Mathematical Principles of Theoretical Physics

(Rick Simeone) #1

2.6. PRINCIPLE OF HAMILTONIAN DYNAMICS (PHD) 101


and the fields of PHD are


A 1 ,A 2 ,A 3 and E 1 ,E 2 ,E 3.

It is clear that
1
4 π


E(−A ̇)−


1


4 π

LEM=H(A,E)−


1


c

φJ^0 ,

andJ^0 =ρ. Hence, the relation (2.6.17) does not hold true in general.


Now, we consider the energy conservation. WhenH contains∇φandJ, which depend
ont, the HamiltonH=H(A,E,t)contains explicitlyt. It implies that the Maxwell fields
have energy exchange with other charged particles. ThenHis not conserved.
As there is no charged particles inΩ, then


(ρ,J) =0 inΩ.

In this case, ∫



∇φ·Edx=−



φdivEdx=−



4 π φ ρdx= 0.

Hence, the Hamilton energy (2.6.42) becomes


(2.6.45) H=


1


8 π



(E^2 +|curlA|^2 )dx.

By (2.6.37), the energy fluxPdefined in (2.6.45) reads

P=


1


4 π

curlA×

∂A


∂t

Note that
∂A
∂t


=−cE, curlA=B (Bthe magnetic field),

Then, by (2.6.34) we have that


dH
dt

=−



∂Ω

P·nds=−


∂Ω

(c
4 π

E×B


)


·nds.

The field
P=
c
4 π


E×B


is the Poynting vector, represents the energy flux density ofan electromagnetic field.


2.6.4 Quantum Hamiltonian systems


In quantum physics, the state functions are a set of complex valued wave functions:


ψ= (ψ 1 ,···,ψN)T, N≥ 1 ,

andψkare as


(2.6.46) ψk=ψ^1 k+iψk^2 for 1≤k≤N.

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