Mathematical Principles of Theoretical Physics

340 CHAPTER 6. QUANTUM PHYSICS

Principle 6.10.A conserved quantum system can be described by a set of conjugate fields

Ψ= (ψ 1 ,···,ψN), Φ= (φ 1 ,···,φN),

and an associated Hamiltonian energy H=H(Ψ,Φ), such that the dynamic equations of this

system are in the form

(6.2.37)

∂

∂t

(

Ψ

Φ

)

=JHˆ(Ψ,Φ), J=

(

0 I

−I 0

)

,

whereHˆ=δH is the variational derivative operator of the Hamiltonianenergy H, I is the

N-th order identity matrix. In particular, if the Hamiltonian H is invariant under the trans-

formation of conjugate fields

(

ψk

φk

)

→

(

cosθ −sinθ

sinθ cosθ

)(

ψk

φk

)

for 1 ≤k≤K,

then the conjugate fields constitute complex valued wave functionsψk+iφk(orφk+iψk), 1 ≤

k≤K, for this system.

Remark 6.11.In classical quantum mechanics, the Klein-Gordon equationencounters a dif-

ficulty as mentioned in (6.2.31) that the operatorHˆof (6.2.32) is not Hermitian, inconsistent

with quantum mechanical principles for describing bosonicbehaviors. But, under the QHD

model (6.2.37) this difficulty is solved. In the Angular Momentum Rule in Section6.2.4and

the spinor BEC (nonlinear quantum system) in Chapter 7 , we can see this point clearly.

6.2.3 Heisenberg uncertainty relation and Pauli exclusionprinciple

The Heisenberg uncertainty relation and Pauli exclusion principle are two important quantum

physical laws.

Uncertainty principle

We first give this relation, which is stated as follows.

Uncertainty Principle 6.12.In a quantum system, the position x and momentum p, the time
t and energy E satisfy the uncertainty relations given by

(6.2.38) ∆x∆p≥

1

2

̄h, ∆t∆E≥

1

2

h ̄,

where∆A represents a measuring error of A value. Namely, (6.2.38) implies that x and p,t

and E can not be precisely observed at the same moment.

The relations (6.2.58) can be deduced from Postulates6.3and6.4. The average values of

position and momentum are

〈x〉=

∫

ψ∗xψdx,

〈p〉=−

∫

ψ∗i ̄h

∂ ψ

∂x

dx.