674 MATHEMA TICS: N. H. McCOY PR.OC. N. A. S. ON THE FUNCTION IN QUANTUM MECHANICS WHICH CORRESPONDS TO A GIVEN FUNCTION IN CLASSICAL MECHANICS By NEAL H. MCCoy DEPARTMENT OF MATHEMATICS, SMrTH COLLEGE Communicated October 11, 1932 Let f(p,q) denote a function of the canonical variables p,q of classical mechanics. In making the transition to quantum mechanics, the variables p,q are represented by Hermitian operators P,Q which, satisfy the com- mutation rule PQ- Qp = 'y1 (1) where Y = h/2ri and 1 indicates the unit operator. From group theoretic considerations, Weyll has obtained the following general rule for carrying a function over from classical to quantum mechanics. Express f(p,q) as a Fourier integral, f(p,q) = f feiC(r+TQ) r(,r)d(rdT (2) (or in any other way as a linear combination of the functions eW(ffP+T)). Then the function F(P,Q) in quantum mechanics which corresponds to f(p,q) is given by F(P,Q)- ff ei(0P+7Q) t(cr,r)dodr. (3) It is the purpose of this note to obtain an explicit expression for F(P,Q), and although we confine our statements to the case in which f(p,q) is a polynomial, the results remain formally correct for infinite series. Any polynomial G(P,Q) may, by means of relation (1), be written in a form in which all of the Q-factors occur on the left in each term. This form of the function G(P,Q) will be denoted by GQ(P,Q). Let GQ(P,q) indicate the function of the commutative variables p,q obtained from GQ(P,Q) by replacing P,Q by p,q, respectively. In a similar manner Gp(P,Q) and Gp(p,q) may be defined. For example, if G(P,Q) = PQP, we find GQ(P,Q) = QP2 + yP, Gp(P,Q) = p2Q - -yP, GQ(P,q) = p2q + 'ypp Gp(p,q) = pq - yP. Our principal result may now be stated as follows. Let f(p,q) be a polynomial in the canonical variables p,q of classical mechanics, and F(P,Q) the corresponding function of the Hermitian operators P,Q in quantum me- chanics. Then

F0(p,q) = e2 apaq > f(p,q) = f(p,q) + 2 + '2+2!I.pa/- (24)pq2 + ... (4) Downloaded by guest on October 1, 2021 VOL. 18, 1932 MA THEMA TICS: N. H. McCOY 675

and Fp(p,q) = e 2a a f(p,q) = f(p,q) - 2 ?q +

x ayp2-)q2 _ . . .(5) In order to obtain FQ(P,Q) from FQ(p,q) we need only to write FQ(p,q) with the q-factors on the left in each term and then replace p,q by P,Q, re- spectively. It has been shown that2

eP+Q = eY/2 eO eP, from which it follows that

gTy ory ei(oP+TQ) = e- 2 eirQ e"P = e 2 eWTP e$ Q- Substituting in (3) we find °' FQ(P,Q) = JJe22 eitQQ e'°P (T)dadr and thus r r7-Y FQ(P,q) = ei(p+TQ) r(O,T)dOdT

=bJU i - z + (a.,)2 - T e (uP+i) r( r)dVdr. (6)

But from (2) we see that

- JJ'J')( _T) r(V,+TT)t(,)dodT, (n = 1,2 ..)

and formula (4) is obtained by substituting in (6). In a similar manner the formula (5) may be verified. Since e2 4taq is a linear operator it is sufficient to consider the case in whichf(p,q) is a single term of a polynomial. Let f'(p,q) = prqS, where r and s are arbitrary positive integers. Then from (4) and (5) we have

F (P,Q) = 2 k! (k) (k) qk pk (7) k = O and a Fp(P,Q) = s (- 2) k! P- Q9- (8) (k) (k) -k=O Downloaded by guest on October 1, 2021 676 MA THEMA TICS: N. H. McCOY PROC. N. A. S. the sum in each case being extended to the smaller of r and s. It is not difficult to verify directly by means of formula (10) below that FQ(P,Q) = Fp(P,Q), which we know must be true as these are different forms of the same function F'(P,Q). A more symmetrical form of F'(P,Q) may be obtained by eliminating 7y in (7) or (8) by means of relation (1). We shall now show that r S r F'(P,Q) = 1 Pr-, p= 1 E QS-, pr =0 '(9) I= O I= o By means of the known formula3 pn-k pnlQ = Zykk!(n) (k) Qm-k (10) k=z O we find

r r rl 1v ( r)p-I spi 1 Ir = ( k! ( V-k _p-k 2r I = 0 I = o k = o

(r-k) k r! (s Qs- pr-k 2r k =E=Eo 1 = o E3 (7)k! (k) (S) QS-k pr-k

k = O But by (7) this is FQ(P,Q). A similar calculation verifies the second part of (9). The Hermitian character of F'(P,Q) is now evident. It may be of interest to note how F'(P,Q) differs from the formula pro- posed by Born and Jordan in 1925.4 They suggested that prqs should go over into

r

r + 1 E Q

This is seen to differ from F'(P,Q) in case both r and s are greater than unity. 1 Weyl, The Theory of Groups and Quantum Mechanics, E. P. Dutton & Co., N. Y., 275 (1931). 2 Kermack and McCrea. Proc. Edinburgh Math. Soc., Ser. 2, 2, 224 (1931); McCoy, Ibid., 3, 121 (1932). 3 Born and Jordan, Zeits. Phys., 34, 873 (1925). 4Born and Jordan, loc. cit., 874. Downloaded by guest on October 1, 2021