V. Fock
Zs. Phys.75, 662, 1932 Fock57, pp. 25–52
In principle, it is known that the method of the quantized wave function is equivalent to the method of the usual wave function in the configu- ration space. However, the close connection between both methods was not observed in a proper way. In the proposed paper, this connection is traced in detail. It appears that it is close to such a degree that on each step the calculation with the quantized wave function admits the direct transition to the configuration space.
The paper consists of two parts. The first one is of introductory character and contains the derivation and comparison of the known re- sults. Here the transition from the configuration space to the second quantization one is considered both for Bose and Fermi statistics, the uniqueness of the definition of the order of noncommutative factors be- ing especially stressed. The starting point of the second part is the commutation relations between the quantum wave functions (operators Ψ). It is demonstrated that these relations are satisfied with some oper- ators acting on the sequences of the usual wave functions of 1,2, . . . , n particles. Thereby, the representation of operators Ψ in the configuration space is obtained (more exactly, in sequences of configuration spaces).
Further, the dependence of the operators Ψ on time is considered and the form of the operator ˙Ψ = ∂Ψ/∂t is defined. With the help of the representation obtained, it is shown that the Schr¨odinger equation for the operator Ψ containing the time derivative can be written as a set of usual Schr¨odinger equations for 1,2, . . . , n particles. As an applica- tion of the representation obtained, the simple derivation of the Hartree equation with exchange is given.
1This paper was reported to a theoretical seminar at Leningrad State University in January 1931.
I Transition From the Configuration Space to Second Quantization2
Let us denote asxrthe set of variables of anr-th particle (for example, the coordinates and spin of an electron,xr= (xr, yr, zr, σr)) and consider the wave function
ψ(x1x2. . . xn;t), (1) which describes the set ofnidentical particles in the configuration space.
For convenience, let us pass by canonical transformation from the initial variablexto a new variableE, taking only discrete values
E=E(1), E(2), . . . E(r), . . . . (2) By quantities (2), one can mean the eigenvalues of the operators with a discrete spectrum. If one denotes the corresponding eigenfunctions as
ψr(x) =ψr(E(r), x), (3) then the transformed wave function
c(E1, E2, . . . , En;t) (4) is related to initial wave function (1) as
ψ(x1x2, . . . , xn;t) =
= X
E1,...,En
c(E1, E2, . . . , En;t)ψ(E1;x1). . . ψ(En;xn), (5) where each summation variableE1, E2, . . . , En runs all the values (2).
The Schr¨odinger equation in the configuration space will be written as follows:3
Hψ(x1. . . xn;t)−i~∂ψ
∂t = 0. (6)
Let the energy operatorH have the form H=
Xn
k=1
H(xk) + Xn
k<l=1
G(xk;xl). (7)
2The reader familiar with the theory of second quantization can skip the first part and begin reading from the second part at once. (V. Fock)
3Here~denotes the Planck constant divided by 2π. (V. Fock)
Here the usual sum gives the energy of separate particles and the double sum gives their interaction energy. For Coulomb forces
G(x, x0) = e2
|r−r0|. (8) In order to get the Schr¨odinger equation for transformed wave function (4) one should substitute series (5) into equation (6), decompose the result over products
ψ(E1;x1). . . ψ(En;xn)
of functions (3) and put the coefficients of each product zero. In this way, we obtain
Xn
k=1
X
W
hEk |H |Wic(E1. . . Ek−1W Ek+1. . . En;t)+
+ Xn
k≤l=1
X
W W0
hEkEl|G|W W0ic(E1. . . Ek−1W Ek+1. . . (9)
. . . El−1W0El+1. . . En;t)−i~∂
∂tc(E1E2. . . En;t) = 0 where the following notations are introduced for the matrix elements
hE|H|Wi= Z
ψ(E;x)H(x)ψ(W;x)dx, (10) hEE0 |G|W W0i=
= Z Z
ψ(E;x)ψ(E0;x0)G(x;x0)ψ(W;x)ψ(W0;x0)dx dx0. (10∗) Let the arguments
E1, E2, . . . , Ek, . . . En
of the wave function c in equation (9) be equal, respectively, to the eigenvalues
E(r1), E(r2), . . . E(rk), . . . E(rn). If we write for brevity
hr|H |si instead of hE(r)|H |E(s)i, hrt|H |sui instead of hE(r)E(t)|H |E(s)E(u)i, c(r1, r2. . . rn;t) instead of c(E(r1)E(r2). . . E(rn);t),
then wave equation (9) will take the form X
r
Xn
k=1
hrk|H |ric(r1. . . rk−1rrk+1. . . rn;t)+
+X
rs
Xn
k<l=1
hrkrl|G|rsic(r1. . . rk−1rrk+1. . . rl−1srl+1. . . rn;t)−
−i~∂
∂tc(r1. . . rn;t) = 0. (9∗) Until now we did not take into account the symmetry properties of the wave function and, therefore, the kind of statistics. But the wave func- tion (bothψandc) is either symmetric (Bose statistics) or antisymmetric (Fermi statistics). In the case of the symmetric wave function, the value c(r1, r2, . . . rn;t) is defined by the set of numbers
n1, n2, . . . nr, (11) which indicate how many times the corresponding arguments 1,2, . . . , r orE(1), E(2), . . . E(r) occur inc. Therefore, we can put
c(r1r2. . . rn;t) =c∗(n1n2. . .;t). (12) Now, the set of valuesr1r2. . . rn (defined independently of their order) corresponds to each series of numbers (11). For example, forn= 3 we havec(4,4,5) =c(4,5,4) =c(5,4,4) =c∗(0,0,0,2,1,0,0. . .).
In the normalization condition X
r1,...rn
|c(r1r2. . . rn;t)|2= 1 (13) one can make the first summation over all permutations of a given set of valuesr1, r2. . . rn and then over different sets
X
(r1,...rn)
X
Perm
|c(r1r2. . . rn;t)|2= 1.
The sumP
Permcontains n n!
1!n2!... equal terms; consequently, we have X
(r1,...rn)
n!
n1!n2!. . .|c(r1r2. . . rn;t)|2= 1
or, introducing quantitiesnras variables according to (12), X
n1n2...
n!
n1!n2!. . .|c∗(n1n2. . .;t)|2= 1. (14) In normalization condition (14), one can bring the weight function
n!
n1!n2!... to unity with the substitution c∗(n1n2. . .;t) =
rn1!n2!. . .
n! f(n1n2. . .;t). (15) For a new wave functionf, the normalization condition takes the form
X
n1n2...
|f(n1n2. . .;t)|2= 1. (16) In the case of Fermi statistics, it is yet not enough to fix numbers nr for the unique definition of c(r1r2. . . rn;t) since the quantity c is defined by them up to a sign. However, we can also keep equations (12) and (15) for Fermi statistics if we impose an additional condition that in this case the arguments inc(r1r2. . . rn;t) form the “natural” sequence, for instance:
r1< r2< r3. . . < rn.
If the sequence of arguments is obtained from the natural one by even permutation then equation (12) remains unchanged. For odd permuta- tion, its sign should be changed, for example:
c(1,4,5) =−c(4,1,5) =c∗(1,0,0,1,1,0,0. . . ).
In what follows, Bose and Fermi statistics will be treated separately.
Bose Statistics
In the case of Bose statistics, few equal arguments can occur in the wave functionc(r1r2. . . rn;t), for example:
c=c(u, u, u, v, v, w, . . . ).
Therefore, in the first sum in expression (9∗), functions can appear that differ by the order of arguments only, namely,nuitems can occur, whose argumentrstays on the placeu,nv items withron the placev and so
on. If we collect the identical items, then for the first sum in (9∗) we obtain the expression
X
r
hu|H |rinuc(r, u, u, v, v, w, . . . )+
+X
r
hv|H |rinvc(u, u, u, r, v, w, . . . ) +. . . .
Introducing quantitiesnk as variables according to (12), we obtain X
r
hu|H|rinuc∗(. . . nu−1, . . . nr+ 1, . . . )+
+X
r
hv|H |rinvc∗(. . . nv−1, . . . nr+ 1, . . . ) +. . . or simpler
X
p
X
r
hp|H |rinpc∗(. . . np−1, . . . nr+ 1, . . . ), (17) where the index pcan run now all the values (but not only valuesp= u, v, w, . . . ), since extra terms vanish due to factor np. Herewith, for r=p the quantity c∗(. . . np−1, . . . nr+ 1, . . . ) should be understood simply asc∗(. . . nr. . . ).
Similarly, one can also transform the second sum in expression (9∗).
Taking into account the number of identical items, we obtain:
X
r,s
{huu|G|rsi1
2nu(nu−1)c(r, s, u, v, v, w, . . . )+
+huv|G|rsinunvc(r, u, u, s, v, w, . . . )+
hvv|G|rsi1
2nv(nv−1)c(u, u, u, r, s, w, . . . ) +. . . }.
Introducing the quantitiesc∗(n1n2. . . ), we can write X
r,s
{huu|G|rsi1
2nu(nu−1)c∗(. . . nu−2, . . . nr+ 1, . . . ns+ 1, . . . )+
+huv|G|rsinunvc∗(. . . nn−1, . . . nv−1, . . . nr+ 1, . . . ns+ 1, . . . )+
+hvv|G|rsi1
2nv(nv−1)c∗(. . . nv−2, . . . nr+ 1, . . . ns+ 1, . . . ) +. . . }
or simpler
1 2
X
p,q
X
r,s
hpq|G|rsinp(nq−δpq)ã
ãc∗(. . . np−1, . . . nq−1, . . . nr+ 1, . . . ns+ 1, . . . ). (18) Here the summation indicespandqcan also run all the values without exception (but not only p, q =u, v, w). The factor 12 should stay with allthe items, since the combinationp=u, q=v is met in (18), e.g., as well as the combinationp=v, q=u. The meaning of the terms in (18), in which two or more numbersp, q, r, scoincide, is clear by itself.
With the help of (17) and (18), equation (9∗) can be written as X
p
X
r
hp|H |rinpc∗(. . . np−1, . . . nr+ 1, . . . )+
+1 2
X
p,q
X
r,s
hpq|G|rsinp(nq−δpq)ã
ãc∗(. . . np−1, . . . nq−1, . . . nr+ 1, . . . ns+ 1, . . . )−
−i~∂c∗(n1n2. . .;t)
∂t = 0.
(19)
Henceforth, it is expedient to introduce the operatorUrwhich trans- forms the functionf(n1, n2, . . . ) into the function
Urf(n1n2. . . nr, . . . ) =f(n1n2. . . nr+ 1, . . . ). (20) The quantityUrconsidered as a matrix with respect to the variable nr
and its conjugate matrixUr† are of the form
Ur=
0 1 0 0 . . . 0 0 1 0 . . . 0 0 0 1 . . . 0 0 0 0 . . . . . . .
, Ur†=
0 0 0 0 . . . 1 0 0 0 . . . 0 1 0 0 . . . 0 0 1 0 . . . . . . .
. (21)
Consequently, the conjugate operatorU†transforms the functionf(n1n2
. . . nr. . . ) intof00, where f00(n1n2. . . nr. . . ) =f(n1n2. . . nr−1. . . ) in the case ofnr6= 0 andf00= 0 at nr= 0. Thus,
Ur†f(n1n2. . . nr. . . ) =
ẵf(n1n2. . . nr−1, . . . ) for nr6= 0,
0 for nr= 0. (22)