Mathematical Principles of Theoretical Physics

6.2. FOUNDATIONS OF QUANTUM PHYSICS 335

6.2.2 Quantum dynamic equations

In quantum mechanics, the following are the four basic dynamic equations:

(6.2.9)

Schr ̈odinger equation, governing particles at lower velocity

Klein-Gordon equations, describing bosons,

Weyl equations, describing massless fermions,

Dirac equations, governing massive fermions.

These four equations were initially derived in the spirit ofPostulate6.5. They can also be
equivalently obtained by the Principle of Lagrangian Dynamics (PLD) or by the Principle of
Hamiltonian Dynamics (PHD).
Although the three fundamental principles: Postulate6.5, PLD, and PHD are equivalent in
describing quantum mechanical systems, they offer different perspectives. In the following,
we introduce the four dynamic equations based on the three principles.

Quantum dynamics based on Postulate6.5

1.Schrodinger equation. ̈ In classical mechanics we have the energy-momentum relation

E=

1

2 m

p^2 +V, V is potential energy.

By the Hermitian operators in (6.2.2), this relation leads to the Schr ̈odinger equation, written
as

(6.2.10) i ̄h
∂ ψ
∂t

=−

h ̄^2

2 m

∆ψ+V(x)ψ,

which is clearly is non-relativistic.

2.Klein-Gordon equation.The relativistic energy-momentum relation is given by

E^2 −c^2 p^2 =m^2 c^4.

From this relation we can immediately derive the following Klein-Gordon equation:

(6.2.11)

(

−

1

c^2

∂^2

∂t^2

+∆

)

ψ−(

mc

̄h

)^2 ψ= 0.

In the 4-dimensional vectorial form, the equation (6.2.11) is expressed as

∂μ∂μψ−

(mc

h ̄

) 2

ψ= 0 ,

which is clearly Lorentz invariant.

3.Weyl equations.For a massless particle, the de Broglie matter-wave relation gives

E=h ̄ω, p 0 = ̄h/λ, c=ω λ,