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arXiv:quant-ph/0001111v1 30 Jan 2000

Noether Theorem and the quantum mechanical operators

Wai Bong Yeung

phwyeung @ccvax.sinica.edu.tw

Institute of Physics, Academia Sinica, Taipei, Taiwan ,ROC

February 1, 2008

Abstract

We show that the quantum mechanical momentum and angular momentum operators are fixed by

the Noether theorem for the classical Hamiltonian field theory we proposed.

Recently, we have proposed a classical Hamiltonian field theory [1],which in the limit of very large Planck

frequency,mimics many aspects of a quantum mechanical system. In particular, the Schrodinger Equation

will follow from the Hamilton’s Field Equation, and the Hamiltonian of the classical field theory will become

the energy expectation value of the corresponding quantum mechanical system.

The Hamiltonian density for the classical field theory we proposed contains (1+2n) pairs of canonical

conjugate variables (p,q) ( Pj,Qj), (πj,ηj ), j=1,...,n.All of these canonical variables are functions of x =

(x1,..,xn) and t.And it reads as

H = (1/2h)(V (x))(p2+q2)−(1/2h)(mc2)(P2

Independent variations of the field variables generate the Hamiltonian Field Equation. If we are interested

in the case in which the field variables oscillate with frequencies far smaller than the Planck frequency h/mc,

then the variables Pj,Qjand πj,ηjwill be related to p,q through [1]

j+Q2

j+π2

j+η2

j)−(c/2)p∂j(Qj+ηj)−(c/2)(Pj+πj)∂jq (1)

Qi

Pi

ηi

πi

= h/2mc∂ip

−h/2mc∂iq

h/2mc∂ip

−h/2mc∂iq,j = 1,..n

=

=

=(2)

In this paper, we will explore the translational and rotational symmetries of this classical Hamiltonian

field theory. It is well known in the literature that these continuous symmetries will lead to conserved

physical quantities ; a result called the Noether theorem [2] .And the aim of this paper is to understand

how these conserved physical quantities coming from a classical field theory are related to the measurable

quantities of the corresponding quantum mechanical system.

Let us first consider the V (x) = 0 case.When there is no external potential present, there will be both

translational and rotational symmetries for the classical field system. And by Noether theorem, there exist

some corresponding conserved quantities. For the translational invariance, the resulting conserved quantities

are the components of the second rank tress-energy tensor Tµ

ν,given by Noether as

Tµ

ν= Σ(∂L/∂(∂µu))(∂νu) − Lδµ

ν

(3)

where L is the underlying Lagrangian density for the classical field theory. u stands collectively for all the

field variables.Noether theorem requires the conservation law

∂µTµ

ν= 0(4)

We are particularly interested in the vector mjdefined as

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mj=

?

dnxT0j

(5)

These mj,other than a multiplicative constant that we shall fix later,are always taken as the mementum

components carried by the classical fields because they are generated by the translational symmetries. A

close look at T0

jwill show that they are independent of the detailed structures of the Lagrangian density L

and has the simple form of

T0

j= Σ(∂L/∂(∂tu))(∂ju) = p∂jq + Pi∂jQi+ πi∂jηi

(6)

Using the result given in Eq(2),T0

jcan be written in terms of p and q as

T0

j= p∂jq − 2(h/2mc)2∂ip∂j∂iq(7)

For a very large Planck frequency , the second term of Eq(7) drops out, and hence

mj=

?

dnx(−p∂jq) (8)

If we define the corresponding quantum mechanical wave function by ψ(x,t)=(q(x,t) + ip(x,t))/√2[1].

It can be seen immediately that

mj=

?

dnxψ ∗ (−i∂j)ψ, (9)

after integration by parts.

The physical meaning of the above result is the following: If we use ⋆pjto denote the quantum mechanical

operator for the j the component of the momentum, and if we use the above ψ to compute the expectation

value of the momentum components, then ⋆pjmust be of the form

⋆ pj= −iβ∂j

(10)

where β is a proportional constant that will be shown to be h later.This result can be regarded as a

derivation of the most fundamental quantum mechanical prescription

⋆ pj= −ih∂j

(11)

For the rotational invariance, we assume that p,q,Pj,Qj,πj and ηj all transform as scalars under the

rotation group.The resulting conserved quantities will then be the components of the third rank angular

momentum tensorMβ

λµ, given by Noether as

Mβ

λµ= xµTβ

λ− xλTβ

µ

(12)

The components of this third rank tensor that are related to the angular momentum components of the

classical fields are M0

lk.Using the result given in Eq(7), it can be shown easily that the integrated components

Llk=

?

dnxM0

lk

(13)

can be written as

Llk=

?

dnxψ ∗ (−ixl∂k+ ixk∂l)ψ(14)

And hence the orbital angular momentum ⋆L in quantum mechanics will have the familiar form

⋆ L = rX ⋆ p(15)

In the presence of the potential V(x), we will no longer have translational invariance, and so no more

conservation law. Instaed we shall have [3]

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d/dxµ(Tµ

ν) = −∂νL = ∂νH (16)

The integrated spatial parts for the above equation read as

− ∂t

?

dnxT0j+

?

dnxd/dxlTlj=

?

dnx∂jH (17)

Throwing away the surface term,and using the results given in Eq(1),Eq(8) and Eq(9), Eq(17) will become

− ∂t

?

dnxψ ∗ (−i∂j)ψ = 1/h

?

dnxψ ∗ (∂jV )ψ (18)

or

∂t

?

dnxψ ∗ (−ih∂j)ψ =

?

dnxψ ∗ (−∂jV )ψ (19)

This is the Ehrenfest theorem [4] that we always encounter in quantum mechanics. And as we have

promised before, we have fixed the proportional constant β that appeared in Eq(10) to be the Planck

constant h.

An equation similar to Eq(18) can also be derived for the angular momentum which relates the rate of

change of angular momentum with the external applied torque.

So we may conclude our paper by saying that the quantum mechanical operators for the momentum and

angular momentum variables will be fixed by the Noether theorem for our classical Hamiltonian field theory.

References

[1] .B.Yeung,The Schrodinger Equation From a Quadratic Hamiltonian Syatem,preprint,

ph/9911001.

quant-

[2] .Noether,Nachrichten Gesell.Wissenchaft.Gottingen 2,235(1918)

[3] .Goldstein, Classical Mechanics,Addision-Wesley Press. Cambridge.Ma 1980

[4] .Ehrenfest,Z.Physik 45,455(1927)

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