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(solution) Explain, in your own words (without going into mathematical


Explain, in your own words (without going into mathematical details, although you may use the odd equation for illustrative purposes), the main features of the Heisenberg-Born-Jordan atom, as described by their new theory of Quantum Mechanics, using quotes from the papers of Born and Jordan and the analysis of that paper by Fedak and Prentis. The explanation should include: what are the new variables describing the states of the atom? What are their fundamental properties? How do they link to observables? The two papers are attached.

Also add references


1925 M. Born, Z. Phys. 34, 858 On Quantum Mechanics M. Born

 

Received 1925

 

?? ? ??

 

Translation into English: Sources of Quantum Mechanics,

 

Ed. by B. L. van der Waerden, North Holland, Amsterdam (1967) 277.

 

?? ? ?? The recently published theoretical approach of Heisenberg is here developed

 

into a systematic theory of quantum mechanics (in the first place for systems having one degree of freedom) with the aid of mathematical matrix methods. After

 

a brief survey of the latter, the mechanical equations of motion are derived from

 

a variational principle and it is shown that using Heisenberg?s quantum condition,

 

the principle of energy conservation and Bohr?s frequency condition follow from the

 

mechanical equations. Using the anharmonic oscillator as example, the question of

 

uniqueness of the solution and of the significance of the phases of the partial vibrations is raised. The paper concludes with an attempt to incorporate electromagnetic

 

field laws into the new theory. Introduction

 

The theoretical approach of Heisenberg 1 recently published in this Journal,

 

which aimed at setting up a new kinematical and mechanical formalism in

 

conformity with the basic requirements of quantum theory, appears to us

 

of considerable potential significance. It represents an attempt to render

 

1 W.Heisenberg, Zs. f. Phys. 33 (1925) 879. 1 justice to the new facts by selling up a new and really suitable conceptual

 

system instead of adapting the customary conceptions in a more or less artificial and forced manner. The physical reasoning which led Heisenberg

 

to this development has been so clearly described by him that any supplementary remarks appear superfluous. But, as he himself indicates, in its

 

formal, mathematical aspects his approach is but in its initial stages. His

 

hypotheses have been applied only to simple examples without being fully

 

carried through to a generalized theory. Having been in an advantageous

 

position to familiarize ourselves with his ideas throughout their formative

 

stages, we now strive (since his investigations have been concluded) to clarify the mathematically formal content of his approach and present some of

 

our results here. These indicate that it is in fact possible, starting with

 

the basic premises given by Heisenberg, to build up a closed mathematical theory of quantum mechanics which displays strikingly close analogies

 

with classical mechanics, but at the same time preserves the characteristic

 

features of quantum phenomena.

 

In this we at first confine ourselves, like Heisenberg, to systems having one degree of freedom and assume these to be ? from a classical

 

standpoint ? periodic. We shall in the continuation of this publication concern ourselves with the generalization of the mathematical theory to systems having ah arbitrary number of degrees of freedom, as also to aperiodic

 

motion. A noteworthy generalization of Heisenberg?s approach lies in our

 

confining ourselves neither to treatment of nonrelativistic mechanics nor to

 

calculations involving Cartesian systems of coordinates. The only restriction

 

which we impose upon the choice of coordinates is to base our considerations

 

upon libration coordinates, which in classical theory are periodic functions

 

of time. Admittedly, in some instances it might be more reasonable to

 

employ other coordinates: for example, in the case of a rotating body to

 

introduce the angle of rotation ?, which becomes a linear function of time.

 

Heisenberg also proceeded thus in his treatment of the rotator; however, it

 

remains undecided whether the approach applied there can be justified from

 

the standpoint of a consistent quantum mechanics.

 

The mathematical basis of Heisenberg?s treatment is the law of multiplication of quantum?theoretical quantities, which he derived from an

 

ingenious consideration of correspondence arguments. The development of

 

his formalism, which we give here, is based upon the fact that this rule of

 

multiplication is none other than the well?known mathematical rule of matrix multiplication. The infinite square array (with discrete or continuous

 

indices) which appears at the start of the next section, termed a matrix, is

 

a representation of a physical quantity which is given in classical theory as

 

2 a function of time. The mathematical method of treatment inherent in the

 

new quantum mechanics is thereby characterized through the employment

 

of matrix analysis in place of the usual number analysis.

 

Using this method, we have attempted to tackle some of the simplest

 

problems in mechanics and electrodynamics. A variational principle, derived from correspondence considerations, yields equations of motion for the

 

most general Hamilton function which are in closest analogy with the classical canonical equations. The quantum condition conjoined with one of the

 

relations which proceed from the equations of motion permits a simple matrix notation. With the aid of this, one can prove the general validity of the

 

law of conservation of energy and the Bohr frequency relation in the sense

 

conjectured by Heisenberg: this proof could not be carried through in its

 

entirety by him even for the simple examples which he considered. We shall

 

later return in more detail to one of these examples in order to derive a basis

 

for consideration of the part played by the phases of the partial vibrations in

 

the new theory. We show finally that the basic laws of the electromagnetic

 

field in a vacuum can readily be incorporated and we furnish substantiation

 

for the assumption made by Heisenberg that the squares of the absolute

 

values of the elements in a matrix representing the electrical moment of an

 

atom provide a measure for the transition probabilities. Chapter 1. Matrix Calculation

 

1. Elementary operations. Functions

 

We consider square infinite matrices, 2 which we shall denote by heavy

 

type to distinguish them from ordinary quantities which will throughout be

 

in light type, a(00) a(01) a(02) · · · a(10) a(11) a(12) · · · a = (a(nm)) = a(20) a(21) a(22) · · · .

 

···

 

···

 

···

 

2 Further details of matrix algebra can be found, e.g., in M. Bocher, Einf¨

 

uhrung in die

 


 

ohere Algebra (translated from the English by Hans Beck; Teubner, Leipzig, 1910) § 22?

 

25; also in R. Courant and D. Hilbert, Methoden der mathematischen Physik 1 (Springer,

 

Berlin, 1924) Chapter I. 3 Equality of two matrices is defined as equality of corresponding components:

 

a=b means a(nm) = b(nm). (1) Matrix addition is defined as addition of corresponding components:

 

a=b+c means a(nm) = b(nm) + c(nm). (2) Matrix multiplication is defined by the rule ?rows times columns?, familiar

 

from the theory of determinants:

 

a = bc means a(nm) = ?

 

X b(nk)c(km). (3) k=0 Powers are defined by repeated multiplication. The associative rule applies

 

to multiplication and the distributive rule to combined addition and multiplication:

 

(ab)c = a(bc);

 

(4)

 

a(b + c) = ab + ac. (5) However, the commutative rule does not hold for multiplication: it is not in

 

general correct to set ab = ba. If a and b do satisfy this relation, they are

 

said to commute.

 

The unit matrix defined by

 



 

?nm = 0 for n 6= m,

 

(6)

 

1 = (?nm ),

 

?nm = 1

 

has the property

 

a1 = 1a = a. (6a) The reciprocal matrix to a, namely a?1 , is defined by3

 

a?1 a = aa?1 = 1 (7) As mean value of a matrix a we denote that matrix whose diagonal elements

 

are the same as those of a whereas all other elements vanish:

 

?

 

a = (?nm a(nm)). (8) As is known, a?1 is uniquely defined by (7) for finite square matrices when the determinant A of the matrix a is non?zero. If A = 0 there is no matrix to a.

 

3 4 The sum of these diagonal elements will be termed the diagonal sum of the

 

matrix a and written as D(a), viz.

 

X

 

D(a) =

 

a(nm).

 

(9)

 

n From (3) it is easy to prove that if the diagonal sum of a product

 

y = x1 x2 · · · xm be finite, then it is unchanged by cyclic rearrangement of

 

the factors:

 

D(x1 x2 · · · xm ) = D(xr xr+1 · · · xm x1 x2 · · · xr?1 ). (10) Clearly, it suffices to establish the validity of this rule for two factors.

 

If the elements of the matrices a and b are functions of a parameter t,

 

then

 

X

 

d X

 

a(nk)b(km) =

 

{a(nk)b(km)

 

?

 

+ a(nk)b(km)},

 

dt n

 

k or from the definition (3): d

 

?

 

(ab) = ab

 

? + ab.

 

dt (11) Repeated application of (11)

 

d

 

(x1 x2 · · · xn ) = x? 1 x2 · · · xn + x1 x? 2 · · · xn + · · · + x1 x2 · · · x? n .

 

dt (110 ) From the definitions (2) and (3) we can define functions of matrices. To begin

 

with, we consider as the most general function of this type, f (x1 , x2 · · · xm ),

 

one which can formally be represented as a sum of a finite or infinite number

 

of products of powers of the arguments xk ; weighted by numerical coefficients. Through the equations

 

f1 (y1 , · · · yn ; x1 , · · · xn ) = 0,

 

............................

 

fn (y1 , · · · yn ; x1 , · · · xn ) = 0 (12) we can then also define functions yl (x1 , . . . xn ); namely, in order to obtain

 

functions yl ; having the above form and satisfying equation (12), the yl (

 

need only be set in form of a series in increasing power products of the xk and

 

the coefficients determined through substitution in (12). It can be seen that

 

one will always derive as many equations as there are unknowns. Naturally,

 

the number of equations and unknowns exceeds that which would ensue

 

5 from applying the method of undetermined coefficients in the normal type of

 

analysis incorporating commutative multiplication. In each of the equations

 

(12), upon substituting the series for the yl ; and gathering together like

 

terms one obtains not only a sum term C 0 x1 x2 but also a term C 00 x2 x1 and

 

thereby has to bring both C 0 and C? to vanish (e.g., not only C 0 + C?). This

 

is, however, made possible by the fact that in the expansion of each of the

 

yl , two terms x1 x2 and x2 x1 appear, with two available coefficients. 2. Symbolic differentiation

 

At this stage we have to examine in detail the process of differentiation

 

of a matrix function, which will later be employed frequently in calculation.

 

One should at the outset note that only in a few respects does this process

 

display similarity to that of differentiation in ordinary analysis. For example,

 

the rules for differentiation of a product or of a function of a function here

 

no longer apply in general. Only if all the matrices which occur commute

 

with one another can one apply all the rules of normal analysis to this

 

differentiation.

 

Suppose

 

s

 

Y

 

xlm = xl1 xl2 . . . xls .

 

(13)

 

y=

 

m=1 We define s

 

s

 

X

 

Y

 

?y

 

=

 

?lr k

 

?xk

 

r=1 m=r+1 xlm m=r?1

 

Y xlm , m=1  ?jk = 0 for

 

?kk = 1. j 6= k, (14) This rule may be expressed as follows: In the given product, one regards all factors as written out individually (e.g., not as x31 x22 , but as

 

x1 x1 x1 x2 x2 ); one then picks out any factor xk and builds the product

 

of all the factors which follow this and which precede (in this sequence).

 

The sum of all such expressions is the differential coefficient of the product

 

with respect to this xk .

 

The procedure may be illustrated by some examples:

 

y = xn , dy

 

= nxn?1

 

dx y = xn1 xm

 

2 , ?y

 

n?2 m

 

n?1

 

= x1n?1 xm

 

x2 x1 + · · · + xm

 

,

 

2 + x1

 

2 x1

 

?x1 y = x21 x2 x1 x3 , ?y

 

= x1 x2 x1 x3 + x2 x1 x3 x1 + x3 x21 x2 .

 

?x1

 

6 If we further stipulate that

 

?y1

 

?y2

 

?(y1 + y2 )

 

=

 

+

 

,

 

?xk

 

?xk

 

?xk (15) then the derivative ?y/?x is defined for the most general analytical functions

 

y.

 

With the above definitions, together with that of the diagonal sum (9),

 

there follows the relation

 

?D(y)

 

?y

 

(mn),

 

=

 

?xk (nm)

 

?xk (16) on the right?hand side of which stands the mn?component of the matrix

 

?y/?xk . This relation can also be used to define the derivative ?y/?xk . In

 

order to prove (16), it obviously suffices to consider a function y having the

 

form (13). From (14) and (3) it follows that

 

s

 

s

 

X

 

X Y

 

?y

 

?lr k

 

=

 

?xk (mn)

 

?

 

r=1 xlp (?p ?p+1 ) p=r+1 ?r+1 = m, r?1

 

Y xlp (?p ?p+1 ); (17) p=1 ?s+1 = ?1 , ?r = n. On the other hand, from (3) and (9) ensues

 

s

 

s

 

X

 

X r?1

 

Y

 

Y

 

?D(y)

 

?lr k

 

xlp (?p ?p+1 )

 

=

 

?xk (mn)

 

?

 

r=1 p=1 ?1 = ?s+1 , xlp (?p ?p+1 ); (170 ) p=r+1 ?r = n, ?r+1 = m. Comparison of (17) with (17?) yields (16).

 

We here pick out a fact which will later assume importance and which

 

can be deduced from the definition (14): the partial derivatives of a product

 

are invariant with respect to cyclic rearrangement of the factors. Because of

 

(16) this can also be inferred from (10).

 

To conclude this introductory section, some additional description is

 

devoted to functions g(pq) of the variables. For

 

y = ps q r (18) it follows from (14) that

 

s?1 r?1 ?y X s?1?l r l

 

p

 

q p,

 

=

 

?p ?y X r?1?j s j

 

q

 

p q .

 

=

 

?q

 

j=1 l?1 7 (180 ) The most general function g(pq) to be considered is to be represented

 

in accordance with § 1 by a linear aggregate of terms

 

k

 

Y z= (psj qrj ). (19) j=1 With the abbreviation

 

pl = k

 

Y (psj qrj ) l?1

 

Y (psj qrj ), (20) j=1 j=l+1 one can write the derivatives as

 

k sl ?1

 

?z = P P

 

psl ?1?m qrl pl pm ,

 

?p l=1 m=0

 

k rP

 

l ?1

 

P ?z =

 

?q l=1 m=0 qrl ?1?m pl psl qm . (21) From these equations we find an important consequence. We consider the

 

matrices

 

?z

 

?z

 

?z

 

?z

 

d1 = q

 

?

 

q, d2 = p

 

?

 

p.

 

(22)

 

?q ?q

 

?p ?p

 

From (21) we have

 

d1 = k

 

X (qrl Pl psl ? Pl psl qrl ), k

 

X (psl qrl Pl ? Pl psl qrl ). l=1 d2 = l=1 and thus it follows that d1 + d2 = k

 

X

 

l=1 (psl qrl Pl ? Pl psl qrl ). Herein the second member of each term cancels the first member of the

 

following, and the first and last member of the overall sum also cancel, so

 

that

 

(23)

 

d1 + d2 = 0.

 

8 Because of its linear character in z, this relation holds not only for expressions z having the form (19), but indeed for arbitrary analytical functions

 

g(pq).4

 

In concluding this brief survey of matrix analysis, we establish the following rule: Every matrix equation

 

F(x1 , x2 , . . . xr ) = 0

 

remains valid if in all the matrices xj one and the same permutation of all

 

rows and columns is undertaken. To this end, it suffices to show that for

 

two matrices a, b which thereby become transposed to a0 , b0 , the following

 

invariance conditions apply:

 

a0 + b0 = (a + b)0 , a0 b0 = (ab)0 , wherein the right?hand sides denote those matrices which are formed from

 

a + b and ab respectively by such an interchange.

 

We set forth this proof by replacing the procedure of permutation by

 

that of multiplication with a suitable matrix.5

 

We write a permutation as

 

 

 



 



 

n

 

0 1 2 3 ...

 

=

 

kn

 

k0 k1 k2 k3 . . .

 

and to this we assign a permutation matrix,

 



 

1 when m = kn

 

p = (p(nm)), p(nm) =

 

0 otherwise.

 

The transposed matrix to p is

 

p

 

? = (?

 

p(nm)),

 

4 p?(nm) =  1 when n = km

 

0 otherwise. More generally, for function of r variables, one has

 



 

X

 

?g

 

?g

 

xr

 

?

 

xr = 0.

 

?xr

 

?xr

 

r 5 The method of proof adopted here possesses the merit of revealing the close connection

 

of permutations with an important class of more general transformations of matrices. The

 

validity of the rule in question can however also be established directly on noting that in

 

the definitions of equality, as also of addition and multiplication of matrices, no use was

 

made of order relationships between the rows or the columns. 9 On multiplying the two together, one has

 

X

 

p?

 

p=(

 

p(nk)?

 

p(km)) = (?nm ) = 1,

 

k since the two factors p(nk) and p?(km) differ from zero simultaneously only

 

if k = kn = km , i.e., when n = m. Hence p? is reciprocal to p:

 

p

 

? = p?1 .

 

If now a be any given matrix, then

 

X

 

pa = (

 

p(nk)a(km)) = (a(kn , m))

 

k is a matrix which arises from the permutation

 

equivalently

 

ap?1 = ( X  n

 

kn  of the rows of a and a(mk)?

 

p(km)) = (a(n, km )) k is the matrix arising from permutation of the columns of a. One and the

 

same permutation applied both to the rows and the columns of a thus yields

 

the matrix

 

a0 = pap?1 .

 

Thence follows directly

 

a0 + b0 = p(a + b)p?1 = (a + b)0 ,

 

= (ab)0

 

a0 b0 = pabp?1

 

which proves our original contention.

 

It is thus apparent that from matrix equations one can never determine

 

any given sequence or order of rank of the matrix elements. Moreover, it

 

is evident that a much more general rule applies, namely that every matrix

 

equation is invariant with respect to transformations of the type

 

a0 = bab?1 ,

 

where b denotes an arbitrary matrix. We shall sec later that this does not

 

necessarily always apply to matrix differential equations. 10 Chapter 2. Dynamics

 

3. The basic laws

 

The dynamic system is to be described by (lie spatial coordinate q and

 

the momentum p, these being represented by matrices

 

q = (q(nm)e2?i?(nm)t , p(p(nm)e2?i?(nm)t ). (24) Here the ?(nm) denote the quantum-theoretical frequencies associated with

 

transitions between states described by the quantum numbers n and m. The

 

matrices (24) are to be Hermitian, e.g., on transposition of the matrices,

 

each element is to go over into its complex conjugate value, a condition

 

which should apply for all real t. We thus have

 

q(nm)q(mn) = |q(nm))|2 (25) ?(nm) = ??(mn). (26) and

 

If q be a Cartesian coordinate, then the expression (25) is a measure of the

 

probabilities6 of the transitions n ? m.

 

Further, we shall require that

 

?(jk) + ?(kl) + ?(lj) = 0. (27) This can be expressed together with (26) in the following manner: there

 

exist quantities Wn such that

 

h?(nm) = Wn ? Wm . (28) From this, with equations (2), (3), it follows that a function g(pq) invariably

 

again takes on the form

 

g = (g(nm)e2?i?(nm)t ) (29) and the matrix (g(nm)) therein results from identically the same process

 

applied to the matrices (q(nm)), (p(nm)) as was employed to find g from

 

q, p. For this reason we can henceforth abandon the representation (24) in

 

favour of the shorter notation

 

q = (q(nm)),

 

6 p = (p(nm)). In this connection see §8. 11 (30) For the time derivative of the matrix g = (g(nm)), recalling to mind (24)

 

or (29), we obtain the matrix

 

g? = 2?i(?(nm)g(nm)). (31) If ?(nm) 6= 0 when n 6= m, a condition which we wish to assume, then the

 

formula g? = 0 denotes that g is a diagonal matrix with g(nm) = ?nm (nn).

 

A matrix differential equation g? = a is invariant with respect to that

 

process in which the same permutation is carried out on rows and columns

 

of all the matrices and also upon the numbers Wn In order to realize this,

 

consider the diagonal matrix

 

W = (?nm Wn ).

 

Then

 

Wg = ( X ?nk Wn g(km)) = (Wn g(nm)), k gW = ( X g(nk)?km Wk ) = (Wm g(nm)), k i.e., according to (31),

 

g? = 2?i

 

2?i

 

((Wn ? Wm )g(nm)) =

 

(Wg ? gW).

 

h

 

h If now p be a permutation matrix, then the transform of W,

 

W0 = pWp?1 = (?nk m Wnk )

 

is the diagonal matrix with the permuted Wn along the diagonal. Thence

 

one has

 

2?i

 

pgp

 

? ?1 =

 

(W0 g0 ? g0 W0 ) = g? 0 ,

 

h

 

where g0 = pgp?1 and g? 0 denotes the time derivative of g0 constructed in

 

accordance with the rule (31) with permuted Wn .

 

The rows and columns of g? thus experience the same permutation as

 

those of g, and hence our contention is vindicated.

 

It is to be noted that a corresponding rule does not apply to arbitrary

 

transformations of the form a0 = bab?1 since for these W0 is no longer a

 

diagonal matrix. Despite this difficulty, a thorough study of these general

 

transformations would seem to be called for, since it offers promise of insight 12 into the deeper connections intrinsic to this new theory: we shall later revert

 

to this point.7

 

In the case of a Hamilton function having the form

 

1 2

 

p + U(q)

 

2m

 

we shall assume, as did Heisenberg, that the equations of motion are just of

 

the same form as in classical theory, so that using the notation of §2 we can

 

write: 1 p,

 

=

 

q? = ?H m

 

?p

 

(32) p? = ? ?H = ? ?U . ?q

 

?q

 

We now use correspondence considerations to try more generally to elucidate the equations of motion belonging to an arbitrary Hamilton function

 

H(pq). This is required from the standpoint of relativistic mechanics and in

 

particular for the treatment of electron motion under the influence of magnetic fields. For in this latter case, the function H cannot in a Cartesian

 

coordinate system any longer be represented by the sum of two functions of

 

which one depends only on the momenta and the other on the coordinates.

 

Classically, equations of motion can be derived from the action principle

 

H= Zt1 t0 Zt1

 

Ldt = {pq? ? H(pq)}dt = extremum. (33) t0 If we now envisage the Fourier expansion L substituted in (33) and the time

 

interval t1 ? t0 taken sufficiently large, we find that only the constant term

 

of L supplies a contribution to the integral. The form which the action

 

principle thence acquires suggests the following translation into quantum

 

mechanics:

 

P

 

L(kk) is to be made an extremum:

 

The diagonal sum D(L) =

 

k D(L) = D(pq? ? H(pq)) = extremum, (34) namely, by suitable choice of p and q, with ?(nm) kept fixed.

 

Thus, by setting the derivatives of D(L) with respect to the elements of

 

p and q equal to zero, one obtains the equations of motion

 

2?i?(nm)q(nm) =

 

7 ?D(H)

 

,

 

dp(nm) Cf. the continuation of this work, lo lie published forthwith. 13 2?i?(mn)p(mn) ?D(H)

 

.

 

?q(mn) From (26), (31) and (16) one observes that these equations of motion

 

can always be written in canonical form, q? = ?H , ?p

 

(35) p? = ? ?H .

 

?q For the quantization condition, Heisenberg employed a relation proposed

 

by Thomas8 and Kuhn.9 The equation

 

J= I Z1/?

 

pdq =

 

pqdt

 

?

 

0 of ?classical? quantum theory can, on introducing the Fourier expansions of

 

p and q,

 

?

 

?

 

X

 

X

 

p? e2?i?? t , q =

 

q? e2?i?? t ,

 

p=

 

? =?? ? =?? be transformed into 1 = 2?i ?

 

X ? ? =?? ?

 

(q? p?? ).

 

?J (36) If therein one lias p = mq,

 

? one can express the p? in terms of q? and

 

thence obtain that classical equation which on transformation into a difference equation according to the principle of correspondence yields the formula

 

of Thomas and Kuhn. Since here the assumption that p = mq? should be

 

avoided, we are obliged to translate equation (36) directly into a difference

 

equation.

 

The following expressions should correspond:

 

?

 

X ? =?? ? ?

 

(q? p?? )

 

?J with ?

 

1 X

 

q(n + ?, n)p(n, n + ? ) ? q(n, n ? ? )p(n ? ?, n));

 

h ? =??

 

8

 

9 W. Thomas, Naturwiss. 13 (1925) 627.

 

W. Kuhn, Zs. f. Phys. 33 (1925) 408. 14 where in the right-hand expression those q(nm), p(nm) which take on a

 

negative index are to be set equal to zero. In this way we obtain the quantization condition corresponding to (36) as

 

X

 

k (p(nk)q(kn) ? q(nk)p(kn)) = h

 

.

 

2?i (37) This is a system of infinitely many equations, namely one for each value

 

of n.

 

In particular, for p = mq? this yields

 

X ?(kn)|q(nk)|2 = k h

 

,

 

8? 2 m which, as may easily be verified, agrees with Heisenberg?s form of the quantization condition, or with the Thomas-Kuhn equation. The formula (37)

 

has to be regarded as the appropriate generalization of this equation.

 

Incidentally one sees from (37) that the diagonal sum D(pq) necessarily becomes infinite. For otherwise one would have D(pq) ? D(qp) = 0

 

from whereas (37) leads to D(pq) ? D(qp) = ?. Thus the matrices under

 

consideration arc never finite.10 4. Consequences. Energy-conservation and frequency laws

 

The content of the preceding paragraphs furnishes the basic rules of

 

the new quantum mechanics in their entirety. All other laws of quantum

 

mechanics, whose general validity is to be verified, must be derivable from

 

these basic tenets. As instances of such laws to be proved, the law of energy

 

conservation and the Bohr frequency condition primarily enter into consideration. The law of conservation of energy states that if H be the energy,

 

? = 0, or that H is a diagonal matrix. The diagonal elements H(nn)

 

then H

 

of H are interpreted, according to Heisenberg, as the energies of the various

 

states of the system and the Bohr frequency condition requires that

 

h?(nm) = H(nn) ? H(mm),

 

or

 

Wn = H(nn) + const.

 

10

 

Further, they do not belong to the class of ?bounded? infinite matrices hitherto almost

 

exclusively investigated by mathematicians. 15 We consider the quantity

 

d = pq ? qp.

 

From (11), (35) one finds

 

?H ?H

 

?H ?H

 

d? = pq

 

? + pq? ? qp

 

? ? qp? = q

 

?

 

q+p

 

?

 

p.

 

?q

 

?q

 

?p

 

?p

 

Thus from (22), (23) it follows that d? = 0 and d is a diagonal matrix. The

 

diagonal elements of d are, however, specified just by the quantum condition

 

(27). Summarizing, we obtain the equation

 

pq ? qp = h

 

1,

 

2?i (38) on introducing the unit matrix 1 defined by (6). We term the equation (38)

 

the ?stronger quantum condition? and base all further conclusions upon it.

 

From the form of this equation, we deduce the following: If an equation

 

(A) be derived from (38), then (A) remains valid if p be replaced by q and

 

simultaneously h by ?h. For this reason one need for instance derive only

 

one of the following two equations from (38), which can readily be performed

 

by induction

 

h n?1

 

p

 

pn q = qpn + n

 

,

 

(39)

 

2?i

 

h n?1

 

.

 

(390 )

 

qn p = pqn ? n

 

q

 

2?i

 

We shall now prove the energy-conservation and frequency laws, as expressed above, in the first instance for the case

 

H = H1 (p) + H2 (q).

 

From the statements of §1, it follows that we may formally replace H1 (p)

 

and bf H2 (q) by power expansions

 

X

 

X

 

as ps , H 2 =

 

bs qs .

 

H1 =

 

s s Formulae (39) and (39?) indicate that

 

h ?H ,

 

Hq ? qH = 2?i

 

?p h ?H . Hp ? pH = ? 2?i

 

?p

 

16 (40) Comparison with the equations of motion (35) yields q? = 2?i (Hq ? qH). h p? = 2?i (Hp ? pH). h H for brevity, one has

 

Denoting the matrix Hg ? gH by g H H H = b + a ; ab a b from which generally for g = g(pq) one may conclude that 2?i H 2?i

 

g? =

 

(Hg ? gH). =...

 


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