NATIONAL BUREAU OF STANDARDS REPORT

2188

QUANTUM DYNAMICS

Part I

by

Julian Schwinger

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
U. S. DEPARTMENT OF COMMERCE

Charles Sawyer, Secretary

NATIONAL BUREAU OF STANDARDS

A. V. Astin, Director THE NATIONAL BUREAU OF STANDARDS

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NATIONAL BUREAU OF STANDARDS REPORT

ms project ms report

1101-10-^100 December 19 9 19$2 2188

QUANTUM DYNAMICS

Part I

by

Julian Schwinger

University of California , Los Angeles

and

Harvard University

PREPRINT

This work was performed on a National Bureau of Standards con- tract with the University of California^ Los Angeles, and was sponsored (in part) by the Office of Naval Research, U5N.

The publication, reprlntl

unless permission Is obta

25, D. C. Such permlssi

cally prepared if that ag

Approved for public release by the

Director of the National Institute of

Standards and Technology (NIST) on October 9, 2015

iart, Is prohibited

lards, Washington

t has been speclfifor Its own use,

ACKNOWLEDGMENT

This material was presented in a series of lectures

during my visit to the National Bureau of Standards

in the summer of 1992 „ I should like to thank

Stephen Gasiorowicz and Walter Futterman, „ whose careful

lecture notes served as a basis for this paper , Part II g which deals with perturbation theory s will be ready in

the near future <>

Julian Schwinger

Quantum Dynamics'"

Part I

-X-fr by Julian Schwinger

Quantum Mechanics developed historically as a set of "quantization rules" superimposed upon the structure of Classical Mechanics

o

In view of the fact that the laws of classical physics are only

limiting laws* it seems advisable to construct a self-contained quantum theory. The development of quantum dynamics to be outlined in

the following lectures will parallel the development of classical

mechanics from the action principle of Hamilton but will not be built

upon it. In addition to improving the logical basis of quantum

mechanics* the theory provides powerful general methods for the solution of problems . The discussion will be confined to systems of

particles* the extension to fields (1.6.* systems with an infinite

number of degrees of freedom) following analogously.

We shall start with the mathematical foundation which will not

be the usual geometrical basis involving vectors in Hilbert spaces*

etc. We shall develop instead an algebraic basis which is in somewhat closer correspondence with the physical phenomena to be

described* and is constructed as a symbolic representation of the

measuring process in the atomic domain with its characteristic statistical features.

-X-This work was performed on a National Bureau of Standards con- tract with the University of California* Los Angeles* and

was sponsored (in part) by the Office of Naval Research* USN«

University of California* Los Angeles* and Harvard University.

2

I 0 ^le Algebra of Measurement

A measurement may be considered as a process by which an assembly

of systems is "sorted" into sub-assemblages characterized by the same

set of numbers representing the property being measured (e 0 go 5 the

Stern-Gerlach experiment) „ Thus if we intend to "measure” the property A whose possible values are a

[t 0 °° (denoted generally by

a") then we symbolically represent by M(a 8 ) the measuring process

which out of an assembly of systems selects those 9 for which the property A has the values a

0 0 The measuring process M(a“) has the

following properties %

(i) Reproducibility ? If a certain measurement is followed by a

second measurement of the same property then the results of the previous measurement are repeated » This is symbolically represented by

M(a”)M(a 8 ) = M(a') (1.1)

(ii) Exc lus ivene ss % If we make a measurement of the property A and

look for the sub-assemblage having the numbers a* 5 and then make a

measurement upon this sub-assemblage and look for systems having the

values a

t8 (a ,i ^ a°) for A then we will expect to find no such systems

and this is symbolically represented by

M(a’)M(a M

) - 0 (1.2)

where 0 stands for the measurement process that selects no system,,

The properties (i) and (ii) may be combined to give

M(a“)M(a 8 ”) - f ( a', a

8D )M(a 5 ) (1.3)
'
3

in which the numbers 1 and 0 represent certainty and impossibility of

agreement respectively9 for the results of the two measurements,,

(iii) Completeness g If we look for all possible values of A* every

system in the assembly will fall somewhere in that classification*

and we then can write symbolically

E8 M(a 8 ) - 1 (1.U)

where 1 stands for the measurement process that selects all systems.

It follows from (l e 3) that

M(a ! )M(a !D

) - M(a 8 8 )M(a 8 ) - M(a 8 )

so that one can consistently ascribe to 1 the algebraic property of

the unit element

„

More precisely* we mean by measurement the determination of the

values of the maximum number of simultaneously determinable quantities* and we take a" to represent the set of numbers corresponding to

such a complete measurement „ We speak of a system so selected as

being in the state characterized by a

8 „ This measurement process is

one that selects systems in a particular state and leaves them in that

state o A more general measuring process is one which selects systems

in the state a\ saj$ and leaves them in the different state a

8 8

associated with the same set of properties A„ Such a process is symbolically denoted by M(a’ 5 a

89 )„ In this notation* the previous

simple measurement corresponds to M(a*a”)® Clearly

M(aV 8 )M(a M a

3 ' 88 ) « S{ a

88

* a

808 )M(a 8 a5888 ) . (1.5 )

h

An even more general measuring process is one in which systems with

properties A characterized by the set of numbers a

3 are selected^ and

are then left in the state characterized by the numbers b” for the

I property where B and A are not simultaneously determinable 0 Such

a measuring process is symbolized by M(a 3 b

I! ) 0 Clearly we have

M(a 8 b

8 )M(b* °c ” ) ~ I (b% b

c “)M(a 8 c

8 )

o ( 1 „ 6 )

The question now iss What can we say about

M(a V)M(c 8 d

B )

This must be proportional to M(a 8 d°) 5 since the sequence of measurements takes us from a

H to dV The constant of proportionality is 1

when c

11 - b% and 0 when c

8 - b

ff 0 ^ b

8 o In general we know that the

state c

! cannot be predicted if the system is known to be in the state

Vo In fact we get the whole spectrum of values of c' s each value

having a certain probability,, Pending a more quantitative probability

interpretation we denote the numerical constant of proportionality in

the above relation by (Vjc 8 ),, and so write

M(a°b 8 )M(Vd 8 ) « (V )c 8 )M(a a d

8 ) . (lo7)

In particular

(b'|b") -S(b', b") . (1.8)

We see that the algebra defined by the measuring process and the

associated numbers is linear,, associative and non-commutative „ The

last two properties can easily be shown to be true since
'
5

M(a 8 b

8 )[M(c 8 d

8 )M(e s f8 )] - M(a 8 b

8 )(d 8 |e 8 )M(c 8 f

8 )

® (d 8

!

e

8 )

(

b

8

j

c

0 )M(a c f8 )

while

[M( a

“b 8 )M(c 3 d

8 ) ]M(e “f 8 ) « (b 8 |c 8 )M(a“d 8 )M(e “f 8 )

= (b 8 |c 8 ) (d 8 |e°)M(a 8 f

D )

also

M(a 8 b

8 )M(c 8 d8 ) = (b 8 Jc

8 )M(a 8 d5 )

M(c 8 d

8 )M(a 8 b8 ) - (d 8 Ja

8 )M(c 3 b8 ) jf (b 8 |c 8 )M(a 8 d8 )

¥e shall now obtain some consequences of this algebra 0 Thus when

M(a“ )M(b“c 9 )M(d 8 ) * (

a

8 |b 8 )(c 8 )d 8 )M(a 8 d8 )

is summed over a" and d

8 , then by virtue of (loU) we get

M(b 8 c

8 ) sT(a 8 jb 8 )(G 8 |d 8 )M(a 8 d

8 ) (l c 9)

a 'd 8

which is a linear relation giving the connection between two sets

of measurement symbols « In particular if B and C are the same physical quantities, and b

8 ~ c% then

M(

b

8 ) s £(a 8 |b 8 )(b 8 |d 8 )M(a 8 d8 ) a

8 d

8

If we now also take A and D to represent the same set of physical

quantities, we then get

6

M(b“) - ^ (

a

5 jb3 ) (b 8

j a

8 8 )M(

a

8 a

8

) o a

E a“ 9

Now taking

M(a°)M(b 8 )M(e“) - (a 8 |b 8 )(b 8 |c 9 )M(a'c 8 )

so that we infer the numerical relation

(a 8 lc 8 ) - ^ (a 8 |b 8 )(b l! |c 9 ) D

If we specialize this to the case where A - C we then get

rU'lb'Kb'U") -&(a', a”)

b

The Trace

It follows from (IdO) that

©

This,, together with (1„9) leads to the result that

MCb'c” ) - (e 0 |b 8 ) - r( a MV)(c'|d') (M(a'd') - (d a “ d.

8 '

(lolO)

(loll)

( 1 . 12 )

7

This indicates that if we associate some number with M(b 8 c’) in a

linear manner,, the choice M(b 8 c

3 ) “=“9* (c 5 |b“) will be invariant under

the transformation (1*9)

We caJLl the associated number the trace of M(b 8 c

3 ) 5 so that

Tr 0 M(b 8 c

8 ) ® (c”jb c ) « (lolU)

We now deduce some properties of the traces

We find that

Tr e M(c 5 d

8 )M(a rb8 ) * Tr. (d 8 |a 8 )M(c 8 b8 )

* (d 8 ja

8 ) Tro H(c ! b3 )

= (d c ja 0 )(b 8 |c *)

Similarly we have

Tr 0 M(a ! b

9 )M(c 8 d

8 ) - (b c |c 8 )(d 8 (a 8 )

so that the trace of a product of two measuring symbols is independent of the order of the multiplicand o

As a consequence of (1*8) we have

Tr» M( a

8 a

9 8 ) - &(a% a88

) (1.15)

and Tr , M(

a

8 ) = 1

In addition we have the relation that

Tr. M(a“)H(b 3 ) ~ (a" |”b* )(b 8 )a s ) ( 1 . 16)
'
The measurement symbol MCa^b") as written implies a certain

sense,, namely the succession of events happens as read from left to

righto Th§ measurement symbol in which the convention is opposite to

the above one is called the adjoint symbol* and is denoted by M(a s b

t ) 4

9 where

M(a°b a ) +

§ M(b»a 3 ) * (l .17)

As a result of this definition

(M(a i b

1 )M(c , dl ))

t = M(d 3 c

11 )M(b 8 as )

M(c 5 d

8 ) +M(a l bt ) +

e (1.18)

This can also be written as

[ (b B |c B )M(a tl d

!! )] +

- (c 3 jb 8 )M(a 8 d

8 ) + (1„19)

so that with a reversal in sense (b ! jc

3 ) is replaced by (c! !b3 )

0 If

we insist that no physical result depend upon this convention^ the

probability of transition between states a

J and b“ must involve

(a" |b 3 ) and (b s |a°) symmetrically 0 A quantity possessing the correct

properties is

p(a«, b») - p(b», a

3 ) - (a 3 |b»)(b 3 |a 3 ) (1.20)

Ep(a s s b

3 ) » 1

b“

where the latter statement,, which follows from (Idl)* is of course
;
necessary for any probability interpretation „ However, a probability

must also be a real non-negative number „ If (a s jb®) is considered to

be defined in the field of complex numbers, this will be satisfied by

the following restriction on the measuring algebra,

(b 5 |a 8 ) « (a 1 |b 8 )'* (l a 21)

©6

p(a% b

8 ) » | (a 3 |b 3 ) | 2 ^ 0

Note the general algebraic property of the adjoint operation deduced

from (lol9) and (l c 21)

[(b 8 |c , )M(a , d®)] +

= (b tt |c“)'‘M(a 5 d

8 ) +

Operators and Matrices

A symbol can be associated with a physical quantity in the following way c We have from (1„16) and (l o 20) that

Tr 0 M(a')M(b») - p(a% b

! ) (1.22)

hence we obtain for the expectation value of the physical quantity B

in the state a

8

< B> -£b“p(a\ b

8 ) . Tr 0 BM(

a

8 ) (1.23)

a” b

e

where

10

Other forms follow from

M(b') = £ (a'|b')(b'U")M(aV) . £ (a'|b')(b' |c')M(a’

aT

a' ( a“c'

l © O O ^

B - £(a'|B|a*')M(a*a”) - £ (a 1 |B|o')M(a’c')

where

(a 5 |B|a st

) a ') .(a 8 jb B )b 8 (b 8 ja 8 8 ) s Tr„ BM(a 88 a

8 ) b

8

(a 3 |Bjc 8 ) m V. (a 8 |b 8 )'b

8 (b s jc8 ) =Tr0 BM(c”a”) e b

3

Thus a physical quantity is characterized in relation to an

measuring process by an array of numbers a matrix 0 From

eral relation between measurement symbols

M(d 8 a

8 ) P(a ! |b s )M(o s b

t )(c'|‘3‘)

we deduce the matrix transformation law

with the aid of the trace formula (1 0 26) 0

For the product of two quantities we have 9 say

IT « £(a 3 |x|b 3 )M(a 8 b°) £ (b 88 |l |c 8 )M(b 5 8 c

8 )

- Yj (a 8

| X 1 b

Q )

(

b

c

1 Y |

c

8 )M(a 8 c

8 )

)

( 1 . 25 )

( 1 . 26)

arbitrary

the gen-

( 1 , 27 )

( 1 . 28 )
'
XL

or

(a 8

| XI |c 8 ) - £(a 8 |x|b c )(b'|lic 8 ) 5 (1.29)

b°

the matrix multiplication law 0 In view of the complete correspondence

between the measurement algebra and the conventional mathematical forinulation5 we shall borrow the usual terminology „ Thus we call the

elements of the algebra operators,, etc 0 We have anticipated this connection in speaking of the trace 0 Thus according to our definition

Tr e B - £b B - £ (a 8 |B Ja®) . (1.30)

b” a

6

Note also our definition of the adjoint of an operator

X - X (& 8

! X |b 3 )M(a 3 b

8 )

namely

X

+

- £(a 8 |x|b°)*M(bV) (1.31)

shows that

(b 5 |X+

|a t ) = (a 8| X | b

*

)

X

. (1.32)

Since the symbols of elementary measurements^ M(a’) are self-adjoint

(Hermitian)

M(a tt ) +

- M(a 8 ) (1.33)

this property extends to the operator representing any physical quantity,, i e e OJ, one with real eigenvalues
Iu

Hpsggs*--

ilpk' : Si

Slip^

IBb-m ;WSBSk

->.
12

Eigenvectors

The measurement symbol MCa'b 8 )^ describing the transition of a

system from the state a

8 to the state b” 5 can be analyzed further by

introducing a hypothetical state of non-existence, 0 o Thus we may

think of a two-step process equivalent to M(a“b“)^

M(a 8 b

9 ) - M(a 9 0)M(0b ! )

where M(a 8 0) symbolizes a measurement which selects systems in the

state a

f and annihilates theirq while M(0'b ! ) describes the creation of

a system in the state b

c 0 We shall use the notation

M(a 8 0) - f (a 8 ) (1„3U)

M(Oa') - f(a')*

so that

H(a'b') -f (a')f (b')

+

. (1.35)

The algebraic properties of the adjoint operator then correctly yield

M(a B b

8 ) +

(b 8 )f(a 8 ) +

- M(b V) •

According to the multiplication law

M(Oa 8 )M(b 8 0) - (a 8 jb B )H(0)

or

f (a') + l(b') - (a'!b')M(0) ( 1 .36 )

13

Thus,

(a'|b') - (0|f(a') + ?(b‘)|0)

or with a simplified notation, in which the null state is

(a'|b ! ) - (f (a')+W(b')) .

In particular

(Y(a ! )’Y(a'')) - 6(a'» a”) o

We infer from. (1<,38) that

(a'lb’f - (Y(b')V(a')) " (b'|a')

and from (1„37) that

(a' |b') = Tr.f (a i )*'f(b') * Ir. f(b ’ )fr( a'

)

+

- Tr. M(b'a')

For a general operator represented by

X = E(a*|x|b-)F(a')f(b>)+

we deduce that

XY(b') -E'f'(a , )(a , |x|b') a

b

'

|X|b')f(b') +

( 1 .37 )

understood,

( 1 .38 )

( 1 .39 )

(lolO)

and

since

¥(a')M(0) -Y(a'), M(0)y(b’)+ -Y(b')*

In particular^ justifying the eigenvector designation,,

Af(a') * a >y(a')Y(a')

+A - Y(a')

+

a'

We can also conclude from (I0I4O) that

Y(a E )*xY(b B ) - (a’ |x|b c )M(0)

whence

(a 8 |X jb ! ) - (Y(a a ) +Xf(b“))

and

(a 5 |l|b“) - TroT (a 8 Tx^Cb 3 )

Tr e XM(b “a 8 )

As a special case of the measurement symbol transformation

(lo9) we have

M(b*0) - £ (a 9 |b t )H(a 8 0)j M(0a 8 ) » £ (a 8 |b ! )M(Ob ! ) a

3 b

5

(lolll)

(1.U2)

(loU3)

equation

(lokh)

in which the transition amplitudes (a 9 jb s ) appear most directly as
.
15

transformation functions <, Conversely the transformation equation

(lo9) follows from (l 0 UU)° Note also the converse derivation of the

multiplication law5

which involves (

0 Ifl )„

Unitary Transformations

¥e now look more precisely at the changes in the manner of description of our system 0 Consider two descriptions of the system* one

in terms of the properties As with eigenvalues a% the other in terms

of the properties B with eigenvalues b

B

<, Since the number of independent states of the system is the same in A as in B* we can establish

a one-to-one correspondence between the states a

8 and b8 « After making the association a*« Vb B we take M(a “b B ) to refer to pairs of

states put in such a one-to-one correspondence 0 We now define the

quantity

M'(a"b 8 )M(c 5 d

s ) - Y (a 8 ) T(b 8 ) + f

r (c “ )lr (d t )

f(a 6 )(y(b s ) +lT(c“))y(d s )

(b“ |e“)M(a 8 d

8 )

o (ioix5)

all pairs

(a 8 b

8 )

Evidently

(1.U6)

16

and

Uba - E H(b'a’) = U

ba (a , b .)

ab

For se queried transformations a —-» b —I c 9 we have

U , U, - f M(a*b 8 ) T M(b 5 c

5 ) ab be

(aVb

i) (bV)

? , M(aV) * °ac

where the c

B written down is the one corresponding to the a

the intermediary of b

8 „

In particular with c - a5 we have

U . U, - 1

ab ba

and similarlys

XL U . = 1

ba ab

so that

U . if, « if, U . * 1

ab ab ab ab

which characterizes U

^

as a unitary operator,,

It follows, fron the definition of U , that

ab

(1*1+7)

( 1 * 1+ 8 )

through

(1.1+9)

(1.50)

(1.5D

U

aby(b") - Y(a”) , f(a»)\b

- Y(b»)+

(1.52.)

9 ? where a and b are corresponding states

The inverse relations are

U

fca ^(a 9 ) - “\j/(b') , ^

r (b’)

+Uba

s (1*^3)

One can construct the transformation function (a ! Jb”) as a. matrix,

element of the operator in the 5 a’ description

(a

1 |b" ) - C^( a ’) + ^(b")) -

( Y(a')\a ^(a" ) OuS'M

- (•

, |oba l«">

or the ’ b

5 description^

(a'|b n

) - (f (a')+ T(b"))“ (Y(b')

+

Oba Y(b"))

( 135 ) (V lojb") .

¥e now remark that

M(b') - ^

a

M(a')Uab (1^6)

which follows directly from the multiplication law of the measurement

symbols, or from the eigenvector construction

M(b') - -f(^)*Vah . (137)

Accord!ngly 5

B - 2 b s M(b 5 ) - 2 b(a ! ) M(a ! ) U

ab

(lo58)

- V U

ab

18

where the correspondence between eigenvalues enters in writing b 3 as

a function of the corresponding eigenvalue a® c We have also used the

general definition of a function of an operator 9

b(A) - S b(a') M(a®) . (l £9 ) a”

In the important situation where A and B have the same spectrum^ we

can establish the correspondence so that

a® = b ®

and therefore

(l©60)

B " V A V » A ” U

ab BD

b a (lo6l)

°s° Conversely9 let U be an arbitrary unitary operator U s

and construct

I - UAXf1 s Za> UM(a 9 )lf’

1

„

This can be written

A 83 2 a” M(a*

)

where >

a® - a* *

}

1

( 1 . 62 )

and

Y(I') “ <!?(»') , U "1

,

(1.63)
'
19

so that A and A possess the same eigenvalue spectrum and corresponding

eigenvectors are related by the operator U,

For an arbitrary operator

X « 2 (a* |X

| a" ) M(a"a ! ' )

we have

I - U X If

1 - Z (a» |Xj a*8 ) M(a 5 S ,? )

so that

(a* |X| I ?s ) - (a« |xja° ) . (l ff 6U)

Furthermore, all algebraic relations are preserved,,

OTTi) « s + ? , (3) - a

and

C£)+

- (?) .

Thus the description resulting from the unitary transformation

is on precisely the same footing as the original description.

Infinitesimal Unitary Transformations

Consider the special situation in which A and A differ infinitesimallys as obtained from a unitary operator U which is in the

infinitesimal neighborhood of the unit operators

U "1 - bF „ (1 .6?)

20

Here F is an infinitesimal operator and $ is introduced as a constant

with the dimensions of action in order that our physical quantities

be measured in conventional units „ Since U is unitary,, we must have

# i + U » 1 + ^

F

equal to

u-1

- 1 + i F ,

that is s F must be an infinitesimal Hemitian operator e We write

?(!') - f (a*) - (U-l) ^(a>) = SY(a ') (1.66)

so that

hf(i') - - |fYU') (1.67)

and

S

1? (a”) s Tf (a')

* F 0 (l«68)

For an arbitrary operator X3

X - U X if1 - X + jjj [XjF] .

This we write as

X - X - 5 X (1.6?)

where

gj [X,F] . U . (1,70)

Now it follows from (l 06U) that

21

(I* |X|1 S * ) - (a' |x|a" ) - (a" |X - I) |l" ) . (l e ?l)

For an infinitesimal transformation this becomes s in our notation,,

S(a' |x|a" ) - (a 8 |S l|a !B ) (l„72)

where the operator is held fixed on the left side

„

An important special case is that in which it is possible to construct B A as an arbitrary infinitesimal multiple of the unit operator,

oA s S a

which requires that

[A, (F/Sa)] - it . (1.73)

A (a*

)

s (A - 6 a'i^Ca’) ® a

8 ^(a*

)

yields

Af (a') - (a 8 + l>a)'Y(a 8 ) ,

which implies that ‘^(I 8 ) is an eigenvector of A with the eigenvalue

a

5 + & a s our assumption can be realized only when A possesses a continuous spectrum ? Notice that ( 1 *,72) reads

S(a*jA|a H

) s Sa S(a',an

)

in agreement with the fact that the change in the eigenvectors is

equivalent to increasing the eigenvalues by £a„

22

We now examine the effect on a transformation function (a 8 ]b 8 )

(a 8 and b 8 again refer to arbitrarily chosen eigenvalues) of subjecting the 8 a

5 states to an infinitesimal unitary" transformation generated

by F $ and the rb

5 states to an independent transformation generated

ct by Since

(a'|b>) - (y(a')+ ^ (b>))

we get

6(a<jV) - | (^(aO* F

a y(b')) - |( (a') * F

fe vp (b’ ))

or

&(a 8 |b 8 ) - |(a'|(F

a - F

b

)|b 8 ) . (lc?U)

Of course., if the same transformation is applied to both types of

states (F s F, ) , the transformation function is unaltered,, a b

9 One may inquire, more generallys what form &(a 5 jb’) must have,

for any conceivable alteration that is consistent with the three fundamental properties of transformation functions 9 namely

Z(a 8 ]b 8 ) (b 8

j c

* ) - (a 5 |c J ) 5 b

!

(a* fa” ) s S(a f s a”) , (l®7£)

(a* |b 5 )* (b 8

|

a

1 )

We shall write

h( a

8 |b 8 ) - |

(a 8 |SW

ab

|b 8 ) ( 1 .76)
-
23

which is the definition of the infinitesimal operator S W.,.^ . According to the first9 composition property 9 changes in (a 9 jb ! ) and (b 9 |c 9 ) imply a change in (

a

9

| c

9 ) given by

&( a

9 |e 9 ) - 2 S(a 9 |b 9 ) (b 9 |c») + 2(a 9 jb 9 ) $ (b 9 |c 9 )

- |

Z (a* |8WA

|b»)(b* |s>) + |

Z (a' |b<) (b> | 6 Wb8 |c')

-^'l(s>wab + swbc |c)

which is the additive composition property

SW. ^ Su e ab b© ae (lo77)

In particulars if c « a s we have from the second fundamental

property*,

SWab

+ Swba

" 0 (1,78)

The third general property of transformation function implies that

-j?( a'l £w*lb ')*"

|

(b’i s w

ba l a,)

or

£W

ab

£Wba

£ W

ab ’

(1.79)

that is 9 W_.. is an infinitesimal Hermitian operator. Of course

these conditions are satisfied by the special form

8Wab -Fa" F

b

• (1.80)

11° The Dynamical Flinciple

¥e introduce the time t as a parameter upon which physical quantities depend,, and require (principle of time homogeneity) that all

values of t be equivalent^, for complete physical systems,, This means

that the spectrum of a plysieal quantity is independent of t- s and that

a change of t corresponds to a unitary transformation. Furthermore 3 we

assert that, in generals compatible physical quantities refer to the

same time. That is,, a state (of maximum information) will be specified

by the values of a complete set of quantities at a given time , !'(t) „

We write the associated eigen-vector as y(f t>. A change in description may consist of choosing a new set of commuting operators at

the time t 9 or of changing fee time for a given set of commuting opera"

tors s or of both alterations. Thus the most general transformation

function is

( 1 1

*1 1 i 2 V

s

( ¥( t Is)* ? ^2 e ( 2

This describes the relation between states at the two times and thus

contains the entire dynamical history of the system in this interval*

It is the object of quantum dynamics to construct all such transformation functions 9 and accordingly s we may expect that the fundamental

dynamical principle will be a differential characterization of this

general transformation function.

According to the work of the last section, we know that for any

change ©f the transformation function (2,1) s be it of the times
*

v.'
2£

and t

25 of the operators t,., and (

9 or of the physical attributes

of the system in the interval from ty to t29 that

S(qti|$2 t

2

) -l( titi|SWi 2 K'2 t

2

) , (2.2)

where £ is an infinitesimal Hermitian operator with the additive

property

& W

12 + SW

23 - S¥13 o

Another additivity property refers to composite systems 5 i„e„ 5 two

dynamically independent systems oc and /3* which are considered in

conjunction e If the states of <X and [3 are described by the eigenvectors

'L|

>' t) and Y ^ t)^ respectively, the composite state

is described by

’ t p

' n = ^(C' t) o - Yty’ t)

^

(t

01 ’ t) .

Accordingly

/9'

1 t

2

) - *3.1? 2" t

2 )(^’ tjicf t

2

)

and

«Vif t

2

) - (r1

, t

1 ix°ty2

" 1

2 )q\' /3 " 2 h)

Q( where X Is a physical quantity of the ot system 0 There is an

analogous statement for X

' Q With the shorthand notation

(l) » (l)^ (1)^ , we find

26

MD - s<D* (i)^ + (i)^ s(i^

{ 1 (

w*

2 +

which is the additivity property for dynamically independent systems:

swA+ $wA - SW. 12 12 12 *

There are two types of infinitesimal changes in the transformation functions „ In the first we adhere to a given dynamical system

and introduce infinitesimal alterations of £

(t^) and £ 0 (tg)

This

includes changes ©f t*, and lies© transformations are generated by

infinitesimal Hermitian operators a and f^ g which, are functions of

dynamical variables at and t0S) respectively. Hence for this ‘type

of change

In the second type of change,, the initial and final states are unaltered,, but some physical characteristic of the system is modified

in the time interval t + dt. Now

2 s

dt)d|'(S*t+ dt|{ ,, t)dj"(s"t|^'t2

) ,

which has been written in the form appropriate to continuous spectra*

Transformation functions referring to an interval that does not include (t,t + dt) will not be altered,, while,, as a special case of (2 0 2) 5

2 ?

S ( |

' t + dt j f

'* t) 55 i( | ’t + dt j SL(t)dt|^'! t)

where 6L(t) is an infinites imal HermLtian function of dynamical

variables at time t s and the differential dt appears to conform with

the vanishing of the left side for equal times 6 We conclude that for

this type of changes

SW12 - 8L(t)dt , (2.3)

or more generallys if we consider a distribution of variations in

physical attributes s

The form of the infinitesimal operator characterizing a general

change in the transformation function is then

SW12 ^ F

1

“ F

2 + 1

or if we construct a function F(t) such that

1 * F(t0 ) s F

2

9

we may write

:w.12

t

dF(t)

” dt

+ SL(t) latd .

We now assume that there are classes of changes for which the

generating operators $ W^ are obtained by appropriate variation of

a single operator W^

28

SWX2“ S <V •

and that ¥1 ?

has the form

¥ f L(t)dt

\

where L(t), the lagrangian operator, to borrow the classical terminology,

is a function of certain fundamental dynamical variables .r , in the

infinitesimal neighborhood of t, i.e..

The limitation to first derivatives can always be achieved by suitable

adjunctions of dynamical variables e ¥e take L to be a Hermitian operator, thus imparting the same property to 0, the action integral operator, and thereby satisfy the requirement that S¥- ,, be Hermitian*

As indicated by the explicit occurrence of t in the Lagrangian, our

treatment will not be restricted to complete systems. One should notics, however, that for a system acted on by time dependent external

forces, not every physical, quantity has a time independent spectrum.

There will occur in the structure of the Lagrangian. certain

parameters* Any alteration of these quantities is a change in the

nature of the dynamical system, (the addition to a Lagrangian of a

new term can be thought of in this way) . The associated S ¥_

has the form (2 .3) with SL « S (L) * On the other hand, for a given

form of the Lagrangian, we may introduce certain infinitesimal

'2

29

changes of the x. (t) , and of tq and •% „ This must correspond to the

possibility of altering the nature of the states,, at t, and tg for a

fixed dynamical system 0 Henee

s w

2 2

o

This is the operator principle of stationary action since must

be independent of dynamical variables in the interval between and

toe ¥e shall obtain therefrom equations of motion for the x. (t) 9 and

expressions for F. and Ft, 0 ¥e may note here that if we were to replace L with

L-L-|Lw 5 ¥ e ¥(x(t) 9 t)

with W-^ g

?

W, 0 - W, 12 1

W, W, ¥(t $ w

2 W(t

2

)

we should be adding to ¥, operators referring to times t^ and tg*

Hence the stationary action principle leads to the same equations of

motion with - 4 as with ¥^ f?

i» and

S %2 63 F

1

" F

2

where

8 ¥

1

“ F

1

" F

1

s ^ W

2

’ F

2 " F

2

*

Hence altering the Lagrangian by the addition of a time derivative

does not change the dynamical system under consideration5 but rather

yields new generators of infinitesimal transformations at t^ and tg«,

30

Concerning the structure of the Lagrangians we require that the

limitation to first derivatives be maintained under any integration

by parts s i 0 e 0|) the addition of a total time derivative e This implies

that the Lagrangian is linear in the time derivatives „ Accordingly«,

we write

1 8

2 2 b.o^ (x±

xf) - H(x,t) (2 „U)

where (b. .) is a numerical matrix 0 This structure remains unchanged

3-3 if an integration by parts is performed on the time derivative terms „

The operators x. can. be chosen Hermit! an without loss of generality c

In order that L be Hermitians it is necessary that H s the Hamiltonian

operators be Hermitian5 and that

ij i dt

V

1

the b-matrix must be skew^Hermitian » We shall decompose b„ . into

anti-symmetrical and symmetrical elements,,

£

which are 5 respectively,, real, and imaginary$

31

&c$ j 1.3 13 3-3

and as sume that the dnyamicaX variables correspondingly decompose into

two kinematically independent setsj variables of the first kind 9 associated "with a. .« and variables of the second kind, associated with

13

s

ec (3 (employing Greek indices to distinguish the second set) s

/ dx, dx,

L "

2

2 a

ij \

Xi IF + Ft

dx „ dx

x, | + ^Z X ar'w x «)' » or t)

We have used the phrase 8 kinematically independent 8 to mean the decomposition of' the time derivative terms s as distinguished from 5 dynamically

independent 5 which refers to an additive structure of the entire

Lagrangian 3 i 0 e 05 of the Hamiltonian also„

The action integral associated with the Lagrangian (2J4) is

On subjecting this to a variation we may keep the t' limits fixed

,

representing variations of t^ and t0 by an alteration of the functional relation between t and V „ Since "C is not varied we need not

write it explicitly

32

6 ¥

12

2 bo c(6xadx 0 “ dXo S x„ + x

2 -D ID ID ;.dix. “ dSx.x.) ~ gH dt - H dSt 1 D ^D

d J Z bo . (x. Sxo - 5 x,i.) - HS t

(t, -‘-D 1 j 3-D

7 | 2 \j ( « x

ite

a S x.) - g H dt + dHg t

J

The stationary- action principle requires the vanishing of the second term,,

which can be expressed as

dH dx- dx„

:H '#E 8t + z VSxi7T" ir s y

* m 5 t * uij( Sl

i

-ar + rn^ sx.) + z a dx. dx

(2.5?)

dx^ dxx~ -ar --ar

Sx* )

¥e also obtain

F

1

" p( t

l) 5 F;» - F(t

2

)

•where

F - * Z Fix. SXi ° ^ x.xJ - H S t

-LJ ± j x j

K 5 2 a. .(x. 5 x. T+ Sxx) L’ +T i 2 (x^, S 2

* ®<*/9 ® x

/3

( 2 . 6 ) 6 x^ x^) - H5 t

The character of the variations to which the principle of stationary

action refers is now made expli.ci t by the statement that the symmetrizations and antd-symmetrizations occurring in (2 S) and (2 C6) are superfluous 3 in virtue of the operator properly a£ &x.„ and 5 x 0 ¥e infer the commatator and anti-commutator relations
'
33

Sx^x.J-0 , &x<3 , x^

8 X/

dx

©s s dt

- 0

ix:» dx

Now we shall obtain from (2 S) expressions for and -r?- as functions

of the dynamical variables 3 i.n terms of the structure of the Hamiltonian* The first of the latter conditions is then satisfied if

which gives $ x. the character of an infinitesimal multiple of the

3 unit operator o The second of the latter conditions is satisfied with

X.

dx

o{ „ provided is an odd function of the variables of the -second kind.

It is thus necessary that the Hamiltonian be an even function of the

variables of the second kind s but- is without restriction in its dependence on the variables of the first kind.

We write

c rr 3H , , r- 3 H , &H -gj t+Z 6^5^ +

2> t h

x,*4*^ *

or an alternative form in which 'left derivatives' are replaced by

'right derivatives'

3 t H

a, " * fix

of

} H

S «

No such distinction occurs for first class variables. The equations

of motion are obtained as
.
3h

and

dH k _£H

dt St 5

dx „ a

ij "at

5"

9

dx

~dF

(3 d7r E d H

L „ _r_

y

a» „ i. Sx „ +

13 i 3 s«„

x« 4 x, H St

dx„i

We now turn our attention to variables of the first class.

The Canonical Form

In order that the equations of motion be solvable for the

the anti-symirietrical matrix (a. „) must be non-singular. This requires

that N s the number of the x^ 5 be even. Indeed

det a.. e det a.. * (~l)

N

det a . . j

J”*"

the determinant vanishes identically for N odd. Hence,

N 85 2n

when the. integer n is the number of degrees of freedom. Now a real

anti-symmetrieal matrix of even dimension can. by real linear trans'

formations^ be reduced to the canonical form

35

To show this we consider the binlinehr f on

2n 2n 2n

A

. L a

ij

x

iy

j • + *1 X Vk +

k®3

*2 XkXB Vt

x *ia y-i - x vt ya k®3 k-3

2n

i 5 j*3

a. .x.y.

13 1 3

We assume that > 0 ('if it is negative 5 then a-., > 0 and we may

satisfy our assumption by a relabeling) and define the quantities s and 1' ? «1 1

(2a10 ) 2 Q1 - x. 1

in

a2k?

ck

in

( 2a!2) W " n - |vk

la,*)” 2 £_» ® x„, + 1

^ a2n

12 3

(Sa^)*"2

in

X

alkyk

Under this transformation A becomes

2n

£

- - J 1 l

,,3,1) ^ 0

* 1' __

Jo cio 3= CJJ ^

aij - “4 (*11*23 - *«»2i>] x

i7j

Since the matrix of the 2n - 2 dimensional, form is again anti-symmetric ale,

we can repeat this process and finally obtain

, n

A

k“l k € .»!

k

36

For the linear combinations of x. variables associated with the i

canonical form we shall write

5 k

" Pk

k

! \ 9 k n

Thus the Lagrangian and the infinitesimal generator F become (we are

considering only the first class variables)

*

dq

k

dp, dq,

'k "dt

’ q k df ‘ dt p 3

dp,

-%r qj “ H(q, p, t)

F

1 E (Pjc 5 %

- <?k SPi> - sst

while the equations of motion in the canonical form read

dqk „ BE dpk _ 3H dH s £H

dt $$ 9 dt ~ 5 dfc Si

It will be noted that the derivative terms in the Lagrangian can

be given less symmetrical but simpler forms by the addition of total

time derivatives o Thus

1

n

P

k^ \

dpk V “ST \ * M l E (pk. %\

and correspondingly

I £ <p

fc £ \

- pk

> E pk

& qk

- £

[ l l {pic> q k }

qk

s p k

+ E E{p k>

37

Hence if we employ a form of L in which only derivatives of the

occur,

da.

2 'k _

1 k

df H

that part of F referring to changes in the and p^ will be

ha " XW

while if L contains only derivatives of the p, >s

1 % V vr H

the relevant part of F is

% B Pk °

The Canonical Commutation Relations

We must evidently interpret F_ as the generator of an infinitesimal

© q

change of tile with no alteration of the p, s and conversely for F

ST

Hence

V F

gq |. - i XSqk , |p k, ‘sq

Pk. J Pk > [>

v

Since 6q S p^ commute with all quantities s i.e® s are arbitrarily infinitesimal multiples of the unit operators we have
'
38

or

[v y ]

* [tv y ]

" 0

[

qk» vi J

1 ^

where the last canonical commutation relation is consistently obtained

from both generators „ Observe that for any change of alone that

is compatible with the commutation relations

3

and similarly with £p^ 0 This is our original hypothesis concerning

the £ q are £ p^ 3 which is thereby shown to be consistent with the commutation relations derived therefrom. It also follows from (l„72) et„ seq,

that the spectra of the q

5 s and p5 s form a continuum

„

If G(q,p) is an arbitrary function^ we have

G. F. 5 &q

$G c

3 \ ^

or

K h t

G ’ pk ]

* I [v G

]

Similarly «,
.
39

[g 9 F * i # $ G

L &P J P ap.

Sp

3d.elds

Complete sets of compatible physical quantities ( commuting operators) are provided by the totality of q’s 5 or of p's s at the same time*,

Thus we have two elementary descriptions s mth the associated eigenvectors (q ! t) and (p 5 i) «, The transformation generated by and

have a particularly simple aspect for these eigenvectors % b P

I

F

5q

^"V

jKP

(p ? t) - S (p 8 t)

whence

~~ ‘T(q't)Sq.k ,

t(p’t)SPk A n it K

pk Y(qs t) “ i H ^(q’t)

\'T’(p , t) - f 4t

fCp’t) .

The adjoint equations are

T(s’t)

+

Pk qk

^(q't)‘ J

(p"t) q.k t

k

If G(qs p) is an arbitrary function of the q’s 5 but a polynomial

in the p’s,, we have

( Y(q3 t)" G(qp) » G(q«* | j|f ) l^(q’t)

This follows by induction from its assumed validity for G-^ and G0

and.

its verification for G^ + G0 and for G^G^s

q*t) G1 (q p) G0 (q p] q*t) G,(qp)

G

l

(

' q?s 2 aqT' °2 2 To^

combined wi th the evident truth of the statement for- G “ G( q) s and

G p^

0 On the other hand ,,

G(qs p) ^(q't) -tq',

where the order of all factors is reversed in G „ Hie significant part

of the induction proof is

G

x

(q p) G

2 (q p q*t) « G1

(q p) G

2 (q% i jA ^(q't)

f^r) G^(q 5 ,i /. )^F(q’t)

- g^

g

2

(q8 ^ * &Y(q *t) -

Notice that if G is a Hermitian function of the q’s and p’s with real

coefficients* G “ G. The analogous statements for a function that is a

polynomial in the q’s are

Notice that the effect of Fr8p on^ (q*t) s and of F on^ (p’t)

is just a numerical phase changes

Sp 1? (q't) (2 q£Spk

) 1

J' (q't) 9

%n "¥ (p 5 t)

q

This indicates that the notation1^ (q’t) } sa

y

5 is really incomplete s since the change in phase does not alter the eigenvalue q® 5 but does

yield a different physical state

Time Displacements

It is evident that

is the generator of the transformation >h ich consists in replacing

dynamical variables at time t by those at t + £t 0 Hence for the fune

tion G of q(t) s p(t) and t 3 we have

F Hfit

[G j -Hit] ee

when $ G is such that

the unitary transformation has no effect upon t as it occurs explicitly

in Go We infer the general equation of motion,,

h2

dG „ 1 r dt " Jt IH [G > H]

By successively placing G = H, q

k

, - pk

, we check the consistency of the

theory by rederiving the equations of motion originally deduced from the

action principle;

dH 2K

cTE sTi s ^

1 r 2H

at ~ w [qh ? H] s Jph

=

i

in ,, i . £2

dt if? 5 *n J ?qh

The time dependence of an eigenvector ^ (£'t) is determined by

-|Fgt f(4't) -S

t f (ft) 1(4 ’t)st

whence

1 tji left) - h Yt’t)

and

i K |f 1 (ftp - YK't)+ k . 2t

In particular, if H is a polynomial function of the p's, we have

itf & ¥ (q't)

+ -H(q’, | J-; , t) Tf(q't)+

and

i K j^Cq’t) s i # t) (q't)

U3

Accordingly if is the eigenvector of some state not involving t in

its specification 3 the 'wave function' of that state

-q^(q't) - C^(q't)

+ Y)

oo o obeys the Schrodinger equations

i t ^ (q’t) - H(q' s |

t)

^ (q't)

and

i K ji Y (q ! t) v* - H(q« , i t t) Y (q ! t) w

.

V* When H is a real function,, H s H, More gene rally5 if Y is a member

g of a complete set of eigenvectors $ 'f(=o 5 the transformation functions

(q E t|oJ) - Yi (q ?t) a (cA 1

|

q

? t) - (q ! t)"'

cx cA09 obeys the Schrodinger equations.

Canonical Transformations

We now consider in more detail the freedom of description for a

given system associated with the possibility of replacing a Lagrangian

L by

£ - L - at w »

the action integral W^ by

^12 " W

12 “ ^W1 ” W

2

^ 5

and the generating operator F by

hh

F ® F = SW

We have seen that one can Introduce a csnonical form for F 5

F « ^ Pk s ” H ^ t s

which implies the canonical commutator relations and the canonical

equations of motion «, We ask for the conditions under which F will

preserve the canonical form, but expressed in terms of new quantities

V Ph’ **(5, Ps

F - 2 pk

5 qk

- H St .

This will yield the canonical form for the commutator relations and

equations of motion obeyed by these new quantities,,

The difference of the generating operators F and F is the varia=

tion of an operator Ws

"5 W - = H B t *= + .

Thus ,, in terms of a function W(qs q, t) s we obtain

p

« .jL. w 9 - p. * w

k d %

k

H-H+ijW ,

as the equations defining such a canonical transformation, provided

it is possible to solve without exceptions for the q’s and p’s.

An elementaiy example is provided by

W - 2 |

16

We have

SW « S qk

S q k +2 q k

S q k

so that

% ‘ Fk > Pk “ ‘ H !

this is the canonical transformation interchanging the q’s and p's 3 with appropriate signs

„

The general linear transformation is generated by

or 5 in a matrix notation

p

« cx q + ^ q .

The explicit equations of the transformation are then

q

- a q

+ b p

p

® c q

+ d p

U6

where

a - - f^T

1 ^ s b *

c - - ft + If /S'"

1 c*

9 d - - m

The four matrices a 3 *b 5 c 5 d satisfy the relation

a d » b o E 1

which, in fact5 is just the condition that

£ V P

?

]-iH s# .

The matrices appearing in W are expressed in terms of the matrices of

the transformation equations by

06 « -» b"1

a 9 (3~b L

3 /

s - d b

^

The fact that the & and qf matrices are necessarily symmetrical implies

that

a b a b o5 b d H d b s cd^dc 3

the first and third, cf which are the conditions on the transformation

imposed by the requirements

[q k

, q^] “ [p ks p^3 “ 0 .

The transformation function

( q’t j q’t) - (q’t)

+

'ip’(q't)

h7

can be constructed from the differential equation

8>(q 5 t jq't) - g

(q’tj(F - F) |q't)

“ (q 5 t|6 W(q5 q5 t) |q ! t) ,

by performing the following process,, Take the differential expression

S ¥ and 5 employing the commutation properties of the q"s and q's, arrange the operators so that the q

s s everywhere stand to the left of the

q

J s„ This ordered differential expression will be denoted by 6

cw(q,qs t) .

That is 5

SW(q,qjt)j^(q,q,t) ,

but the ordered operator ^(q,qs t) obtained by integration is not

equal to W(q5 q5 t) 5 and indeed is not a Hermitian operator „ With this

ordering 5 we have

g(q«t|q s t) - |

(q't| £V(qs q,t) |q’t)

* |S‘^(q ! s q

? 5 t)(q , t|q’t)

since the operators now act directly on their eigenvectors. The solution of tills differential equation is

(q»t|q't)

gW(q' s q',t)

where the constant of integration is additively incorporated in TV.

It is to be determined from normalization requirements such as

(q’tjq’t) dq" (q5 1 |q”t) “ S(q ? - q") ( 2 . 8 )

U8

For the example of the general linear transformation we have

SW “ S 5 _ q,. + (3 ±

. q.) + 2 ( q± (3 ±j +

q±

lf ± ^) S q

^ * 5 ^ 3

the ordering operation here is trivial. Hence

V - Z (i«1} q± qj + qx t

. 1

q. q.) + Const2

and

(q' |q') - C ((3 ) e’

gSC^-or . .q'q! +

ft. -q- ql +2 7$ - .q'q

1.)

in which we have anticipated that the integration constant does not

depend upon the matrices ^ and \ Notice that the inverse transformation is obtained from the substitutions q5 p

^

q3 pj cL

/3 — {3 s so that

Cq’ |

q

f ) * C(- p) e ^(4« ljq!q^/»i;j qfq^Ti

.5;q')

This should also be the complex conjugate of the original transformation function , which is indeed true if

C (~p) - C((3 f 0

¥e now compute

(q* j q® ) d q

s (q ? |q n

) 2 Y

iZcx (q’q'-q^q")

|C(

(5) f e

1 J J J

d q e

;

wp>l2 wwl s( q' - <*')

h9

whence

|0( (3) | 2 |det (3|

(2'lTK)

The condition (2,8) is now satisfied with

C(/3)

The explicit appearance of i is demanded by the requirement that in

the limit of the identity transformation,, the transformation function

approach S(q ! - q’). In this limits d ~ ($ s - (3 9 (3 ^ —t 0 5 and

(q } |q’)

3

“£ S (

q

! ~ q! )

as it should „ For the special case provided by q^ * p^3 * -q^ we

have ci if TM 0 9 (3 “ ls so that

A simple connection between the HeHaitian operator ¥ and the nonHermitian ordered operator can be established by treating $ as a

variable parameter. We must then write the differential characterization of a transfor mation function as

whence
'
?0

(j/aj) a) - K|w|)

provided W does not involve $ explicitly e However^ the ordering process that defines s

S ® S ( "^ w

)

introduces $ into the structure of ~yfs so that

¥ « (<? fd |)(| V )

. = V - # V .

For the example of the general linear transformation

V- £ (I “‘ipiO, + /hpi'ij * i * |j log \

n

2Hty

det ft

which is non-Hermitian?

- "W 38

L, qj5 q± iK log det (3 (2irX) n

iK n(log 2 7fp{ + 1 ) ~ log det ft ,

according to the commutation relation

Now

(2.9)

det

ft

so that

$1

n

2

which is indeed equal to ¥ in virtue of the commutator (2$) „

The Hamilton-J acobi Transformatlon

A canonical transformation ~ the Hamilton-Jacobi transformation

is generated by the action integral itself e If we put ¥ ® ¥~

L

~, and.

write t^ « t, * t 5 where t

Q

is an arbitrary fixed times we have

Accordingly, the action integral induces a canonical, transformation

from q^t), p^(t) 5 H(t) to q(t

Q

) a p(tj s 0 e The vanishing of the new

Hamiltonian is required by the fact that the new canonical variables

are independent of t. Thus s the equations describing this canonical

transformation are

the Hamilton-Jacobi equations „ Incidentally, the new Hamiltonian,

H « 0, should not be confused with H(t ) which determines the dependence of ¥ on t ,

g ¥ - F - F , o

3

that is.

P

£2

A simple illustration is provided by the system of one degree of

2 freedom H “ p /2m 0 This is a conservative system,, so that W depends

only on t = t 9 and we shall place t

Q

28 0* The equations of motion

have the solution

q(t) ' %

+ I P0 »

which is a linear transformation* Accordingly the action integral

operator has the value

2

¥

2m

m / i ^ w lq {

q - V V -2m~ “ 2t

which is of the general form. (2 0 ?) with

« ® ® = p

1=5 j

Thus we have the commutation relation

m

the ordered operator

V m

2q q0 + qo

k

) + ^ log (2if

and the transformation function

m

(q't |q

!

* 0) - 3

1 V(qf ,q" ,t) m

1 i m / , > a t 2 g 3t(q q ^ e

f

.

which satisfies the requirement

(q’O j

q

n 0) - S (q* - q

H

)

It is often convenient to employ rather than qv (t^) as an k o independent variable in the Hamilton-Jacobi transformation, i.e.

3 ¥ W

k ^qk

s J k

s "o " 3 p k

( FT

H + 2J1 « 0

The connection between the two generators ¥ and ¥ is provided by q%

¥.

q p ir o

» pk

(t0

)

qp.

namely

¥ ® ¥ + ¥

qqo Vo

For our example,

¥

qp. {

q> p

°}

P

_2_ t

dm s

which again possesses the form ('17) 5 with ck ® °> P = ^ r

Hence

t

2m ‘

v/(q, p0

s t)

[p0

q] - 1 3

po

+

Vi , i

q po _ ar - n •Log tfk >

and

(q 1 1 |p^) * e'|V(q' 3 p%t)

Another example is the one dimensional system with

TT p , mto c H ® $r- + —w— q 2m H

The equations of motion have the solution

1

q “ q cos w t + — p sin a» t ^ x> mu?

1 o

mw q sin uu t + p cos toi

a linear transformation,, On substituting these solutions 9 the action

integral is obtained as

2

/ P,

¥ 2 V Pq r sin cot

m.u) cot cut q

[ COS iufc % r + %

Hence ck * ^ m\u cot wt3 ft

33 =» row esc u>t 5 and

Constrained Transformations

A special situation is encountered when the canonical transformation involves one or more relations between the q's and q's 3 so that

they are not actually susceptible to independent variations. The

simplest example is the identity transformation

qk

=

3 Pk “ Pk

where W(q5 q) has the value z ero , indicating a relation between the

q’s and q's. Nevertheless., one can treat the q

! s and a

? s as independent

variables s and derive the transformation equations from a suitable W,

provided one introduces an intermediate transformation not so handicapped

and refrains from eliminating the intermediate variables, Thus 5 describe the identity transformation as q —> p —

$

q for which

We have

fmm which follows the desired equations e

For the general T? point transformation”,

3

the appropriate Hermitian operator W is

since

£6

yields the desired relation between the q

s s and q’s 5 and the infonnation

V £

i ] ^

k

(q)

5 pk

The latter expression can also be written

3qk

( q) y pkTi7 + r

7

H

d \ r%

pv

+ # 7

=~ det

k

, 5 $ a , - B Qn/ V o q

,

% H

9

~t\ Pk ~ T log det

To construct (

q

5 jq n

) wp e order gW with respect to q.^ and q^.

9 V - £ & Pk

“ * pk<y n n + £q, vq)S q. p, - £ P, — - - 6

k

Fk ^ k rq^ ~ P X- (log det j s

^ £ ? q% \

thus obtaining the differential equation

B(q' |q

?! ) E^Pk(^ - Qj^q'OKq’ |q") + £ 6(o^ - qk

(q

,! ))|TMr (q ! k') M k

+ \

g (log det |«L) (q ’ |q") . 9 q

¥e infer the conditions of constraint on the transformation function

q£ - c^(qn )J

(q 5 |q

!! ) 0 3
-
and a simpler differential equation whose solution in terms of an

arbitrary function is fixed by the constraint equations as

In connection with this example 3 note the strict requirement that

the q

5 s and. q’s be uniquely connected by an everywhere non-singular

transformation , Should these conditions be violated,, the new variables

will not possess all the canonical attributes. We may then speak of

a quasi-canonicai transformation, A familiar example is the transformation from rectangular to spherical coordinates 9 where the angle 0 is

only defined mod 2 'Tf's and the determinant vanishes at r “ 0 and at

Thus £ spherical coordinates are quasi-canonieal

A simple dynamical illustration of a constrained transformation

2 * is provided by the one-dimensional system with H « p /2m ~ Fq5 described in terms of the transformation function (p 8 t|p"0) e The

equations of motion have the solution

so that there is a relation between the variables of the transformation function^ p and p^. Now

pJr « pSt-q + F t

£8

which requires no explicit ordering to write it as 8W a We thus obtain the differential equation

S(p’tj p"0) * £(p 8 - p

!! - Ft) (p 8 t|pH 0)

2 ,3

I

s Sr * i* TT TT

which is supplemented by the constraint condition

(pt «, pi? _ Ft) (p 3 tjp!! 0) « 0

The solution is

(p 5 t|p 1! 0) * S(p ! - p* J ~ Ft) e

F

2

t

3^)

" &(p 3 o

On placing F * 0 5 we obtain the transformation function for the system

with H « p

2

/2ms

. ,2 _

i PL t

(p t t|p,, 0) “ £(p ? - p

n

) e ^

Non-Unitary Transformations

Canonical transformations are representable as unitary transformations

qh

- u qh

U"

1

, Ph

- D Ph

U"

1

in virtue of the identical spectra of all canonical variables,, However 5 for the purpose of preserving the algebraic structure of the

£9

canonical commutation relations, and thereby the canonical equations

of motion, it is not necessary that U be a unitary operator. Of

course, other features of a canonical transformation Trri.ll be sacrificed.

An example is provided by the point transformation of the previous

section.. We have

For this canonical, non-unitary transformation

U -

and

A 1

Now

(q0 * U (q ? ) * Y (

q

? ) 5 q

s = q

(qi) - q(q>)

and

J

However

60

T(q’)

+

- -fiq’f U (/ xf (5>)

+ D'

1

)

so that the eigenvector orthonormality conditions read

( § (q')^(q") - *(q'

where

$ (S') - <?(q')

+ det ia

i?q'

Hence the dual and Hermitian adjoint eigenvectors are no longer the

This non-unitary transformation corresponds to the familiar procedure

of replacing one set of coordinates by another, without transforming

the eigenvectors „ The determinant of the transformation then enters

as a weight factor in all integrals and orthonormality statements.

Non- Hermitian canonical variables are useful in discussing the

harmonic oscillator. Thus

same. In virtue of the non-Hermitlan nature of p^5 it is the dual

eigenvector that satisfies

1

are canonical variables,

61

in terms of which this Hamiltonian can be written

H 1 » t (q» p j

• tiwfi (a, a

4 }

The canonical equations of motion.

dt

1 2M= * «.

« i u> a

are solved by

da 1 d H . + at in „ +

' da

a * a e o

“iuit + + iu)t a “ a e o

A convenient Hamilton- Jacobi transformation employs a^ and a as independent variables 9 Thus

^W^-i^afia^-i^a'Sa oo = H 6 t

whence

S y/ m - i )k S a

+ a . e

aiot . .j “ i M a 6 a e o

f o

vlo/t i/ f “ICUt 1\ r . “ # o)(a a, e + t

and

yfia , a . t)

0 , t + ”lU)t 1 Mf a a e

o

ijr Jz( co t + Const.

If we introduce eigenvectors of a and a

Q

, in a purely heuristic

manner, we can express the latter result as

(a

+I

tla"0) - 1"*)

-|wt

a

4 !"®-1"*

e e
*
62

choosing the multiplicative constant to be unity. In particular 5 for

t “ 0,

i- + » (a a”

+ 3 a a'

9 n'l *p'q'

*8 e

The transformation functions connecting th,e eigenvectors of a and

a with the eigenvectors of q can be obtained from the theory of the

general linear transformation. We find

(q 5

j

a

5 ) « C e

(a a ®

)

« C e

*y[ A q +a''

- c - 2

a

fTx q

3 a ’ ]

y[A q'^+a' ~2v^Txq,, a' ]

( a ,| qt) . c„ e^Uq'

V

2-2vT*r q'a']

(q> la*') - C" e

. t 9 + 8 0 „ *J[ A q +a -2VTK q» a ]

where "A » mu>/$.

Accordingly (q’ja*) is not equal to (a ! jq') 5 but rather can be

-f>

f 4* identified with (a jq

? ) 5 provided the eigenvalues of a and a are

complex numbers related by

(a 5 )*'"

.

The constant C® ® C" can then be fixed from the requirement

(a‘ |a") “ / (a* jq

5 ) d q

? (q> |a,! ) + 9

(2 ( 7f

i +

|0| Ijl ea a"
'
63

This is satisfied with

C s c' “

On the othel hand, note that

(a* |q ? )d q'(q ? la”

1 ")

does not exist

Infinitesimal Canonical Trans formations

An infinitesimal canonical transformation

% " \

* 5 \

p

k

“ Pk

- s

can be generated by a ¥ which differs Infinitesimally from the generator

of the identity transformation^

¥ - F(q ? pj t)

¥hether one writes q or q in the infinitesimal operator F is immaterial

for its value,, but is relevant in the derivation of the canonical transformation. Now

sw - £pk £qk

- X

-

3F(liy

+ 1 (® \

” s A " H st

X Pk 5 qk

- £ pk

8

5k - ( H - H) « t
-
6U

whence

8 \

m aF(gpt)

<?Pk

^F(qpt)

5qk

H(qpt) - fi('qpt) - ^ F(qpt)

characterize a general infinitesimal canonical transformation. We can

also write

Sqk

“ [qks F] 5 6 Pk “ ^ [Pk5 F]

which shows that F is the infinitesimal Hermitian generator of the

equivalent unitary transformation,,

The effect of the transformation on an arbitrary function G(qpt)

can be computed directly9

SG G(qpt)

G(qpt)

G(qpt)

G(q n

dV

il

<?q

t) a

or

, r, [ 3G n SC 3f\ 5 , 30 , E!/% ^

(G' F)

which defines the Poisson bracket of two operators. The notation is

2 F symbolic in that £—

} sa

y

5 occurs in definite places in the structure

^ pk

of G. We also have

S G “ -nv [G. F]
'v
6£

which expresses the Poisson bracket in terms of the commutator

(Gj, F) a ^

[G5 F]

From this connection it follows that

(G, F) - - (F, G) ,

although this is not quite evident from the definition.

We obtain from these results

H(qpt) ~ H(qpt) + (F 5 H)

- H(qpt) - F

or

H(qpt) * H(qpt) - F - (F, H)

- H(qpt) - ~~ ,

in virtue of the Poisson bracket form of the general equations of

motion. This implies that the generator of any transformation that

leaves the form of the Hamiltonian unchanged is a constant of the

motion.

Parameterized Transforma tions

Let us suppose that the infinitesimal, transformation is that associated with an infinitesimal change - d'^ of certain parameters t

so that F has the form

F 2F

(r)

dt

.

and

66

dq,k

d t « d, r \

Thus

¥ , „

0. Q k s dq

k Z F / N dt(r) r

and

8(q , T|q"T- dt) - \ (q 5 r U w

d -c |q" t'- dt)

A finite canonical transfomati on 5 (q 5 X jq

n Z ) f can now be character1 ^ ized by adding the generators of an infinite sequence of infinitesimal

transformations „

W,

k {Pk> d 1c} + l F

(r)

dt '

In particular,, with the single parameter T ® t 3 and F = - H, we regain the original action principle

„

We compute $

6 ¥12 r

l(S Pkdq^ Pk 6qP I ®F

(r)

dr

In order that a finite transformation be generated,, the coefficients

of the intermediate £

q^ and S p must be zero. This yields the

equations of motion
'
67

<iq v 2F

(r) dp, 3F f N _ k - . (r)

dr

r apk

3 d-r * qk

3

which repeat the original assertion that F^ dX is the generator of

the infinitesimal change dr in Hence

6 W

12

" F

1

” F

;

dF

"dt ~-±2±\6T dr

dr v

i r s

where

F s + Z F

( r) 5 r

3

The last term of S allows for the possibility that the transformation function may depend upon the integration path of the r variables

Now3 according to the significance of F ^

df/ s we have for any

operator G,

1$ L

~ G * F

(s) s

1 " 5

sG

dG 9 G , , ^

dt

g “ ns

I s

or

dG d G

dr

m + (F/ X. G)

In particular,

dF/ x d¥,

. -V1/- w „.i£l + (p , p . , ) dr gz

s u (s) 3 (r)

Hence

dF/ x 0F, % (n) (s) rs dT dT

n, x 9?, x

- ^ _ ^JLeI +. (f- , F/n) ns

( s )

5 (*•)'

68

is anti- symmetrical with respect to the indices r and s „ The change

in the transformation function produced by an alteration of the integr

tion path is thus given by

S(q ! r-j jq

!i Tg) “ g (q’^-L

K

TS (6V dt -6 r dr ) in'

' r> «=s s r5 ' I ^ *

2

>

The simplest possibility is A^ s Oj the transformation function

is independent of the integration path® Second in the hierarchy of

complications is A^_ ® a.^ _ (T ) , a numerical function e Here the transformation function depends upon the path only to the extent of a phase

constant which is independent of q® and q% etc 0 We shall be content

with the first situation - independence of path„ In particular if the

do not involve the parameters, they must satisfy

[F (r)

5 "(s' 1

Now suppose that the form a complete set of commuting operators

so that we may introduce the eigenvectors '^(F®x)o The transformation function (F’Z^ jF" t-,) is determined by

dr (F ' x

i

|F"V -|(F ’ T

ilF

(r)lF" T2> -gF^r'rpp.rp

„L X

in conjunction with the boundary condition

(F»t

? |F%) - g(F», F” ) ,

(assuming discrete eigenvalues) * Hence
*

.
69

-1 F ' t

(F'T., |F"r9 ) - r ” $(F ? S F" ) 9 r m x - X

But the canonical transformation function (qh, Jq” Z0 ) can be written

(q

! < |q'^

2

) -£ ( q' r

l l F ,Ti)

(

F ,r

i l F' ,r2

) (F" r

2

\

2

)

«2F

s E (q* i Ft ) * ~ (F 5 |q

n

) 5

or, with a notational change

(q 5 X, |q

n To) " C ^ e

' r-1 F

!

ySF ’ X Lr r r

Accordingly if one can construct the transformation function describ

ing the finite canonical transformation generated by the F

( r

)<> the

expansion of that transformation function in exponentials of the

will yield all the eigenvalues and eigenfunctions of the arbitrary

complete set of commuting operators.

We illustrate this with two transformation functions already ob

tained for a system of one degree of freedom and T “ t5 F = ~H 0

For the harmonic oscillator5

(a

+?

t|; *0)

fWt =iwt

T e" (a")

n

m36© VnX \/nT

so that the eigenvalues of the Hamiltonian are

En

- (n + k) Mw 9 h * 0 5 I,

" b ° °

70

and

(a+

' |n) -

(a+ 'F

(nla 8 ) “

»n

which satisfy

+ * ® in a T

The eigenfunctions (q ? In) “ ^

(q 5

1 8 ) can then be constructed from the n transformation function

/ \i mu) i2 i >2. / 2imo

. \ V -yi q -5“ 'V -’* q

' a

'*’> ' Ur? J e '

(q' n

which is, essentiallys the well-known generating function of the

Hermite polynomials 0

For the particle exposed to a constant force,, we found

(p 9 t|p5! 0) » & (p 5 - p*! - Ft) e

1 / >3 »3\

(p ~p ;

If one inserts the integral representation of the delta function,

»oo S(p* - p" - Ft)

dE |

|( P '-p"-Fir)

00

one obtains
J
71

p

00 _ ig

t

(p

5 tjp"0) * / (p 8 |E)dE e ^ (E jp"

)

oo

where

(p'|E) - (2tr e

for this problem the Hamiltonian has a spectrum ranging continuously

from -oo to oo 0 Hence H is a canonical variable,, In fact, with

p

- H

2m Fq

1 rfP *

we have

The transformation function (p® jp") can now be constructed from

SW““q6p+q6p® ~(p = l~) & p + ^ p b p

1

We get

(p» p) “ i (p P “ fe) + " onst

6m

and, writing p® a E<

(p ® ]E) * G e

:

But

(E jp

? ) dp f (p * jE 1 ) * jc

.00 i

! .

00

|El(E-E»)

* * ?

dp® - |C f2f 6(E - E®
'
72

whence

c - ( 2 irjt f)"2 .

Notice that the transformed function (p'jE) has a singularity at

F ” 0 5 corresponding to the fact that the Hamiltonian H E is not

a canora.cal variable e

Greenes Func tions

A general method for constructing the transformation function

(q ,/tjq"0) is based upon the differential equation

\-§r ( q " 1 1 q” 0 ) - (q^|F

(r) (q p) |q»0)

f^)(<1 ' r l q

'' 0)

in which the use of the differential operator F^

illustrative* integral operators can also occur

„

to be supplemented by the boundary condition

(q S i

is only

These equations are

(q?0|qI! 0) ® <S(q 8 = q"

)

In particular,

i % fi (q

B t|q"0) * H(q» s | j|T)(q t t|q,, 0) (q 8 0 |q"0)

(q’O jq"0) - S(q ? - q») .

Turning to the simpler situation of a single parameter,, we note

that the boundary condition can be incorporated into the differential

equations by defining the discontinuous Green’s function:
3
73

G(q” q% t) “ (q’t | q" 0) s t > 0

t < 0

Indeed,

i rf L „ H I a

i M JL~,

' 0t \ q 5 I 0q® G(q 5 q

ss t )

22 B(t) S' (q ! - q

!! )

and we now seek the solution of this inhomogeneous equation which

•vanishes for negative Ifs as we have tacitly assumed5 the Hamiltonian is time-independents the Green's function equation can be given

anothers convenient form in terms of the Fourier transform

,oo G(q’q% E) * j

dt e*

1 G(q ? ? q

n

s t)

“00

I m S > 0

namely

E - H(q* a f £?) j G(q 5 s q\ E) * S(q« - q")

3 q

:

¥e now desire a solution which9 as a f‘unction of the complex variable

E, is regular in the upper-half plane. Since

G(q'q% E) [ dte^l l q (q-) e (q.")*

J, ^ E’ff

9 E

! /» E'y!

jE't

f (q ? ) ^

(qn

) * e ! y

9 E* r

*

E - E

here in conjunction with the Hamiltonian forms a complete set,,

we see that the poles of G(q 5 q

,! E) as a function of E are the eigenvalues E" 5 and the residues yield the eigenfunctions.

7h

For the general problem of n parameters T 9 we define

Gtq'q-'t) - (|)

n

(q’ *|q" 0 ) , -T, > 0

0 any T

r < 0

Hence

" F

(l) iq ’> \ /q’

t . F G(q 1',T) - SXtp fi

) (q’T|q"0)|

h-1

T^=o

and finally

nT

r^l

" t 3

i dZr dr) l q 3 5 dq* )

G( q q'.l) - S(tT) S(q s - q”)

The Fourier transform

G(q 5 q% f) „ ^ 2 f t

die ^ r

r 1 G(q s q

n fC) , I m f <0

obeys

nTT

r

33!

, fr

“ F

(r)^ q s 2 £q

G(q ! q% f) - 6 (q ? - q")

and

G(q', q", f) “

tt T p

f- TT (fr“Fp

7*

The Asymptotic Spectrum

If the operations are polynomials in the p^, one can easily

construct the transformation function (q ? t' + d 'Tjp , 't) 0 The appropriate W is 1

W - 2 l|qk

(T + at) - , P

k

(r)j + s F

(r)

dt

r

+ 2 i{^(r)

* h

;r)}

- 2 i|qk

(t + dt) , p

k (t)} * 2 F(

r ) dr .

¥e compute SW and order it into & >“/ s which must be explicitly possible if the are polynomials in the p^ «, Thus,,

'W “ 2 q^Ct + dT) pk

(t) + S 3f r

(q(r + dT), p(t)) dt

r

and

n/2 q’oVZ^ (q» p»)dT ] P k k k r r

VH

r (q ! T + dtjp’t)

With the aid of this transformation function, one obtains

(q’T+ dtfqu t) ® / (q ? T + dt|p ’T)d p

5 (p 1 jq"

)

1

(2 UK);\n dp1 eJ[2(q'”q'

! ) k

P

k

+2?

r (q’p')dXr

]

A general application of this formula involves the computation of the

quantity yielding all the eigenvalues.
'

V
76

dq'(q 5 r i |q

5 T

2

) a / d q

! J (q« |F ' ) 3

1

' r 1 (F 9 |q ? )

e-a S F ? r nr r r F

’

in the limit of infinitesimal^ =* We get

dq*(q s r+ dr|q3

/C)

dq.dp. JPr(^')4t

r

(2ffjf£) n

£ 2 F’dr J r r r

If we write

F'*3< F 5 +dF 8 p{¥ ? )dF *

this result becomes

^2 F’dT

h r r F1

f(F - ) dF« e 2 F*dr

r r

Evidently for infinitesimal dX the sum is dominated by the dense 9 essentially continuous part of the spectrum and /

?(F I ) dF ! is the

number of states in the eigenvalue range dF 5 0 This can be expressed

by the familiar rule that there is one state per volume (2 irtf) n

of

phase space

December 19 <, 1952

THE NATIONAL BUREAU OF STANDARDS

Functions and Activities

The functions of the National Bureau of Standards are set forth in the Act

of Congress, March 3, 1901, as amended by Congress in Public Law 619, 1950.

These include the development and maintenance of the national standards of

measurement and the provision of means and methods for making measurements

consistent with these standards; the determination of physical constants and

properties of materials; the development of methods and instruments for testing

materials, devices, and structures; advisory services to Government Agencies on

scientific and technical problems; invention and development of devices to serve

special needs of the Government; and the development of standard practices,

codes, and specifications. The work includes basic and applied research, development, engineering, instrumentation, testing, evaluation, calibration services

and various consultation and information services. A major portion of the

Bureau’s work is performed for other Government Agencies, particularly the

Department of Defense and the Atomic Energy Commission. The scope of activities is suggested by the listing of divisions and sections on the inside of the

front cover. Reports and Publications

The results of the Bureau’s work take the form of either actual equipment and

devices or published papers and reports. Reports are issued to the sponsoring

agency of a particular project or program. Published papers appear either in the

Bureau’s own series of publications or in the journals of professional and scientific

societies. The Bureau itself publishes three monthly periodicals, available from

the Government Printing Office: The Journal of Research, which presents complete papers reporting technical investigations; the Technical News Bulletin, which

presents summary and preliminary reports on work in progress; and Basic Radio

Propagation Predictions, which provides data for determining the best frequencies

to use for radio communications throughout the world. There are also five series

of nonperiodical publications: The Applied Mathematics Series, Circulars, Handbooks, Building Materials and Structures Reports, and Miscellaneous Publications.

Information on the Bureau’s publications can be found in NBS Circular 460,

Publications of the National Bureau of Standards ($1.00). Information on cali- bration services and fees can be found in NBS Circular 483, Testing by the National

Bureau of Standards (25 cents). Both are available from the Government Printing Office. Inquiries regarding the Bureau’s reports and publications should be

addressed to the Office of Scientific Publications, National Bureau of Standards,

Washington 25, D. C.