THE ENERGY-MOMENTUM TENSOR OF QUANTUM MECHANICS

Jose Lopez-Cervantes

Centro de Investigacion en Fisica

Universidad de Sonora,

Apartado Postal A-88 83190 Hermosillo, Sonora, Mwxico

Short title: Energy-momentum tensor

PAC. 03.65.Ca- Quantum theory; quantum mechanics. Formalism

Abstract.

We consider the Schrodinger equation as a field equation and work in the

frame of canonical Field Theory. One can therefore introduce a Lagrangian

density and define an associated energy- momentum tensor. The components

of this tensor coincide with the usual definitions of energy, linear and angular

momentum, and stresses, associated to a quantum mechanical system.

We deduce: a Newton-type equation for the momentum density; and the usual

expression for the momentum operator, with no heuristic rules. We consider

the wave function as a field variable associated with a physical reality,not with

a probability amplitude, and interpret the Schrodinger wave equation as the

sum of two physical energy densities: kinetic plus potential.

I. Introduction

Is well known the importance of the Principle of Least Action in Physics [1,2]

and the fact that the Schrodinger equation could be derived from a variational

Principle[3], however, the last point is not discussed in the most of the textbooks

on Quantum mechanics. It is the purpose of this paper, to make emphasis that

Quantum Mechanics is a Field Theory, with the Schrodinger equation as the field

equation. The field variable is the wavefunction Ψ(r,t). By the canonical method

of Field Theory, we can construct a lagrangian density from which, by applying

the Euler-Lagrange equation one gets the Schrodinger equation. Then one can

calculate the energy-momentum tensor, which carries information on the

following physical densities: energy, linear momentum, stresses of the

field, flux of energy. In section II, we calculate this tensor and show its

correspondence with know physical quantities. In section III, we calculate

the physical stresses of the Hydrogen atom which according to Brillouin [4],

they had not been calculated yet, because of the lack of direct observability of

this field. (by the quantum field Ψ, we mean the field representing the wave

nature of matter, not a second-quantized field) We thought that the analysis of

this stresses could been helpful in a more deep physical understanding of the

yet mysterious atomic transitions [5]. Some authors [6] have stressed the conti-

nuity of this transitions versus the discontinous image, using the usual formalism

of Quantum Mechanics. In section V, we calculate the force and angular-

momentum densities, and derive an Newton-type equation for the momentum

density of the field : the time derivative of the momentum density is equal to the

force density. We stress the physical advantages of studying physical densities

over the abstract operator approach in having a intuitive image of atomic and

sub-atomic process. Hestenes stress this point for the case of Relativistic Quantum

Mechanics [7]. We note a contradiction in the probabilistic interpretation

of Q M: which is the probability operator which corresponds to the probability of a

physical event?, if the probability density is an observable, it should have an asso-

ciated operator. In this formalism, the “quantum potential“ of Bohm [8], which is

derived in a semiclassical limit, from the kinetic energy density,is not strictly a po-

tential,but a kinetic term coming from the variation of the amplitude of the WKB

wavefunction with the coordinates : Ψr = A(r)exp(iS(r)/h).

II. The Lagrangian of the Schrodinger equation and its Energy-Momentum

Tensor.

Let us start by considering the Schrodinger equation for a particle in a potential

in certain coordinates:

– ℏ22m

∇2 ψ + V(r) ψ = = ih ∂∂t

Ψ , (1a)

which, by convenience we rewrite as

– ℏ22m

Ψ + V(r) Ψ - ih ∂∂t

Ψ = 0 . (1b)

The left-hand side of this equation can be considered as the

field equation obtained by minimising the Lagrangian [3]

L = – ℏ22m(∇ Ψ∗ )⋅(∇Ψ) - ℏ

2 i(Ψ∗ ∂

∂tΨ - Ψ ∂

∂tΨ∗ ) - Ψ V(r) Ψ (2)

where, for brevity of notation, we write: ψα =∂ψ∂xα

∂o = ∂∂t

(3)

This Lagrangian density is obtained directly from the

Schrödinger equation (1b) multiplying it from the left by Ψ∗ and integrating

by parts the first term.

The canonical energy-momentum tensor is defined as [3]

T βα = Ψβ Dα L + Ψβ

∗ D∗α L - δβα L (4)

where we have used the notation: Dα = ∂∂ψα

. Its components are :

Too = s (∇Ψ∗ ⋅ ∇Ψ ) + Ψ∗ V(r) Ψ = H (5a)

Tio = ℏ

2i(Ψ∂iΨ∗ – Ψ∗∂iΨ ) (5b)

T oi = - s δij[∂oψ ∂jψ∗ + ∂oψ∗∂jψ ] (5c)

T ji = - s δi k[∂k ψ∗ ∂jψ+ cc ] – δj

iL (5d)

The components of the energy-momentum tensor are easily interpreted as

follows: as usual, T oo is the energy density of the field; T i

o is the

“probability“ current density, usually denoted by: Si [9]

T oi is the momentum density vector, written as: P i ;

T ji are the physical stresses of the quantum field ψ.

Another relevant quantity is the angular momentum density

Mαβγ = T αβxγ – T αγxβ . (6)

The scalar quantities associated to the previous densities are: the energy

E = ∫T oo d3x = [s (∇ψ∗ ⋅ ∇ψ) + ψ∗ V(r) ψ] d3x (7a)

the linear momentum

pi = ∫T io d3x = ∫[ ℏ

2i(ψ∂iψ∗–ψ∗∂iψ)] d3x (7b)

the Poynting vector, representing the flux of energy

∫T oi d3x = ∫- s δij[∂oψ ∂jψ∗ + ∂oψ∗∂jψ ] d3x

and the physical stresses

∫T ji d3x = ∫−s δi k[∂k ψ∗ ∂jψ+ cc ] – δj

iL}d3x (7d)

Associated with equation(6), one has the angular momentum

Mi j = – (½)iℏ ∫{δi k(ψ ∂kψ∗– cc )x j – δjk (ψ ∂kψ∗– cc )x i}d3x (8)

One inmediately sees from eq. (7a), that the

terms in the integral can be interpreted as kinetic and potential energy

densities. This reflects over the Schrodinger wave equation where the first

term is associated to the kinetic energy while the second one is associated

to the potential energy.

This agrees with the interpretation of the Hamilton-Jacobi

equation as a sum of kinetic and potential energies. Let us note that the

momentum operator is deduced without heuristic rules as is seen in equation

(7b).

III. The Energy-Momentum Tensor for the Hydrogen Atom

In spherical coordinates, the spatial part of the Lagrangian is

L =- s [∂rψ∗ ∂r ψ + ( 1r2

(∂θψ∗ ∂θψ + sin2θ ∂ϕψ∗∂ϕψ+ e2r ψ∗ψ (9)

In these coordinates, the components of the energy-momentum tensor are

T r θ = s ∂rψ∂θ ψ∗ (10a)

T θ ϕ = s ∂θψ∂ϕ ψ∗ (10b)

T r ϕ = s ∂rψ∂ϕ ψ∗ (10c)

T r r = s ∂rψ∂r ψ∗ - L(ψ,∇ψ) (10d)

T t r = - i ℏ ψ∗ ∂rψ (10e)

For states having zero angular momentum,(l =0) the wave function ψ

does not depend on angles, therefore, these states have no physical

stresses. (except the rr-component) As an example, let us calculate the

stresses for the (n,l,m)=(2,1,1) and (2,1,0) states: the wave functions are:

ψ211 = R21(r)Y11(θ,ϕ)=(1/2a)3/2 (r/a 3 )exp(–r/2a)[-(3/8π) sinθ exp(iϕ ] (11)

Ψ210 = R21(r)Y10(θ,ϕ)=(1/2a )3/2 (r/a 3 )exp(–r/2a)[(3/4π1/2 cosθ] (12)

so, the components of the tensor are:

T rθ[ψ211] = s ∂r (R21Y 11) ∂θ(R 21Y 11∗ ) (13)

T θφ[ψ211] = s ∂θ (R21Y11) ∂φ(R21Y 11∗ ) (14)

Trr [ψ211] = s ∂r(R21Y11) ∂rR21Y11∗ – L [ψ211 ] (15)

as in fluid mechanics, Tr r gives the radial stresses: dilatation and compression.

In cartesian coordinates, the stresses of the Hydrogen atom are:

(applying the chain-rule to calculate partial derivatives in Cartesian coordinates

and using the matrix of transformation of the diferentials from Cartesian to

spherical coordinates).The derivatives of wavefunctions are:

∂rR21 = (A21 /a√3 )(1– r/2a) exp(– r/2a) (16)

∂θY11 = b cos θ exp(iφ (17)

∂θY10 = – f sin θ (18)

where the constants are (a is the Bohr radius)

b = – (3/8π)1/2 , f = (3/4π)1/2 , A21 =(1/2a)3/2 (19)

by the chain-rule the partial derivative with respect to x, are

∂xψ = sinθ cosφ ∂rψ + 1r cosθ cosφ ∂θψ − sinφ

r sinθ∂φψ

∂yψ = sinθ sinφ ∂rψ + 1r cosθ sinφ ∂θψ +

cosφr sinθ

∂φψ

∂zψ = cosθ ∂rψ − 1r sinθ ∂θψ

substituying for the state (211), we obtain :

∂xψ211 = sinθcosφ Y11(A21 /a√3) (1– r/2a) exp(-r/2a)

+ (1/r)cosθcosφ R21bcosθ exp(iφ − i/rsinφsinθ

R21bsinθ expiφ

#for the Ψ210 state we have:

d j = sin\’e cos\’i d j + (1/r) cos\’e cos\’i d j

= sin\’e cos\’i (A\’y1/a r3) (1-r/2a) e

- (1/r) cos\’e b cos\’i R sin \’e

and similar expressions for the y and z components

IV. A Newton-type equation for the Field Densities

We calculate the time derivative of the momentum density :

∂t P = ∂t ( ψ∗P ψ) = -(i/h)ψ∗(P H– H P)ψ (20)

as the commutator is, for the momentum and the Hamiltonian:

[P,H] = [P,V(r)] = -ih ∇ V(r)

∂t P = ψ∗(–∇V) ψ = ψ∗F ψ (21)

the last term of eq.(21) is the force density, which is equal to the time derivative

of the momentum density, thus we get a Newton-like equation: the time derivative

of the momentum density is equal to the force density And of course, an

abstract commutator is equivalent to a partial derivative of a local density

associated with an operator,in the previous case, the momentum operator.

In this density formulation, the problem of the time operator [10] it has

not physical sense, since a density of time does not exist.

# The angular momentum density is defined as L = r × P

its components are, for example:

Lx = – ih ψ∗ (y ∂z – z ∂y) ψ (22)

and analogous expressions for the y and z components.

#V. Stresses for the atomic transitions

VI. Physical Justification of the Lagrangian Density

¿# In a paper [11], the author writes that: “this approach (the Lagrangian

method) is unsatisfying since the Lagrangian in use has no physical

justification other than it gives the right answer to this problem...“ This

is not exact, historically, as Schrodinger himself wrote the Lagrangian

density from the point-particle Lagrangian, then he applied the variational

Euler-Lagrange method to obtain his equation [6].

We justify the Lagrangian following the original method that Schrodinger

employed :

Take the classical Lagrangian for a point-particle

L = p∘q- H =

p2

m – H (23)

substitute: p = ∇S, and S = -ih log ψ

L = − ℏ22m

∇ψ2

ψ2- V(r) (24)

multiply it by ψ2 k = ℏ / i

L ψ2 = ℏ2im

(∇ψ)2 - V(r) ψ2 (25)

If the wave function is normalized, ∫ ψ2 d3x = 1,

then ψ2 has units of [length]−3, giving to Lψ2, units of

energy/volume: (density of energy) and it has the form of the Lagrangian

density (4), without the time dependent terms.

VII. Conclusions

# Clearly, a possible future detection of the stresses, would be a

experimental confirmation of the physical reality of the ψ, against the

probabilistic interpretation. electronic microscope

We stressed the advantages of the continuous field formulation of quantum

mechanics based on the least action principle over the formal operator and

heuristic approach. Details to be studied further would be: the stresses of

electron waves in conductors and superconductors; a relation between the

microscopical and the macroscopical stresses (of a solid, for example).

Explain the “exchange“ forces due to the anti symmetry of the wavefunction j

in terms of the quantum stresses. To study the Gauge theories in the Higgs

sector (without the second-quantified formalism) in order to visualize in a

co ntinuous way the “spontaneous symmetry breaking“ will be worked in other

paper. Also we can think of potentials for which the quantum stresses turn

to be observable

To understand the process of photon emission (absorption) it could help to

study the time development of the stresses in the two sectors: that of the

quantum field of the atom (nucleus-electrons) and of the photons

(electromagnetic field), and of course, if the electron it has an structure

one would have to write an energy-momentum tensor for such structure: the

proper field of the electron, and visualize the continuous process of the

absorption (emission) of the electromagnetic waves into the

electron’s structure. And to explain the Poincare’s stresses from it.

At this point one remembers the model of Broglie: the electron is made of

electromagnetic waves trapped in closed paths [12].

We speculate that, a microscopic level the energies of the quantum fields do

cause curvature of space, that is, the stress tensor for the relativities,

Diary case, must be included in the Einstein field equations Rαβ = k T αβ[ψ]

If one hydrogen atom cause a negligible curvature, the 10^N hydrogen atoms of

a star like our sun, will cause an appreciable curvature. For consistency,

the macroscopic energy-stress tensor of a star, must be the total sum of all

microscopic energy tensors -atomic and nuclear-.

Acknowledgments.

The author wishes to thank to Dr. Victor Tapia for a critical reading of the

manuscript and for his encouragement.

References

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[2] Landau,L.D., E.M. Lifshitz, Mechanics, (Addison-Wesley, Reading,

Massachusetts, 1966).

P.M. Morse, H. Feshbach, Methods of Theoretical Physics

(McGraw-Hill, New York, 1953). part I

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[4] L. Brillouin, La Informacion y la Incertidumbre en la Ciencia, (UNAM,

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[5] see, for example, R. Kidd, J. Ardini, A. Anton, Am.J.Phys. 57, 27(1989).

[6] E. Schrodinger, Phys. Rev. 28, 1049 (1926), Collected Papers on

Quantum mechanics (Chelsea Publ. Co., New York, 1978).

[7] D. Hesteness, Am.J. Phys. 39, 1028 (1971); J. Math. Phys. 14, 893(1973).

[8] see, for example, Max Jammer, The Philosophy of Quantum Mechanics,

(Wiley-Interscience, New York, 1974), p.278 ff

[9] H.C.Corben, P.Stehle, Classical Mechanics (Wiley, New York, 1964) p.275

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[11] B.Roy Frieden, Am. J. Phys. 57, 1004 (1989)

[12] B. Kivel, Bulletin APS II, 8, 18 (1963)