Mathematical Principles of Theoretical Physics

92 CHAPTER 2. FUNDAMENTAL PRINCIPLES OF PHYSICS

Hence we derive from the Noether Theorem that

I=

〈δL

δx ̇

,

d

dλ

∣

∣

∣

λ= 0

Aλx

〉

=

N

∑

k= 1

mk〈x ̇k,~r〉

is conserved, and is the total momentum in the direction~r. Thus we have shown that the
translation invariance corresponds to momentum conservation.

Example 2.41.Consider the Schr ̈odinger equation, the action is given by (2.5.33), and

(2.5.57) L(ψ,ψ ̇) =

∫

R^3

[

−i ̄hψ∗ψ ̇+

1

2

(

h ̄^2

2 m

|∇ψ|^2 +V|ψ|^2

)]

dx.

LetGbe the phase rotation group

G={Aλ=eiλ|λ∈R^1 }.

It is clear that (2.5.57) is invariant under the phase rotation:

ψ→Aλψ=eiλψ.

We see that
δL
δψ ̇
=−ih ̄ψ∗,

d

dλ

Aλψ|λ= 0 =iψ.

Thus, the quantity (2.5.51) reads as

I(ψ,ψ ̇) =

〈δL

δψ ̇

,

d

dλ

Aλψ

〉

=

∫

R^3

h ̄|ψ|^2 dx.

Hence, the modulus ofψ

(2.5.58)

∫

R^3

|ψ|^2 dxis conserved.

The property (2.5.58) is just what we need because in quantum mechanics, we have

(2.5.59)

∫

R^3

|ψ|^2 dx= 1.

Here, the conservation (2.5.59) corresponds to the invariance of phase rotation of the wave
functionψ.

Remark 2.42.In classical mechanics, the energy conservation can be deduced from the time
translation invariance

(2.5.60) t→t+λ forλ∈R^1.

However, instead of the formula (2.5.51), it is derived using following relation:

(2.5.61) dH=

∂H

∂p

dp+

∂H

∂q

dq−

∂L

∂t

dt,