86 CHAPTER 2. FUNDAMENTAL PRINCIPLES OF PHYSICS
2.5.4 Lagrangian actions in quantum mechanics
PLD is also valid in quantum physics. In this subsection we shall introduce the actions for ba-
sic equations of quantum mechanics: the Schr ̈odinger equation, the Klein-Gordon equation,
and the Dirac equations.
Action for the Schrodinger equation ̈
A particle moving at lower velocity can be approximatively described by the Schr ̈odinger
equation, given by
(2.5.32) i ̄h
∂ ψ
∂t
=−
h ̄^2
2 m
∆ψ+V(x)ψ,
wheremis the mass of the particle,V(x)is the potential energy, andψ=ψ 1 +iψ 2 is a
complex valued wave function. By the Basic Postulates2.22-2.23, the equation (2.5.32) is
derived using the following the non-relativistic energy momentum relation:
E=
1
2 m
~P^2 +V.
Hence the Schr ̈odinger equation (2.5.32) is a basic equation in non-relativistic quantum me-
chanics.
The Lagrange action for the Schr ̈odinger equation is
(2.5.33)
L=
∫T
0
∫
R^3
L(ψ,ψ∗)dxdt,
L=ih ̄ψ∗
∂ ψ
∂t
−
1
2
[
h ̄^2
2 m
|∇ψ|^2 +V|ψ|^2
]
We now computeδLto show that (2.5.33) is indeed the action of (2.5.32). Take the
variation forψ∗for (2.5.33):
∫
QT
(δL)ψ ̃∗dxdt=
d
dλ
∣
∣
∣
λ= 0
L(ψ,ψ∗+λψ ̃∗)
=
∫
QT
d
dλ
∣
∣
∣
λ= 0
L(ψ,ψ∗+λψ ̃∗)dxdt
=
∫
QT
[
i ̄h
∂ ψ
∂t
ψ ̃∗−
h ̄^2
2 m
∇ψ∇ψ ̃∗+Vψψ ̃∗
]
dxdt,
whereQT=R^3 ×( 0 ,T), andψ ̃∗satisfies
ψ ̃∗( 0 ,x) =ψ ̃∗(T,x) = 0 ∀x∈R^3 ,
ψ ̃∗→ 0 as|x| →∞.
Then by the Gauss formula, we have
∫
QT
∇ψ·∇ψ ̃∗dxdt=−
∫
QT
∆ψψ ̃∗dxdt,