M. Born and V. Fock
G¨ottingen
Received 1 August 1928
Zs. Phys.51, 165, 1928
In old quantum mechanics the adiabatic theorem established by Ehren- fest had the meaning that the quantized action variables J = n~ are invariant relative to an infinitely slow (adiabatic) change of a mechan- ical system.1 From this one can guess that if the system is in a state with definite quantum numbers before the adiabatic change started, then after the changes it will be characterized by the same quantum numbers.
The adiabatic theorem in new quantum mechanics has analogous meaning. If we enumerate the states of a system with numbers corre- sponding to the energy levels, the adiabatic theorem holds claiming that if the system was initially in a state with a definite number, then after the adiabatic change the transition probability of the system to be in the state with a different number would be infinitely small, although the energy levels after the adiabatic interaction could differ from the initial levels by finite values.
The adiabatic theorem was transferred into new quantum mechanics by one of the authors2 already in 1926. Nevertheless, the proof given there and also the proof by Fermi and Persico3are mathematically not indisputable. Both proofs consider only the case when at an adiabatic change none of the frequencies vanish, i.e., no degeneration takes place.
So there was no generalization, which was already performed for old quantum mechanics by M. von Laue.4
In this paper we try to give a proof which is more general and more satisfactory from the mathematical standpoint.
1The literature about the adiabatic theorem in old quantum mechanics is given at the end of this paper. (M. Born and V. Fock)
2M. Born,Das Adiabatensatz in der Quantenmechanik, Zs. Phys. 40, 167, 1926.
3E. Fermi and F. Persico,Il prinzipio delle adiabatiche e la nozione de forza vivo nella nuova meccanica ondulatoria. Lincei Rend. (6)4, 452–457, 1926.
4M. v. Laue, Ann. Phys. 76, 619, 1925.
§1. We consider a mechanical system with the energy operator con- taining time explicitly and changing with time slowly. We express this slowness in a way that the time dependence will be
H =H(s); s= t
T , (1)
whereT is a large parameter with the dimension of time, and the deriva- tives bysof the coefficients in the operatorH and in its eigenfunctions must be finite.
The eigenfunctions of the operatorH(s) are ϕ1(q, s), ϕ2(q, s), . . . . They satisfy the equations
H(s)ϕn(q, s) =Wn(s)ϕn(q, s). (2) Because this equation contains time (as a parameter), we can multiply the eigenfunctions by arbitrary phase factors containing time. We can fix these phases by the condition that each eigenfunction is orthogonal to its time derivative
Z
ϕ∗n(q, s) ∂ϕn
∂s %(q)dq= 0, (3)
(%(q) is the density function in q-space). Then the eigenfunctions are defined up to a constant phase factor.
Together with the eigenfunctions of the energy operator let us con- sider a system of functions
ψ1(q, t), ψ2(q, t), . . . , which satisfy the Schr¨odinger equation
H ψn+~ i
∂ψn
∂t = 0 (4)
and coincide withϕn att= 0:
ψn(q,0) =ϕn(q,0). (5) The systems of functions ψn(q, t) are normalized and orthogonal for eacht.5
5V. Fock, Uber die Beziehung zwischen den Integralen der quantenmechanis-¨ chen Bewegungsgleichungen und der Schr¨odingerschen Wellengleichung, Addenda, Zs. Phys.49, 323, 1928. (See also [28-2] in this book. (Editors))
§2. Now we go over from eigenfunctions to matrices. We can con- struct the matrices using three complete sets of eigenfunctions:
ψn(q,0) =ϕn(q,0), (a) or
ϕn(q, s), (b)
or
ψn(q, s). (c)
We indicate the matrices formed by the set (a) by the superscript 0, those formed by the set (b) by the superscriptsand those formed by the set (c) by no superscript. For instance, we write
Hmn0 = Z
ϕ∗m(q,0)H(0)ϕn(q,0)% dq , (6a)
Wn δnm=Hmns = Z
ϕ∗m(q, s)H(s)ϕn(q, s)% dq , (6b) Hmn=
Z
ψm∗(q, t)H(s)ψn(q, t)% dq . (6c) The representation (a) has the property that time-independent matrices will correspond to constant (explicitly independent on time) operators;
in the representation (b) the energy matrix has the diagonal form and in the representation (c) the equations of motion have the form:
˙ q= i
~ (H q−q H),
˙ p= i
~ (H p−p H).
(7)
The transition from one representation to another can be done by a unitary matrix that we shall callU, V andY. (As usual we will callU† the “transposed-conjugate” or “adjoint”matrix
Umn† =Unm∗ ,
so the matrixU is unitary if it satisfies the conditions U U†= 1; U† U = 1.
The matrix representation ofH0, W and H of the operator H is con- nected by the equations
H =U† H0U , (8ac)
W =V† H0 V , (8ab)
H =Y† W Y . (8bc)
The matrixY can be expressed throughU andV as follows:
Y =V† U . (9)
All these relations can be easily verified using explicit expressions for the matrix elements
Umn=R
ϕ∗m(q,0)ψn(q, t)% dq , Vmn=R
ϕ∗m(q,0)ϕn(q, s)% dq , Ymn=R
ϕ∗m(q, s)ψn(q, t)% dq .
(10)
§3. As it was indicated by one of the authors,6 the entries Ymn
have the following physical meaning. The square modulus|Ymn|2 is the probability that the mechanical system, being att= 0 in staten(having the energy levelWn(0)), will be in statemat timet(having the energy level Wn(t/T)). This meaning of |Ymn|2 follows also from the Dirac statistical equation, which was recently shown by the second author.7 The adiabatic theorem states now that at an infinitely slow change of the system, i.e., at an infinitely large value of the parameterT in (1), the transition probability|Ymn|2(m6=n) (which is the function of time) remains infinitely small even for finite values ofs= t
T.
We will prove this statement under some restrictions investigating the differential equation to which the valuesYlm satisfy.
§4. First let us establish the differential equation for the transfor- mation matrixU.
The matricesqandpsatisfy the equations of motion
˙ q= i
~(H q−q H),
˙ p= i
~(H p−p H)
(7)
6M. Born,The Adiabatic Principle in Quantum Mechanics, Zs. Phys. 40, 167, 1926.
7V. Fock,Verallgemeinerung und L¨osung der Diracshen statistischen Gleichung, Zs. Phys.49, 339, 1928. (See [28-3] of this book. (Editors))
and the constant matricesq0 andp0 are connected withpandq by the relations
q=U† q0 U , p=U† p0U .
(11)
The connection between the representations of the energy operatorsH andH0 is
H =U† H0U . (12)
Now we use the expressions for q, p and H through q0, p0 and H0 and calculate the derivatives ˙qand ˙p:
˙
q= ˙U† q0 U+U† q0 U ,˙
˙
p= ˙U† p0 U+U† p0U .˙ BecauseU†U˙ = 1, we have
U˙†=−U† U U˙ † and, therefore,
˙
q=U† −U U˙ † q0+q0 U U˙ † U ,
˙
p=U† −U U˙ † p0+p0U U˙ † U .
(13)
Otherwise, due to the equations of motion,
˙ q= i
~ U† H0 q0−q0 H0U ,
˙ p= i
~ U† H0 p0−P0 H0U .
(14)
Let us denote byK0 the Hermitian matrix K0=H0+~
i U U˙ †, (15)
and compare expressions (13) and (14); then we come to the equations K0q0−q0K0= 0,
K0p0−p0 K0= 0.
(16)
The matrixK0 commutes with q0 and p0. Then, if q0 and p0 form an irreducible system of matrices, the matrixK0must be a multiple of the
unity matrix, and we can put it simply equal to zero. Then for the transformation matrixU, we get the equation
~
i U˙ +H0 U = 0, (17)
which is nothing else but the Sch¨odinger equation in a matrix represen- tation.
If the matrices q0 and p0 are not constants as stated before, they would satisfy the equations of motion
˙ q0= i
~ (K0 q0−q0 K0),
˙ p0= i
~ (K0 p0−p0 K0)
(18)
with the “Hamilton matrix”K0. The considerations of this section also contain the theory of general (time-independent) canonical transforma- tions of the quantum mechanical equations of motion.
§5. From the differential equation (17) for the matrixU it is easy to get a corresponding equation for the matrixY. We have
Y =V† U
and Y˙ =V† U˙ + ˙V† U . (19)
If we now expressH0 in (17) using (8ab) through the diagonal matrix W, then we have
~
i U˙ +V W V† U = 0 or
U˙ =−i
~ V W Y . (20)
On the other hand,
U =V Y . (21)
Putting expressions (20) and (21) for ˙U andU into (19), we obtain Y˙ =−i
~ W Y + ˙V† V Y , (22)
i.e., the needed differential equation.
Here the superscript points mean (as usual) the time derivative. Now instead of time we want to introduce the quantitys= t
T from formula (1):
dY
ds =−i T
~ W Y +dV†
ds V Y . (220)
Further we introduce:
Q=−idY†
ds V =i V† dY
ds . (23)
Because the transformation matrixV is unitary, the matrixQ just in- troduced is Hermitian. Its entries can be expressed through the eigen- functions of the energy operator:
Qmn=i Z
ϕ∗m(q, s) ∂ϕn(q, s)
∂s % dq . (24)
We note that due to normalization (3) of the eigenfunctionsϕn(q, t) all diagonal elements of the matrixQvanish.
Now we want to find another expression for the elements of Q. Let us differentiate equation (8ab) by parameter s. Then we get
dW ds =dV†
ds H0 V +V† H0 dV
ds +V† dH0
ds V . (25)
In the representation where the energy matrix is diagonal, the expression V† dH0
ds V is the matrix for s-derivative of the energy operator; for brevity we denote itH0:
H0=V† dH0
ds V . (26)
Now from (8ab) it follows
H0V =V W ; V† H0=W V†. (27) If one puts (27) into (25) and takes into account the differential equations (23) and (26) forQandH0, then
dW
ds =i(Q W−W Q) +H0. (28) We consider now the nondiagonal element of matrix (28). BecauseW is diagonal, from (28) form6=nit follows
iQmn(Wn−Wm) +Hmn0 = 0
or
Qmn=− i Hmn0
Wm−Wn , (29)
whereHmn0 has the meaning Hmn0 =
Z
ϕ∗m ∂H
∂s ϕn % dq . (30)
Form=nas we already established,Qnn= 0.
It should be noticed here that Qmn remains finite also in the case when for some special value ofsthe differenceWm(s)−Wn(s) vanishes;
this follows from expression (24) forQmn.
Let us return to the differential equation (220) for the matrix Y, which we will write now in the form
dY
ds =−i T
~ W Y +i Q Y . (31)
If we rewrite (31) for matrix elements, then we get a system of equations dYmn
ds =−i T
~ Wm Ymn+iX
k
Qmk Ykn. (32) We consider also the system of equations8
dym
ds =−i T
~ Wmym+iX
k
Qmk yk . (33)
Let us take into account that due to (5) and (10) the matrixY at t = 0, s= 0 is a unity matrix; then we can consider the matrix elements of a column
Y1n, Y2n, . . . Ymn. . . as such a solution
y1, y2, . . . ym. . .
of the system of equations (33), which satisfies the initial conditions ym=Ymn=δmn for s= 0 . (34) The quantitiesymare uniquely defined by the differential equations (33) and the initial conditions (34).
8See footnote7; compare formulas (18) and (19) and theorem 1. (M. Born and V. Fock)
§6. Now we want to indicate the method for solution of equations (33). We put for brevity
ωk(s) = 1
~ Zs
0
Wk(s)ds , (35)
and introduce in (33), instead ofyk, new variables
ck =yk eiT ωk . (36)
The quantitiesck satisfy the differential equations dcm
ds =iX
k
Pmk ck , (37)
where for brevity we denote
Pmk=Qmk eiT(ωm−ωn). (38) The difference of new equations from the original ones is first that now the coefficient at cm is equal to zero, whereas the coefficient at ym is proportional to a large parameterT; second, thatPmk loaded with the largeT in the exponent oscillate rapidly, whereasQmkare slowly varying quantities.
Now we denote by cmn(s) those solutions of (37) that satisfy the initial conditions
cm(0) =δmn, (39)
i.e., the quantities
cmn(s) =Ymn eiT ωm . (40) Their square moduli are equal to those of Ymn, so they are transition probabilities.
As it can be easily checked, the differential equations (37) with the initial conditions (39) are equivalent to the system of integral equations
cmn(s) =δmn+iX
k
Zs
0
Pmk(σ)ckn(σ)dσ . (41)
One can solve these integral equations by iterations. As the zero approx- imation, we can put
c(0)mn=δmn ,
and in the first approximation as the result of inserting the zero approx- imation into the right-hand side of (41), one gets
c(1)mn=δmn+i Zs
0
Pmn(σ)dσ ,
and generally
c(l)mn=δmn+iX
k
Zs
0
Pmn(σ)c(l−1)kn (σ)dσ . (42)
As the final result, we obtain an infinite series cmn(s) =δmn+
+ X∞
k=1
ik Zs
0
dsk sk
Z
0
dsk−1. . .
s2
Z
0
ds1
h
P(sk)P(sk−1). . . P(s1) i
mn. (43) Up to now we did not take into account the convergency considerations.
To ensure the convergency of the method, we have to introduce prelimi- nary requirements on the matrixP(s) to be absolutely restricted9for all sand a constant-restricted matrixM can be found that is a majorant:
|Pmn(s)|=|Qmn(s)| ≤Mmn; (Mmn) restricted. (44) Then10the majorant system of equations
dbmn
ds =X
k
Mmk bkn, (45)
9A matrix (Pmn) is called restricted if for each system of numbersxn, yn that satisfies the normalization conditions
X
n
|xn|2= 1; X
n
|yn|2= 1,
the double sum X
mn
Pmnxmyn
converges and its absolute value stays within some limit independent of the choice of xn, yn. The matrix is called absolutely restricted if the matrix consisting of absolute values|Pmn|is also restricted. (M. Born and V. Fock)
10W.L. Hart, Amer. Journ.39, 407–424, 1917.
with the initial conditions
bmn(0) =δmn, possesses the solution
bmn(s) = (esM)mn=δmn+ X∞
k=1
sk
k! (Mk)mn , (46) which presents the uniformly convergent series in powers ofs. It is easy to see, replacing in (43)Pmn by Mmn, that the absolute value of each term of the series (43) is no larger than the corresponding term in the series (46). From this it is immediately clear that conditions (44) are sufficient for the convergence of series (43).
Whether the matrixQmnin some problem is really restricted, one can decide using the following (sufficient) criteria: according to the theorem by Schur11 it is really the case if the series
zm=X
k
|Qmk| (47)
converges and remains independent of m within a limit. According to (29) forQmn, this series is equal to
zm=X
k
0 |Hmk0 |
|Wm−Wn| , (48) where the prime means that the term withk=mis omitted.
Now if we accept that the set αm=X
k
0 1
(Wm−Wk)2 (49)
converges, and denote byβm the expression βm=X
k
¯¯
¯Hmk0
¯¯
¯2=Z ¯¯
¯¯∂H
∂s ϕm
¯¯
¯¯
2
% dq , (50)
then we can estimate the sumzm using the Schwarz inequality zm=p
αm βm. (51)
11J. Schur,Restricted Bilinear Forms, Crelles Journ. 140, 1, 1911 (Theorem I).
Thus, we obtain the following sufficient condition for the matrix Q to be absolutely bounded: the productαm βm should lie within a limitA independent ofm:
αmβm≤A . (52)
When the eigenvaluesWngrow proportionally ton, which is the case for a harmonic oscillator, the set (49) converges and its sum remains smaller than a number independent ofm. Then forQto be absolutely bounded it is sufficient forβm (50) to be finite, which always will be the case if the time derivative of the perturbation energy is a limited function.
If a mechanical system is restricted by a volume so that theq-space is finite, then for a single degree of freedom the eigenvalues Wn grow proportionally ton2. Thenαmdecrease as 1/m2, and forQto be abso- lutely bounded it is sufficient to admit thatβmincrease not faster than proportionally tom2, which is the case for a very general assumption about the perturbation energy.
§7. Now we come to the initial problem: the proof of the adiabatic theorem. We must prove the following: if the parameter T [formulas (1), (27), (32) and (34)] is sufficiently large, then the square moduli
|Ymn|2=|cmn|2for finitesdiffer arbitrarily little from their initial values δmn. The exact conditions needed for the theorem to be valid will be formulated later.
Next we formulate the Lemma:
Lemma. If in the interval 0 ≤ s ≤ s0 the following assumptions are fulfilled:
1. The following inequality is valid:
|Qmn(s)|=|Pmn(s)| ≤Mmn . 2. Within the interval each function (frequency)
dωm
ds −dωn
ds = 2πνmn(s)
has maximallyN1 zeroes of maximally ther-th order (i.e., for degener- ated states of a mechanical system) and in the vicinity of the zero point s0the estimation is valid
1
|2π νmn(s)| < A
|s−s0|r . 3. The real and imaginary parts of the function
Qmn(s) νmn(s)
are piecewise monotonous; the largest number of segments where they are monotonous isN2.
Then the estimation is valid:
¯¯
¯¯
¯ Z s0
0
Pmn(s)ds
¯¯
¯¯
¯=
=
¯¯
¯¯
¯ Z s0
0
Qmn(s)eiT(ωm−ωn)ds
¯¯
¯¯
¯<4Mmn(N1+N2)r+1 r4A
T . (53) The proof of this statement will be given in the Appendix.
Using this Lemma it is easy to prove the adiabatic theorem.
We perform the first integration (overs1) in thek-th term of the set (43) and estimate the result using formula (53).12
We do the remaining integrations replacing Pmn byMmn. Then we get
|cmn−δmn|<4 (N1+N2)r+1 r4A
T X∞
k=1
sk−1
(k−1)!(Mk)mn=
= 4(N1+N2)r+1 r4A
T dbmn
ds , (54)
where dbmn
ds is the derivative of the solution (46) of the auxiliary equa- tions (45). This quantity is finite for finite s just as the factor of the radical in (54); however the radical tends to zero for infiniteT.
Therefore, we have proven a mathematical theorem:
Theorem. If the matrixQis absolutely restricted and all the conditions for finites and infinitely largeT are valid, the differencecmn−δmn has the orderT−r+11 :
cmn=δmn+O à
T−r+11
ả
. (55)
Thus, this difference tends to zero whenT grows infinitely.13
It follows directly from this theorem that the probability of the tran- sitionn→mto another energy level is of the order of magnitudeT−r+12 :
|Ymn|2=|cmn|2=O(T−r+12 ), (56)
12One should keep in mind thatQnnis zero. (M. Born and V. Fock)
13The notationx=O(α) means thatxis of the order ofα. (M. Born and V. Fock)
or, e.g., of the order 1
T2 if none of frequencies νmn vanish in the course of adiabatic evolution.
Using the normalization relation X
n
|Ymn|2= 1
for the probability |Ymn|2 for the system to stay in the same state m, we obtain the expression
|Ymm|2= 1−X
n
0
|Ymn|2= 1−O(T−r+12 ). (57) This probability differs from unity by the quantity of the same order T−r+12 .
Up to now we considered as the initial state the “sharp” (pure) one, i.e., that at timet= 0 the system is in the stateWnwith the probability 1, whereas all the other states have the zero probability. If, on the contrary, at timet= 0 all the energy levelsWn are populated with the probabilities|bn|2, then we calculate the probabilities |b0m|2 of different levels at timetusing the formula
b0m=X
n
cmn bn . (58)
From (55), we have
b0m=bm+O à
T−r+11
ả
(59) and, therefore,
|b0m|2=|bm|2+O à
T−r+11
ả
if bm6= 0,
|b0m|2=O à
T−r+12
ả
if bm= 0.
(60)
Thus, the deviation of the probability|b0m|2of statemfrom its initial value|bm|2is of different order whether its initial value is zero or nonzero, and actually in the first case it is generally14 smaller, i.e., of a higher order of 1/T.
14Compare on the contrary formula (57). (M. Born and V. Fock)
Finally, we would mention that the adiabatic theorem can be valid in cases for which it was not proven here. As an example we can present a perturbed harmonic oscillator considered by one of the authors,15where matrixQis not restricted and the method considered in§6is not appli- cable.
Appendix
Proof of the Lemma in §7 To estimate the integral
s0
Z
0
Qmn(s)eiT(ωm−ωn)ds
we denote briefly the real or imaginary part of the functionQmn(s) by f(s) and the difference ωm(s)−ωn(s) asg(s) and consider the integral
J =
s0
Z
0
f(s)eiT g(s) ds .
We split the integration interval in two groupsE1 andE2 of segments, namely, the first group is the vicinities
αk−ε < s < αk+ε
of the zeroesαkof the derivativeg0(s) and the second group is the other parts of the segment (0, s0).
Evidently, the integral overE1
J1= Z
E1
f(s)eiT g(s)ds
satisfies the inequality
|J1|< M Z
E1
ds= 2M N1ε ,
15V. Fock,Uber die Beziehung zwischen den Integralen der quantenmechanischen¨ Bewegungsgleichungen und der Schr¨odingerschen Wellengleichung, Zs. Phys. 49, N5–6, 323–338, 1928 (this book [28-2]). Here the perturbation energy forx → ∞ diverges asx2. (M.B. and V.F.)
whereN1is the number of zeroesαk of theg0(s) andM is the maximum of the absolute value off(s).
We write the integral overE2 in the form J =
Z
E2
f(s)
g0(s) eiT g(s)g0(s)ds .
InE2the 1
g0(s) is finite, whereas in the vicinity of the zeroesαk we can estimate
1
|g0(s)| < A εr.
Now we apply the second averaging theorem of the integral calculus Zβ
α
ϕ(s)ψ(s)ds=ϕ(α) Zσ
α
ψ(s)ds+ϕ(β) Zβ
σ
ψ(s)ds ,
α≤σ≤β with
ϕ(s) = f(s) g0(s) and
ψ(s) =g0(s) cos [T g(s)]
or
ψ(s) =g0(s) sin [T g(s)]
to each of theN2 intervals where f(s)
g0(s) are monotonous, then due to f(s)
g0(s) < M A εr
and ¯
¯¯
¯¯
¯
σ2
Z
σ1
g0(s)sincos [T g(s)]ds
¯¯
¯¯
¯¯=
¯¯
¯¯
¯¯
g2
Z
g1
sincos [T g]dg
¯¯
¯¯
¯¯< 2 T we come to the inequality
|J2|<8 M A εr T N2 .
Together with the inequality for the first integral we get
|J|<2 M N1ε+8 M A N2
εr T .
Up to now the choice ofεremained arbitrary (it should only be small).
Now we choose
ε= à4A
T
ả 1 r+ 1 , and then we obtain the estimate
|J|<2M(N1+N2)r+1 r4A
T .
The imaginary part of Qmn(s) can be taken into account simply by multiplication of this formula by a factor 2. So formula (53) is proven.
List of Papers on the Adiabatic Theorem in Quantum Mechanics
1. P. Ehrenfest, Adiabatische Invarianten und Quantentheorie, Ann.
Phys.51, 327, 1916.
2. J.M. Burgers, Die adiabatischen Invarianten bedingt periodischer Systeme, l.c. 52, 195, 1917; further: Verslagen Amsterdam 25, 25, 918 and 1055, 1917.
3. G. Krutkow, Contribution to the Theory of Adiabatic Invariants, Proceedings Akad. Amsterdam21, 1112, 1919; further: On the De- termination of Quantum Conditions by Means of Adiabatic Invari- ants, l.c. 23, 826, 1920.
4. P. Ehrenfest,Adiabatische Transformationen in der Quantentheorie und ihre Behandlung dureh Niels Bohr, Naturwissensch. 11, 543, 1923.
5. V. Fock, Conditionally Periodic Systems with Commensurabilities and Their Adiabatic Invariants, Transactions of the Optical Institute in Petrograd3, N16, 1–20, 1923.
6. G. Krutkow and V. Fock, Uber das Rayleighsche Pendel, Zs. Phys.¨ 7. H. Kneser,Die adiabatische Invarianz der Phasenintegrals bei einem
Freiheitsgrad, Math. Ann.91, 155, 1924.
8. M. v. Laue, Zum Prinzip der mechanischen Transformierbarkeit (Adiabatenhypothese), Ann. Phys. 76, 619, 1925.
13, 195, 1923. (See [23-1] in this book. (Editors))
9. P.A.M. Dirac, The Adiabatic Invariance of the Quantum Integrals, Proc. Roy. Soc.A 107, 725, 1925; further: The Adiabatic Hypothesis for Magnetic Fields, Proc. Cambridge Phil. Soc.23, 69, 1925.
10. A.M. Mosharrafa, On the Quantum Dynamics of Degenerate Sys- tems, Proc. Roy. Soc.A 107, 237, 1925.
Translated by Yu.N. Demkov
29-1
On “Improper” Functions in Quantum Mechanics
V. Fock
G¨ottingen
Received 22 November 1928
JRPKhO61, 1, 1929
1. The application of the so-called improper functions in quantum me- chanics, e.g., the function δ(x−y), originally introduced by Dirac [1], does not meet minimal requirements of mathematical rigor. Sometimes it is not even clear what is meant by a particular mathematical relation expressed in terms of these functions. Certainly, the simplest way to gain some level of rigor is to avoid the use of improper functions, which is quite possible due to the notion of the Stieltjes integral. It was the way that J. von Neumann [3] followed. On the other hand, it is sometimes very convenient to apply improper functions, and the authors of many valuable works make wide use of them. The correct use of the improper functions, despite the lack of rigor, leads to true results. This can be explained only by the fact that manipulations with these functions are essentially legitimate and contain nothing paradoxical, and, in an ap- propriate formulation, they can be made rigorous in the mathematical sense.
2. LetK be a linear operator. If the action ofK on a function f can be represented in the form
Kf(x) = Z
K(x, y)f(y)dy,
we callK an integral operator andK(x, y) the kernel ofK.
Any “improper” function in quantum mechanics appears as the kernel T(x, y) of an integral operatorTwhich has no true kernel. The statement that an improper functionT(x, y) is the limit of a sequenceTn(x, y) of