= = o (r yj Quantum Mechanics and a Preliminary Investigation of theHydrogen Atom.

561

Quantum Mechanics and a Preliminary Investigation o f the Hydrogen Atom.

By P. A. M.

1851 E xhib ition Senior Research Student, St. Jo h n ’s College, Cambridge.

(Communicated by R. H. Fowler, F.R.S.—Received January 22, 1926.)

§ 1. The Algebraic Laws governing Dynamical Variables.

Although the classical electrodynam ic theory m eets w ith a considerable am ount of success in the description of m any atom ic phenomena, i t fails completely on certain fundam ental points. I t has long been th ought th a t the way out of this difficulty lies in the fact th a t there is one basic assum ption of the classical theory which is false, an d th a t if this assum ption were removed and replaced by som ething more general, the whole of atom ic theory would follow quite naturally. U ntil quite recently, however, one has had no idea of w hat this assum ption could be.

A recent paper by Heisenberg* provides the clue to the solution of this question, an d forms the basis of a new quan tum theory. According to Heisen­

berg, if x and y are two functions of the co-ordinates an d m om enta of a d y n a ­ mical system, th en in general xy is n o t equal to yx. In stead of the com­

m utative law of m ultiplication, the canonical variables p r (r l...u ) of a system of u degrees of freedom satisfy the quantum conditions, which were given by the a u th o rf in the form

M s — M r = 0

PrPs — TsV r = 0

qrps — psqr = o

qrPr — p/lr

ih

where iis a root of — 1 and hi s a real universal constant, equal to (2rc)_1 times the usual Planck’s constant. These equations are ju st sufficient to enable

one to calculate xy — yx when x and y are given functions of the p ’s and s, and are therefore capable of replacing the classical com m utative law of m u lti­

plication. They appear to be the sim plest assum ptions one could make which would give a workable theory.

* ‘ Zeits. f. Phys.,’ vol. 33, p. 879 (1925).

t

1

(r 7* s) y j

(i)

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The fact th a t the variables used for describing a dynamical system do not satisfy the comm utative law means, of course, th a t they are not numbers in the sense of the word previously used in m athematics. To distinguish the two kinds of numbers, we shall call the quantum variables q-numbers and the numbers of classical m athematics which satisfy the commutative law c-numbers, while the word num ber alone will be used to denote either a q-number or a

c-number. When xy = yx we shall say th a t x commutes with y.

A t present one can form no picture of w hat a q-number is like. One cannot say th a t one q-number is greater or less than another. All one knows about

q-numbers is th a t if zx and z2 are two q-numbers, or one q-number and one c-number, there exist the numbers zx + zxz2, z2zx, which will in general

be q-numbers b u t m ay be c-numbers. One knows nothing of the processes by which the numbers are formed except th a t they satisfy all the ordinary laws of algebra, excluding the com m utative law of multiplication, i.e.,

zx + z2 = z2 + zx, (zx - f z2) + Zz = z x + (z2 + z3),

(zxz2) z3 = zx {z2zs),

Z1 (Z2 + Z3) = Z1Z2 + Z1Z3> ( + = + *2*3>

and if

zxz2 = 0, either

zx = 0 or 0 ; b u t

ZXZ2 5^ Z2ZX,

in general, except when zx or z2 is a c-number. One may define further numbers, x say, by means of equations involving x and the z’s, such as = z, which defines z%,or xz = 1, which defines z~1. There m ay be more than one value of x satisfying such an equation, b u t this is not so for the equation xz = 1, since if xxz = 1 and x 2z — 1 then (xx — x 2) z = 0, which gives x x = x 2 provided Z 7^ 0.

A function / (z) of a q-number 2 cannot be defined in a manner analogous to the general definition of a function of a real c-number variable, but can be defined only by an algebraic relation connecting / (z) with (z). When this relation does not involve any q-number th a t does not commute with z and / (2), one can define df/dz without am biguity by the same algebraic relation

as when 2 is a c-number, e.gi f / (2) = zn, then df/dz where n is a c-number.

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In order to be able to get results comparable w ith experim ent from our theory, we m ust have some way of representing q-numbers by means of c-numbers, so th a t we can compare these c-num bers w ith experim ental values.

The representation m u st satisfy the condition th a t one can calculate the

c-numbers th a t represent x + y , x y , and yx when one is given the c-numbers

that represent x and y. If a q-num ber x is a function of the co-ordinates and momenta of a m ultiply periodic system , and if it is itself m ultiply periodic, then it will be shown th a t the aggregate of all its values for all values of the action variables of the system can be represented by a set of harm onic com­

ponents of the type x(nm). exp. i co ( n m )t, where and oi(nm) are c-numbers, each associated w ith two sets of values of the action variables denoted by the

labels n and m,and t is the time, also a c-number. This representation was taken as defining a q-num ber in the previous papers on the new theory.*

I t seems preferable though to take the above algebraic laws and the general conditions (1) as defining the properties of q-numbers, and to deduce from them th a t a q-num ber can be represented by c-num bers in this m anner when it has the necessary periodic properties. A q-num ber thus still has a m eaning and can be used in the analysis when i t is n o t m ultiply periodic, although there is a t present no way of representing it by c-num bers.

§ 2. The Poisson Bracket Expressions.

If x and y are two numbers, we define their Poisson bracket expression [x, y] by

xy — yx — ih [x, y]. (2)

I t has the following properties, which follow a t once from the definition and make it analogous to the Poisson bracket of classical mechanics.

(i) I t contains no reference to an y particular set of canonical variables.

(ii) I t satisfies the laws

l>i + x 2,y] = [aq, y] + [x2, y],

[xxx 2, y] = X! [x2, y] + [aq, ] aq, [a:, y] = — [y, x].

(iii) I t satisfies the identity

[[®> y l A + \[y, z], -I- [[z, x], y] 0.

* See particularly, Bom and Jordan, ‘Zeits. f. Phys.,’ vol. 34, p. 858 (1925). Also-

Born, Heisenberg and Jordan,

loc. cit.

Quantum Mechanics and Investigation o f Hydrogen Atom. 563

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(iv) The elem entary P .B .’s (Poisson brackets) are given, from (1), by [Pr, Vs\ — 0, [?r, Qs] = 6,

[qr, p s] = 0 (r s), or 1 (r = and also

[Pr, c] = [?r, c] == 0, when c is a c-number.

If a; and y are given functions of the p ’s and s, then, by successive applications of the laws (ii) the P.B. y\ can be expressed in terms of the elem entary P .B .’s occurring in (iv), and thus evaluated. I t is often more convenient to evaluate a P.B. in this way th an by the direct use of (2). For example, to evaluate [q2, p'2] we have

lq2,v2] = q [?> p 2] + [q, p 2] q, and

[q, p 2] = p [ q, p\ + [q, 2 p,

so th a t

[q2, f ] = 2qp + 2 pq.

One m ay greatly reduce the labour of evaluating P .B .’s of functions of the

p ’s and q’s in certain special cases by observing th a t the classical theory expression for the P.B. \pc, y\, namely ^ ^ may usually be

taken over directly into the quantum theory when this does not give rise to any am biguity concerning order of factors of products, e.g., we can say a t once th a t

I f (x), x] = 0,

when f ( x)does not involve any num ber th a t does not commute with x, and also

[f(qr),Pr]= df/dqr, (3)

when / (qT) does n o t involve any num ber th a t does not commute with qr.

The conditions th a t a set of variable Qr P f shall be canonical are defined to be th a t from the relations connecting the Qr, P r with the qr, pr (which are given to be canonical) one can deduce the equations

[Qr, Qs] = 0 , [Pf, P s] = 0 , [Qr, p j = 0 O’ s* s) or 1 s).

One could evaluate the P.B. of two functions of the Qr, P f, either by working entirely in the variables Qr, P f, or by first substituting for these variables in term s of the qT, p r. The relations connecting the Qf, Pr with the qr, pr t)e p u t in the form

Qr = bqj)-1, P r = bpfb"1,

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where bi s a q-num ber which determ ines the transform ation, b u t these formulae do not appear to be of great practical value.

A dynamical system is determ ined on the classical theory by a H am iltonian H. which is a certain function of the p ’s and s, an d the classical equations of motion m ay be w ritten

(4) We assume th a t the equations of m otion on the quantum theory are also of the form (4), -where the H am iltonian H is now a q-num ber, and is for the present an unknow n function of the p ’s an d q's. The representation of a q-number by c-numbers when it is m ultiply periodic m ust be such th a t if x is represented by the harm onic components x (non) exp. ico t, defined by (4) has the components fee (nm) x (nm) exp. uo t.

Quantum Mechanics and Investigation o f Hydrogen Atom. 565

§ 3. Some Elementary Algebraic Theorems.

In all previous descriptions of n atu ral phenomena the two roots of — 1 have always played sym m etrical p arts. The occurrence of a root of — 1 in the fundam ental equations (1) means th a t this is n o t so in the present theory.

For m athem atical convenience we shall continually be using in the analysis a root of — 1, j say, which is independent of the in (1), th a t is to say, from any equation one can obtain anoth er equation by w riting — for j w ithout a t the same tim e changing the sign of i. From these two equations one can obtain two more equations by reversing the order of the factors of all products occurring in them an d a t the same tim e w riting — h for since if this operation is applied to equations (1) i t will give correct results, so th a t i t m ust still give correct results when applied to any equation derivable from (1). To avoid

having two symbols iand j,both denoting roots of — 1, we shall take = i,

and m ust then m odify the above rules to read :—From any equation one m ay obtain another equation by "writing — i for i wherever it occurs and a t the same time writing — h for h, or by reversing the order of all factors and w riting

— h for h, or by applying the two previous operations together, which reduces to reversing the order of all factors a nd w riting — i for i. This th ird operation applied to any num ber gives w hat m ay be defined as the conjugate im aginary number. A num ber is defined to be real if it is equal to its conjugate imaginary.

The rem ainder of this section will be devoted to some simple analytical rules which will be of use in the subsequent work.

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When forming the reciprocal of a q u antity composed of two or more factors,

one m ust reverse their order, i.e.,

1 _ 1 1 (xy) * '

This equation m ay be verified by m ultiplying each side by xy either in front or behind.

To differentiate the reciprocal of a q u an tity x one m ust proceed as follows :_

I ( I )

Hence, dividing by x behind, one gets

dt ± H \ = - l , l .

The binomial expansion for (1 + x )n when n is a c-number is the same as in ordinary algebra. Also one defines ex by the same power series as in ordinary

algebra. The ordinary exponential law, however, is not valid, ex+v is no t in general equal to exev, except when x commutes with y.

If (<x.q) denotes (argv), where the a (r = 1 are e-numbers, then from (3)

[eiUq\pr\ = i x / (aq).

Hence, since

elUq)p r — = p r\,

we have

el (aQ)pr (pr — a e*{aQ).

More generally, if / ( qr, pr) is any function of the s and p ’s,

eiUa)f(q„ pr) = / (q„ pr — *rh) eiUq), 1 ^

f(qr> Pr) j(aq) = e* (aQ)f Pr, + <*#)• J To prove this result, we observe th a t if it is true for any two fu n c tio n s/,/!

and / 2, say, it m ust also be true for ( /, + / j ) and f 1 f 2. Now we have proved it true w h e n / = p r, and it is obviously true w h e n / = qr since the s commute with each other. Hence it is true when / i s any power series in the p ’s and q s so th a t we m ay take it to be generally true.

E quations (6) show the law of interchange of any function of the p ’s and s w ith a quantity of the form ei(aQ). They are of great value in the theory of m ultiply periodic systems. There are, of course, corresponding equations for any set of canonical variables, Qr, P r.

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Quantum Mechanics and Investigation o f Hydrogen Atom. 567

§ 4. M ultiply Periodic Systems.

A dynamical system is m ultiply periodic oil the quantum theory when there exists a set of utiiformising variables J r, having the following properties :—

(i) They are canonical variables, i.e.,

[Jr, J « ] = 0, [wr, ivs] = 0,

[w,., J s]

0

s), or 1 s).

(ii) The H am iltonian H is a function of the J ’s only.*

(iii) The original p ’s and q's th a t describe the system are m ultiply periodic functions of the w’s of period 2

, the condition for this being defined to be th at a po r qc an be expanded in either of the forms

2 aCa exp i (apiq + a 2w2 + --- + a = S aCa exp i (aw) or

S aexp i (apry + a 2w2 + ---+ a !{ C / = £ a exp i (oav) C'a.

C'a, where the Ca’s and C / ’s are functions of the J 's only and the a ’s are integers.

We have taken the w’s2

times as great an d the J ’s 1/2

tim es as great as the usual uniformising variables in order to save writing.

We have a t once

J r = [ J r, H] = 0 from (ii), and

wr = [wr,H j = 8H /0 J r,

using (3). The quantities wr are, therefore, constants and m ay be called the frequencies. There are, however, other quantities th a t have claims to be called frequencies. We have

x j i

_eitBW) H He1 <aw)

dt ~ l ’ J If,

From (6) applied to the J ’s and w’s,

, H a W )

H ( J r) = H (J,. - a ei(aW\

and Hence

H (Jr) eiUw) = e i{aW) H (Jr + ar/i)- (,i (o.w)

dt = i(aco) et(oW> = iel (aW) (aco)',

where

(y.oi)h H (Jr) — H (Jr — ar h), 1

(aco )'h = H (Jr + a ,^) — H (Jr), J (7)

* He is not necessarily the same function of the J ’s as on the classical theory with the present definition of the J ’s.

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The quantities wr correspond to the orbital frequencies on Bohr’s theory while the (aw) and (aw)' correspond, when the a ’s are integers, to the transition frequencies. I t m ust be remembered though th a t the wr, (aco) and (aw)' are q-numbers, and, therefore, they cannot be equated to Bohr’s frequencies, which are e-numbers. They are merely the same functions of the present J ’s, which are q-numbers, as B ohr’s frequencies are of his J ’s, which are c-numbers.

Suppose x can be expanded in the form

x = 'LaXo.ei iaW) = Z ae i (a,c) (S) where the a ’s are integers and the xa, are functions of the J ’s only.

Also

xa (J,*) Xa (J,* j <xrh).

x = X axa i(aw) ei(aW) S ae t(aW) (aw )'*,' From (6)

(9) If x and the J ’s are real and if xa denotes the conjugate imaginary of xa, then by equating the conjugate imaginaries of both sides of (8) we get

x = 2 ae - i(“w)£a (J,) = S a^a (Jr + a Comparing this with equation (8) we find th a t

x a (J r + a rh) = x _ a (J r).

This relation is brought o u t more clearly if we change the notation. For xa (Jr) write x (J, J — txh).

Then

x (J — (— < xh ,J) — x (J, J -j—

which shows th a t there is some kind of sym m etry in the way in which the

am plitude x( J , J — a li)is related to the two sets of variables to which it explicitly refers. Our expansion for x is now

* = E a x (J, J - a h )ei (“w> = 2 a e1 (aU,) x (J + a J).

The expressions (7) for the transition frequencies suggest th a t we should put (aw) (J) = co (J, J — a h),

and

(aw)' (J) = w (J + ah, J).

We should then have from (9)

* = Z z (J, J — ah) U, (J, J - a h) el“"> = i<a (J + ah, J) * (J + ah, J).

(10)

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Suppose y can also be expanded in the form

y = 'Lpy(J,J - W e****.

Then

xy = Zapx (J, J - a h) eiUw) (J, J - pA) ei(^>,

= Z ^ x (J, J - a h) . y ( J — aA, J - a - $h)

by again using (6), and the fact th a t the w’s commute ; or, the am plitudes of xy are given by

xy (J, J — rh) = S a x(J, J — a A) . (J — aA, J — y A ) . (11) These formulae provide a way of representing q-numbers by means of e-numbers. Suppose th a t in the expressions (J, J — a A) and to (J, J — a

considered merely as functions of the J ’s, we su b stitu te for each J r the c-num ber nr h, and denote the resulting c-numbers by x (n, n — a) an d co — a).

We m ay consider the aggregate of all the c-num bers x (n, n — a) exp. (n, n — a) t,in which it is sufficient (but n o t necessary) for the to take a series of values differing successively by u n ity , as representing the values of the q-number x for all values of the q-num bers J r. E q u atio n (10) shows th a t

x(n, n — a) = ice (n,n — a) x (n, n — a),

while equation (11) gives

xy (n, n — y) == 2 a x( n, n — a) — a, — y),

which is ju st Heisenberg’s law of m ultiplication. Also we have obviously

( x- j- y) (n, n — a) = x( n , n — a) + — a).

Our representation thus satisfies the conditions m entioned in §§ 1 and 2, which proves the sufficiency of this discrete set of .

One gets different representations of the q-numbers x by c-numbers x (n exp. ice ( n m )t by taking different values for the c-numbers, 7)r, say, by which the nr’s differ from integers. Only one of these representations, though, is of physical importance, this being the one (assumed to exist) for which, every x (nm) vanishes when an mr is less th an a certain value, nor, say, which fixes the normal state of the system on B ohr’s theory, and each nr Sr nor. This requires th a t every coefficient £ (J, J — cch) in the expansion of x shall vanish when for each J r is substituted the c-number {nor -J- where the mr are integers not less th an zero, a t least one of which is less than the corresponding ar.

Quantum Mechanics and Investigation o f Hydrogen Atom. 569

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§ 5. Orbital Motion in the Hydrogen Atom.

I t is necessary a t this point to make some assumption of the form of the H am iltonian for the hydrogen atom .* We m ay assume th a t it is the same function of the Cartesian co-ordinates x, y and their corresponding momenta

px> py as on the classical theory, i.e.,

H = 2

m

Pv^ ~{xz

y 2)* ’

where e and m are c-numbers.

We transform to polar co-ordinates r, G by means of the equations x = r cos 0, V — r sin G,

where cos G and sin G are defined in term s of eie by the same relations as on

the classical theory. The m omenta p r and k conjugate to and 0 are giveD by the equations

p r = h(px cos 0 + cos G px + 1 ( sin 6 + sin 0

k = x p y — ypx.

To verify th a t r, G, p r, k defined in this way are canonical variables, we m ust work out all their P .B .’s taken two a t a time. We have a t once that x, y, rand G commute with one another. Also

[r, Px] = [(.«2 + y2)*, Px] = %l(x2 + y2f = cos 0, w ith the help of (3), and sim ilarly

so th a t

and

[r,pv\ = sin 6,

[r, k ] = x [r, pv] — y [r, px],

= x sin 0 — y cos G = 0,

[r, pr] = i [r, px]cos G + | cos G [r, px] + | [r, p v] sin G + | sin 0 [r, pv],

= cos2 G -)- sin2 G = 1.

F urther

r \eie, k] = [reie,k] = + xpy —

= ixly, Py] — y [x,px] e'9,

so th a t

[eie, k] = i ei9.

* The hydrogen atom has been treated on the new mechanics by Pauli in a paper not yet published.

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Quantum Mechanics and Investigation o f Hydrogen Atom. 571

The remaining equations, [ew, pr\ = 0 and [A*, pT\ — 0, m ay be likewise verified by elem entary quantum algebra.

If we solve for px, p y in term s of pr, k, we find th a t Px + iP v = (Pr + i h M = eie (pr + i

Px — i P v = (Pr — ih/r) e~ie = (pr —

where

k

k

hf =— k

2 Ji, so th at

k2 e10

e1 0 kv

k2

by an application of (6). We thus have

f t 2 + iV = (ft — ♦ f t ) (Px + » f t ) = (Pr — » * l/r) (ft + * *h/»0>

( 12 )

— Pr2 i hi *■$ "S i h -L

1 11 —

Now

1 1 1 , v 1

P r - — ~Pr = - " =

r r r r 11 n* l &

Hence

2 i 2 9 i h

Px + Pv = Pr2 + /2 2 ,

= Pr“ + and

H = U ? ' + k t ) r ’

k\ h2 rz

(13) If we had originally assum ed th a t the H am iltonian was the same function of the polar variables as on the classical theory, we should have had instead

H = 2 ^ ' 2 + I ) - 7 - (13')

The only way to decide which of these assum ptions is correct is to work ou t the consequences of both and to see which agrees w ith experim ent.

The equations of m otion w ith either H am iltonian are

r — [r,H] = p rlm,

k = [k, H] = 0,

0 = [0, H] = k

which give pr — mr, k = constant and mr20 = k, as on the classical theory, and finally

b, b . *2

f t = [ f t H] = Z g - £ with (13) (14)

•— 3 _ ~‘> (13')

m r r “ (14')

2 Q VOL. CX.—A.

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We try to find an integral of the equations of motion of the form

1/r = «o + a\e%9 + a

e *■> (15^

where a0 , ax and a2 are constants, corresponding to the classical equation of elliptic motion

l/r — 1 + s cos (0 — ot) in which l is the latus rectum and s the excentricity.

The rate of change of is given w ith either H by

| e ‘» = [e« H ] = [e«, P ] ^ ,

= { k i>" k] + [«“, k] k}l'2mr,

= { hie16 + i2,

__ pi8 __ ^2

m r 2 By changing the sign of both i and h we find

p-ie - _ j_ „ __ _ -i0

dt m r2 m r 2

Hence if we differentiate (15) we get

— - r ~ = — la x etd — a2

r r m\] ^ or

l / r prr — — i (ax — a 2 e~l9kz),

which, using reduces to

pr — l / r . prr = ih/r,

Pr r (®i &x9k x u 2 e g) “I- rA/r,

= — i (ax el9kx — a2 e~l0k2) + f/i («o + + ^2 e_l0)>

= i («0A — aq + a2 e~i9k x) (16)

Now differentiate again. The result is

mpr = eskxk 2/r2 -j- « 2 e 2 / I \ _ Aq&2 a^kibi

- [ - a°) - 7 - ~ - ^ r - - 7 - ’

which agrees with the equation of motion (14) if one takes a0 — , but will not agree with (14').

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Quantum Mechanics and Investigation o f Hydrogen Atom. 573 We can easily obtain an integral of (14') by m aking a small change in (15).

We transform from the variables r, 0, pr, k to the variables r, 0', pr, , where

¥ = (k? +

' =

which are canonical since

[0', ¥] = [0, *'] £ and take

k + W )* k 1 Jr -f- -f- a2 Proceeding exactly as before, we find th a t

*;=i,

eie '__ g h ' d _

— e id' _ * C -id' O J m

where

and further th a t

dt~ m r

’dt

k x ' = k ’ + \h , = - \h,

_ k f k f a k2 a0k2

mpr r 3 r r 3 r 2

(15')

which agrees with (14') if we take a0 = me2/k2.

W ith the H am iltonian (13') the o rb it of the electron is thus an ellipse with a rotatin g apse line. If the Cartesian co-ordinates are now expanded in m ultiple Fourier series, two angle variables will be required, which will give two orbital frequencies. There would therefore necessarily be a two-fold infinity of energy- levels, which disagrees w ith experim ent (when one disregards the relativity fine-structure of the hydrogen spectrum). The assum ption of the H am iltonian (13') is thus untenable.

We therefore assume the H am iltonian (13), which does give a degenerate m otion, and proceed to evaluate the frequencies.

§ 6. Determination of the Constants of Integration.

The equation of the orbit is now given by (15), or

1 /r — m & \k f2 + ai e*9 +

e~l0> (17)

and from (16) pr = i (me2h jkx — + (18)

We m ust determ ine the form of the constants of integration a± and a2.

Since k commutes with r and jpr, it follows from (17) and (18) th a t it com­

mutes with («i eM -f- a2 e~t9) and (ax ex9¥ — a2 e~l9k 1). Hence k m ust commute with a1 ex0 and a2 e~t9 separately.

2 Q 2

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From (17) and (18) we find

- + *P’ = - r ? 2 K + «1 ^ « - " * ! - ^ + «1 <s“ fe - a 2 e -« fe,

= me2/fa + 2kai ete,

= me2/fa + 2ci eie, (19)

where ax = k~xcx. Multiplying this equation by eie in front and e~ie behind we get, since eief (fa) e~ie = f (fa),

k 2 /r + ipr = m + 2e^

Hence

Cle« _ = i b ^ z b _ i ( J L _ 1 \ = J A +

r U i r 2&

&2

Similarly, if a 2 = c2, it m ay be shown th a t

^

1 /e/

<?2

e — e C

— — IT — — me2h

(

20

(

21

(

22

so th a t

r

C1 ^ + c2 e_l61 = ^ C1 H~ c2-

Hence, as k commutes with cx eld and c2 e~id,

- = _r

_fafa + 4 («>e" + < _k % e_“) = r r + («"«i + c_"c _fafa

) I • _k < 23) We could, of course, have obtained directly from the equations of motion an integral of the form

1 jr = a0' + eie - f e~i9 a2 .

Equations (23) show the relations between the and the s. From (21) and (22) the following two additional forms for 1 jr are easily obtained :—

- = _{k A} + _ki r (ci e" + e'% ) = + T- («"<* + «a «■“)• ⁽²⁴⁾

The equations

k 2/r — ipr = me2jk -f- 2c2 e~M, (25)

and

kx\r — ipr = me2/k x -f- 2e_l0c2 (26)

may be obtained in the same way in which (19) and (20) were obtained. Multiply corresponding sides of equations (19) and (26), putting (19) first. The left- hand side of the result is

( b + iPr) ( h - i Pr)

= fa fair2 + p

= 2m (H -f- e2/r),

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Quantum Mechanics and Investigation o f Hydrogen Atom. 575

i

}

using (12) and (13), while the right-hand side is

m2e4 . 2 m e-, ie. _ ifl . , . 2me2 m2e4 , .

-r-g- + -v— (cie + e % ) + 4

!C2 = — — + 4clC2,

A/J_ A /] T t C \

using the first of equations (24). Hence

2mH = 40^2 — m2e f k f

Similarly, by m ultiplying corresponding sides of equations (20) and (25) taking (25) first, we find th at,

2mH = 4c2cx — m2e4/A\2 P u t

2mH = — m2e4/P 2.

P, of course, commutes with k, and c2. We then have 1 1 \

Cic2 = £ m V ( — — - 2) = \m 2ei

where

1

r (27)

c2ci = 4m2e4 ( , — '=5) = l-m2e4

=

* / V 1

\ A - ¥ -

The excentricities s x an d s 2 are constants, and commute with P and &

and with each other.

P u t

cx = ^me2 zxf kx. e-,x.

X is a constant and so commutes w ith P. Since k commutes with cxei9 and with Sj/Aq, i t m ust commute w ith e~ix eie, so th a t

k e_ *x el9 = e~ix el6k = e_*x (k — h) Hence

k e~% x = e-tx (k —

This law for the interchange of e~*x and k shows th a t y is canonically conjugate

to k. y corresponds on the classical theory to the angle between the major axis of the ellipse and the line 0 = 0. We now have

cx — \ me2 sJAq. e_ix = \m e2 s 2/&2, and from (27)

c2 = \m e

e% x zl/k1 = \m e2 s

/A

. e<x.

The expression (17) for 1 thus takes the form 1 me2 |

r kxh>

1 + | y- si e* + |

k

ie

}• (28)

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§ 7. Calculation of the Frequencies.

The easiest frequency to determine is the orbital one w, whose evaluation closely follows the classical calculation of the period. The relation between 6 and the angle variable w is of the form

6 = w + i:&B eHiw= w + 2 hn'e nie, where the 6’s are constants. On differentiating, this gives

0 = w + S ' 6 / - ( - enie ! •

m 2

Where S ' denotes th a t the term corresponding to = 0 is om itted from the summation. Multiplying both sides by r2 behind, we get

Or

= wr

+ S ' bn"

which gives, since mr20 — k,

r2 = —

\ b n" enie.

mw w

Hence if r2 is expanded as a Fourier series in 6 with each of the factors enie behind its respective coefficient, the constant term will be as on the classical theory.

From (28) we have r2 _ I me2

l&i k2

e <x J h S l ,,‘x

C rC

2

\ k

= {oc0 + ai e~ix eie + a 2 e** e~i0} - 2,

say. We can expand the right-hand side by the binomial theorem. This will

give a series of term s containing ei0’smixed up with a ’s, which cannot easily be evaluated. A more satisfactory way of proceeding is as follows :

I t can be shown th a t rn is equal to the expression obtained by expanding (a0 + e~ix + a 2 eix)~n in powers of elx, and inserting after each term of the form (},estx, where is independent of / , the requisite power of et0, namely e~sie. To prove this theorem we assume it to be true for some value of n, and

deduce from th a t, th a t i t is true for n + 1. Suppose for instance th a t and

Let then

(a0 + <x1 e~ix + <x2eix)~n = £(3S esix, rn = H$seaixe -Si0.

rn+1 = Z ysesixe~sie,

rn = Z r sesixersi0 - = 2ysesix - e“sW,

r r

(29) (30)

= X

/;"

(ao + oc^-^e" + z 2e'xe~“) er m.

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Comparing this with (30), we see th a t

fV% = Y / iXao + Y.+iec,+1)<xa1«- ^ + r s - i e (s~1)ix* f x ; but this is ju st the condition th a t

2 ( 3 / * = 2 y sesi* (a0 + Xie~ix + a2C*).

(Note th a t a term like Ys+ie(s+1)tx<x-ie~tx is equal to something independent of x multiplied by estx, owing to the special nature of the laws of interchange

of the a ’s with the e*x’s.) Hence from (29)

2 y / ' x = (a0 + ccxe~ + oc2eix)'

which proves the theorem.

Quantum Mechanics and Investigation o f Hydrogen Atom. 577

n—1

Our problem thus reduces to the determ ination of the term independent of

■/ in the expansion of (a0 + &xe~Xx -j- a 2e*x)-2 . To do this we first factorise the expression (a0 + v.xcT% x + a 2e*x). We have

(oco d- ocjfi *x

~\~ oc2eix) ^ ( 2 f + $1e-<x + £2| l e.x

Lkk\ \ k% k%

+ s1e - i* + e**e1^ - ± ^ ) , me2

2k h ki

1 + | + V l - | e - ‘x}

me2

2 kk - ' \ J 1 -j- — f l + — —---—eix\

V + P l h Z’2 V

/

t l + e - ^ ^ Y + r )

h

p * We m ust now express (a0 -j- a xe lx -f- a 2 etx) 1 in the form of partial fractions.

P utting for brevity (P + Ay)* = X1? (P + Ay)4 = X2, (P — Ay)4 = fjy, (P — Ay)4

= [x2, we have, remembering (5),

1 ' 1

(a o + a!e"ix + a 2eix)_1 = . — .

we2 Ai 1 d- ⁶x(j.2/Xi 1 -f- Ayp,2/AyX]Cx * Xj

2P 1 1 1 Jch

me

. e

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Now it is easily verified th a t

Hence

Jx _i_ h

^ X 2

( I . h m iX 2 h \ ^ jfeXi

(ao + a xe-ix + a 2eix)-1 = — , ^

me2 fci 1 + ki [xtjhi A i. eix Xi

P 1

me2 Xi eix + (x2/Xi ^ (31)

We m ust now square the expression on the right, which will give four terms, and m ust expand each of them by the binomial theorem an d take out the part independent of y . The term

f P Xi kk{

L Me2 1 d- k\ 0.2/k-j_ X] . eix Xi will contribute (Pk/me2)'z. The term

P 1

,W6Z Xj c'x -j~ fX2/ Xi

will contribute nothing, since ( elx + £Jt2/Xi)—1 = e~*x (l + p,2/ X i . e-ix)_1, which, when expanded, consists only of terms of the form e~nix with n > 0.

The remaining two term s m ay best be evaluated by using partial fractions again. I t is easily verified th a t

P kk\ P 1

7Yia“ k\

k\

2

p.2 k P 2 Xx

m2e4 k\

and

17

1 /

“ r + i ^ k j 1 ^

p 1

[L‘zk

p Xx

kk\

me1 Xx eix d- 42/ Xi me2 k\ 1 42/^2 Xx •

X

P 2 1 / _ Xx2ptx2 _______ 1_ , 1

w 2e4 Xx t 2 + k\ 1 d- ki \i-ijk-± Xx . eixd~ p^/Xx The first of these thus contributes

Xx X22 [J-21 kki 42) kki 1 J Xx

P 2 Xi _ m 1. 1

P 2*

m2e4 k\ 2 Xi

P 2 1 m2e4 Xx

X i V 2*x

2m2e4

M'x P2k Xi 2m2e4

(P —*2),

( P - *1).

and the second

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Quantum Mechanics and Investigation of Hydrogen Atom. 579

We therefore have for the term independent of y in the expansion of (

-j- <x1 e~% x + a 2elx)“ 2 the sum of the three contributions

P 2 A

n

2 m2e4 ( P — ^2) + P 2fc 2m2e4

P 3&

m2e4’

This is the value of kjmw. Hence

w — me4/P 3,

which happens to be the same function of P as on the classical theory. Now since H = — me4/2P 2, we have

w = o H /a p ,

which proves th a t P is canonically conjugate to w and is therefore an action variable. The tran sition frequencies are now given by

H (P + nh) — H (P) = me_4 fj_ _ _ J

h 2 I P 2 (P K }

The expansion of r is given by the expansion of the right-hand side of (31) in which the appropriate power of eie has been inserted behind each term . The coefficient of every term in i t vanishes when one puts P = 0, which makes each of the am plitudes x (nm) in the c-num ber representation of the Cartesian co-ordinates vanish when n or m is zero. This suggests, according to the

principles of § 4, th a t the state J = hi s the norm al state. (To prove this completely it would be necessary to show th a t x {nm) = 0 when n is a

negative integer and m a positive integer.) If this is so, one would have to put P equal to an integral m ultiple of h in (32), an d one would then obtain the observed frequencies of the hydrogen spectrum .

The author is greatly indebted to Mr. R. H. Fowler. F.R .S., for m uch valuable discussion and criticism of this paper.

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