2.5. PRINCIPLE OF LAGRANGIAN DYNAMICS (PLD) 87
which implies that
∫QTδLψ ̃∗dxdt=∫QT[
ih ̄∂ ψ
∂t+
̄h^2
2 m
∆ψ−Vψ]
ψ ̃∗dxdt.Sinceψ ̃∗is arbitrary, we derive that
δL=ih ̄∂ ψ
∂t+
̄h^2
2 m
∆ψ−Vψ.Hence we have
(2.5.34)
δL
δ ψ∗= 0 ⇔ i ̄h∂ ψ
∂t=−
h ̄^2
2 m∆ψ+V(x)ψ.We can derive in the same fashion that
(2.5.35)
δL
δ ψ
= 0 ⇔ −ih ̄∂ ψ∗
∂t=−
h ̄^2
2 m
∆ψ∗+V(x)ψ∗.It follows from (2.5.34) and (2.5.35) that
δL
δ ψ∗=
(
δL
δ ψ)∗
.
In other words, (2.5.34) and (2.5.35) are equivalent, and are exactly the Schr ̈odinger equation.
Action for Klein-Gordon fields
The field equations governing the spin-0 bosons are the Klein-Gordon equations:(2.5.36)
1
c^2∂^2 ψ
∂t^2−∆ψ+(mc
h ̄) 2
ψ= 0.It is easy to introduce the Lagrange action for (2.5.36):
(2.5.37) L=
∫M^4[
∇μψ∇μψ∗+(mc
̄h) 2
|ψ|^2]
√
−gdx,where∇μand∇μare 4-dimensional gradient operators as defined by (2.2.19),M^4 is the
Minkowski space, andg=det(gμ ν).
It is readily to see that
δL
δ ψ∗=
(
δL
δ ψ)∗
= 0 ⇔
1
c^2∂^2 ψ
∂t^2−∆ψ+(mc
̄h) 2
ψ= 0.Hence, the Klein-Gordon equation is a variational equationof the action (2.5.37).
Action for Dirac spinor fields
To introduce an action for the Dirac equations introduced inSubsections2.2.5and2.2.6,
we first recall the Dirac equations:
(2.5.38) iγμ
∂ ψ
∂xμ−
mc
̄hψ= 0 ,