Mathematical Principles of Theoretical Physics

2.6. PRINCIPLE OF HAMILTONIAN DYNAMICS (PHD) 103

whereψ= (ψ 1 ,ψ 2 )T, andψk=ψk^1 +iψ^2 k( 1 ≤k≤ 2 ). The Hamilton energyHof (2.6.51)
is in the form

H=

∫

R^3

iψ†(~σ·∇)ψdx

(2.6.52)

=

∫

R^3

[

∂ ψ 12

∂x^1

ψ 21 +

∂ ψ 22

∂x^1

ψ 11 +

∂ ψ^21

∂x^2

ψ^22 +

∂ ψ 11

∂x^2

ψ 21 +

∂ ψ 12

∂x^3

ψ 11 +

∂ ψ 21

∂x^3

ψ 11 +

∂ ψ^12

∂x^3

ψ^22

]

dx.

The Hamilton equations are

1

c

∂ ψk^1

∂t

=

δ

δ ψk^2

H,

1

c

∂ ψk^2

∂t

=−

δ

δ ψk^1

H,

fork= 1 , 2 ,

which, in view of for (2.6.52), are in the form:

(2.6.53)

1

c

∂ ψ 11

∂t

=

(

∂ ψ^12

∂x^1

+

∂ ψ^22

∂x^2

+

∂ ψ 11

∂x^3

)

,

1

c

∂ ψ 12

∂t

=

(

∂ ψ^22

∂x^1

−

∂ ψ^12

∂x^2

+

∂ ψ 12

∂x^3

)

,

1

c

∂ ψ 21

∂t

=

(

∂ ψ^11

∂x^1

−

∂ ψ^21

∂x^2

−

∂ ψ 21

∂x^3

)

,

1

c

∂ ψ 22

∂t

=

(

∂ ψ^21

∂x^1

+

∂ ψ^11

∂x^2

−

∂ ψ 22

∂x^3

)

.

It is readily to check that (2.6.53) and (2.6.51) are equivalent.

3.Dirac equations:

(2.6.54) ih ̄

∂ ψ

∂t

=−ihc ̄ (~α·∇)ψ+mc^2 α 0 ψ,

where~α,α 0 are as in (2.2.58),ψ= (ψ 1 ,ψ 2 ,ψ 3 ,ψ 4 )T, and

ψk=ψk^1 +iψk^2 for 1≤k≤ 4.

The Hamilton energy of (2.6.54) is defined by

(2.6.55) H=

∫

R^3

[

−i ̄hcψ†(~α·∇)ψ+mc^2 ψ†α 0 ψ

]

dx.

It is clear that the expansions of (2.6.54) in terms of real and imaginary parts are in the form

h ̄

∂ ψk^1

∂t

=−

δ

δ ψk^2

H,

h ̄

∂ ψk^2

∂t

=

δ

δ ψk^1

H,

for 1≤k≤ 4 ,