MOMENTUM OPERATORS in QUANTUM MECHANICS by A. M

MOMENTUM OPERATORS IN QUANTUM MECHANICS

by A. M. Arthurs

Mathematics Research Center and Theoretical Chemistry Institute University of Wisconsin, Madison, Wisconsin

ABSTRACT

A representation for momentum operators corresponding to real observables is obtained by using functional integration in phase space. 1. Introduction

In quantum mechanics, real observables such as energy and . momenturr are represented by operators that are self-adjoint on the

domain of physically acceptable bound state wave functions. A well-

known example is provided by the form for the momentum

operator conjugate to coordinate , where the domain is the entire real line R . However, if the domain is an interval P,p] of R , a more general representation of P is necessary to ensure self-adjointness. One such general form for ?

has been suggested by Robinson and Hirschfelder c11 , while an

equivalent integral form, which can also be extended to functions of ?

has been proposed by Robinson and Lewis [23 . These forms were

II obtained in the context of conventional Schrodinger quantum mechanics.

An alternative way to define operators is possible, however,

through an extension of the Feynman approach to quantum mechanics

due to Davies [S] . In this work the equivalence of the Feynman and

Schrgdinger approaches was established by introducing a suitable

inner product in function space. This inner product was defined for

functions on -GO < , and the usual self-adjoint operators

appropriate to this dandn were therefore obtained. But the definition

is readily extended to other domains , and hence can be used

to obtah general self-ad joint representations for various operators.

This is done here for any real fuilction of 'P The resulting

representation leads to those given in [l] and [2] e

1 2. Representation of operator F(?)

Following Davies [3] we introduce a function space with elements f($) , , etc. which are here 1 defined on the interval [d,/3] of the real line. Then the appropriate

inner product (f,Q) is defined as follows:

where

J is a functional integral over all phase space histories from %lo)=+

and to v(7)=+‘I , and involves only the classical variables p + . If a division 3 .

This is readily evaluated and we find that

where 6 is the Dirac delta function. Thus the inner product (f,?)

b ecome s

which is the frequently used inner product of function space.

Now we define the operator corresponding to a real function F(P)

of the momentum variable. This is done in terms of the inner product

where .

in which a time T has been associated with FCP(t9 such that

0 <74 < 7 , and once more the functional integration is over all phase space histories subject to the restrictions

Making the same division of the time interval @,TI as before, we

may write B($:7;${0) as

where I < j 6 YI . It is then found that

where Hence equation (8) becomes .

a result which is independent of the choice of t and . When

the identity operator, equation reduces to (7) as F(p) t I , (14) it should.

The condition for FR) to be self-adjoint on the domain of

the elements of the function space is

Using (14) and the fact that F is a real function, we find that (15)

implies the relation

that is, (14) is independent of the order of integration. Hence we may represent the self-adjoint operator F(P) by the inner product 6

where

subject to condition (16) for self-adjointness.

3. Alternative forms

Equation (17) may be written as

where

and where the order of the 2’ and %’ integrations has been interchanged. Hence F(?) may be defined by

with G(3) given by (20). This is the representation proposed by

Robinson and Lewis [23 . 7

Next we consider the special case F[T) =? . This operator can be represented by the appropriate form of (21), but there is another

form, which can be obtained directly from (17) and (18). For F(?)=P

equation (18) may be written as 8 li

and

and impose condition (16) for self-adjointness, we find that

Thus (24) becomes

we may write (28) in the form 9

This is the representation of ? suggested by Robinson and

Hirschfelder c 13 using quite a different set of considerations.

4. Concluding remarks

A representation for real functions of the momentum operator

P has been obtained by using functional integration in phase space.

The resulting operators are self-adjoint on the interval [X,/3] of the real line. For the operator P itself, the representation involves the first derivative of the delta function, in agreement with the treatment of momentum given by Kramers E43 .

The present work has dealt solely with Cartesian coordinates.

It would be of interest to extend the method to other coordinate systems, though this may prove difficult in the light of wrk by

Edwards and Gulyaev [5] on functional integrals in polar coordinates. i' 10

ACKNOWLEDGMENTS

I should like to thank Professor J. 0. Hirschfelder and

Professor B. Noble for helpful discussions on this subject.

This work has been sponsored by the Mathematics Research Center,

United States Army, Madison, Wisconsin, under Contract No. DA-11-022-ORD-2059 and the University of Wisconsin Theoretical Chemistry Institute under

National Aeronautics and Space Administration Grant NsG-275-62.

REFERENCES

1. Robinson, P. D. and Hirschfelder, J. 0. J. Math. Phys. 4 (19631,338.

2. Robinson, P. D. and Lewis, J. T. Wave Mechanics Group, Mathematical

Institute, Oxford, Progress Report No. 9 (1963), 8.

3. Davies, H. Proc. Cambridge Phil. SOC. 59 (1963), 147.

4. Kramers, H. A. Quantum Mechanics, North-Holland Publishing Company,

Amsterdam, 1957, p. 106.

5. Edwards, S. F. and Gulyaev, Y. V. Proc. Roy. SOC. A279 (1964), 229. THE UNIVERSITY OF WISCQNSIN

MATHEMATICS RESEARCH CENTER

f - MATHEMATICS RESEARCH CENTER, UNITED STATES ARMY m €1u N I VERSI TY OF w I s co N s I N

Contract No. : DA-11-022-ORD-2059

- MOMENTUM OPERATORS IN QUANTUM MECHANICS .

A. M. Arthurs

MRC Technical Summary Report a679 I July 1966

Madison, Wisconsin ABSTRACT

A representation for momentum operators corresponding to real observables is obtained by using functional integration in phase space. MOMENTUM OPERATORS IN QUANTUM MECHANICS

A. M. Arthurs

1. Introduction

In quantum mechanics, real observables such as energy and momentum are

represented by operators that are self-adjoint on the domain of physically

acceptable bound state wave functions. A well-known example is provided by

the form P= -i a/ a q for the momentum operator conjugate to coordinate q,

which is self-adjoint on the real line (-m, m) . However, if the domain is an

interval CY,^] of the real line, a more general representation of P is necessary

to ensure self-adjointness. One such general form for P has been suggested

by Robinson and Hirschfelder [l], while an equivalent integral form, which can

also be extended to functions of P, has been proposed by Robinson and Lewis

[Z] . These forms were presented in the context of conventional Schrodinger

quantum mechanics.

An alternative way to define operators is possible, however, through an

extension of the Feynman approach to quantum mechanics due to Davies [3].

In this work the equivalence of tne Feynman and Schrodinger approaches was

established by introducing a suitable inner product in function space. This

inner product was defined for functions on the real line, and the usual self-

adjoint operators appropriate to this domain were therefore obtained. But the

definition is readily extended to other domains. and hence can be used to obtain Sponsored by the Mathematics Research Center, United States Army, Madison, Wisconsin unde; Contract DA-ll-022-ORD-2059, 'and the University of , h.:, Wisconsin Theoretical Chemistry Institute under National Aeronautics and Space Administration Grant NsG- 2 7 5- 6 2. general self-adjoint representations for various operators. This is done here for any real function of P . The resulting representation leads to those given in [l] and [2].

2. The inner product

Following Davies [3] we introduce a function space with elements f(q), g(q), etc., which are here defined on the interval CY,^] of the real line.

Then the inner product (f, g) is defined as follows:

where

is a functional integral over all phase space histories from q(o) := q 1 to q(T) = q" , and involves only the classical variables p and q . If a division

0-t

tnen A(q", T; q' , 0) is given by

A(q", T; q', 0) = lim dPn ,I dql . . . dqn-l ,I 3 ... J n-+m -CO -00 --co 27T -0-l ZIT n

This is readily evaluated and we find that

- 2- P679 I I' where 6 is the Dirac delta function. Now q = q and q = q , and hence 0 n

A(q", T; q', 0) = 6(q" - 9') , (7 1

which is the frequently used inner product of function space.

3. Representation of operator F(P).

Now we define the operator corresponding to a real function F(p) of the

in which a time T has been associated with F P(T )] such that 0 < T < T , and once more the functional integration is over all phase space histories subject to the restrictions

Making the same division of the time interval (0,T) as before, we may write

B(q", T; q' ,0) as

B(q", T; q', 0) = lim dql.. . n+W dqn-l -cQ -cQ --co -cQ

#679 where j is an integer such that 1 ,< j -Cn . It is then found that

L. B(q", T;q', 0) = F(q" - 9') , where

Hence equation (11) becomes

a result which is independent of T and T . When F(P) is equal to the identity operator, equation (15) reduces to (8), as it should.

The condition for F(P) to be self-adjoint on the interval [a,p] is

Using (15) and the fact that F is a real function, we find that (16) implies the relation P- P P P- f (9") { ?'(q'' - 9') g(q') dq' } dq" = { f (9") F(S" - 9') dq" I (I(@)dq' , CY CY CYCY (17) tnat is, (15) is independent of the order of integration. Hence we may represent

00 1 F(q) = J eiCqF(5)d5 -00 subject to condition (17) for self-adjointness.

4. Alternative forms.

Equation (18) may be written as

- 4- #679

-~ in wnic'n

and where we have interchanged the order of the 5 and q' integrations. Hence

F(P) may be defined by

with G(6) given by (21). This is the representation proposed by Robinson and Lewis [2].

Next we consider the special case F(P) = P . Tnis operator can be represented by the appropriate form of (22), but there is another form, wxich can be obtained directly from (18) and (19). For F(P) = P , equation (19) may be written as

where 6' is the first derivative of the delta function. Hence (18) gives P p- (f, Pg) = -i 1 1f (9") 6' (9" - q') g(q!)dq' dq" (24) aa

- 5- #679 If we write

and

LY and impose condition (17) for self-adjointness, we find that

1 =p=2

Thus (25) becomes

we may write (29) in the form P (f, Pg) = /k)(-1 aga ti-id- (p-q) - iib+(q-ru))g(q)dq , CY or

This is the representation of P suggested by Robinson and Hirschfeldc-r [l]

using quite a different set of considerations.

5. Concludinq remarks.

Arepresentation for real functions of the momentum operator P has been

obtained by using functional integration in phase space. The result.ing

operators are self-adjoint on the interval [CY,p I of the real line. For the

$679 operator P itself, the representation involves the first derivative of the delta function, in agreement with the treatment of momentum given by Kramers [4]. Some related work on operators, from quite a different standpoint, can also be found in a paper by Fuchs [5]. The present work has dealt solely with Cartesian coordinates. It would be of interest to extend the method to other coordinate systems, though this may prove difficult in the light of work by Edwards and Gulyaev [6] on functional integrals in polar coordinates. I should like to thank Professor J. 0. Hirschfelder and Professor B. Noble for helpful discussions on this subject, and Professor C. A. Coulson for bringing the paper of Fuchs to my attention.

#679 - 7- REFERENCES

1. Robinson, P. D. and Hirschfelder, J. O., J. Math. Phys. 4(1963),

338 - 347.

2. Robinson, P. D. and Lewis, J. T., Wave Mechanics Group,

Mathematical Institute, Oxford, Progress Report No. 9 (1963), 8 - 10.

3. Davies, H. , Proc. Cambridge Philos. SOC. 59(1963), 147-155.

4. Kramers, H. A. , Quantum Mechanics, North- Holland Publishing Company,

Amsterdam, 1957, p. 106.

5. Fuchs, K., Roc. Roy. SOC. London Ser. A, 176(1940), 214-228.

6. Edwards, S. F. and Gulyaev, Y.V., Proc. Roy. SOC. London Ser. A, 279(1964), 229-235.

#679