Mathematical Principles of Theoretical Physics

338 CHAPTER 6. QUANTUM PHYSICS

Remark 6.9.Historically the four basic quantum dynamic equations werediscovered all
based on Postulate 5.5, and then their actions (6.2.20)-(6.2.23) were established by the known
corresponding equations. However, the Lagrange actions have played crucial roles in devel-
oping the interaction field theories.

Quantum dynamics based on PHD

In Section2.6.4, we derived all quantum dynamic equations from the Principle of Hamil-
tonian Dynamics (PHD). PHD amounts to saying that each conservation system is character-
ized by a set of conjugate fields

(6.2.24) ψk, φk, ( 1 ≤k≤N),

and the conjugate fields (6.2.24) define the Hamiltonian energy

(6.2.25) H=

∫

R^3

H(ψ,φ)dx.

Then the dynamic equations of this system are given by

(6.2.26)

∂ ψk

∂t =

δ

δ φkH,

∂ φk

∂t =−

δ

δ ψkH,

whereψ= (ψ 1 ,···,ψN),φ= (φ 1 ,···,φN)are as in (6.2.24), andHis the Hamiltonian en-
ergy.
In the classical quantum mechanics, if the dynamic equationof a quantum system is in
the form

(6.2.27) ih ̄

∂ ψ

∂t

=Hˆψ, Hˆ an Hermitian operator,

thenHˆis called the Hamiltonian operator of this system, and the physical quantity

(6.2.28) H=〈ψ|Hˆ|ψ〉=

∫

R^3

ψ†Hˆψdx

is the Hamilton energy of the system.
Three systems: the Schr ̈odinger system, the Weyl system andthe Dirac system, are in the
form of (3.1.27) and (3.1.28). Namely the equations can be expressed in the form (3.1.27),
with the Hamiltonian operators given by

(6.2.29)

Hˆ=− ̄h

2

2 m

∆+V(x) for Schr ̈odinger

Hˆ=ihc ̄(~σ·∇) for Weyl spinor fields,

Hˆ=−i ̄hc(~α·∇)+mc^2 α 0 for Dirac spinor fields,

and the associated Hamiltonian energies given by

(6.2.30)

H=

∫

R^3

[

̄h^2

2 m

|∇ψ|^2 +V|ψ|^2

]

dx for Schr ̈odinger system,

H=

∫

R^3

[

i ̄hcψ†(~σ·∇)ψ

]

dx for Weyl System,

H=

∫

R^3

[

−i ̄hcψ†(~α·∇)ψ+mc^2 ψ†α 0 ψ

]

dx for Dirac system.