical systems (6.2.9), can be expressed in the stan-
dard PHD from, which we call the Quantum Hamiltonian Dyanmics (QHD), which is stated
in the following.