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Quantum Mechanics of Collision Processes∗ [Scattering?]
Max Born
21 July 1926
Abstract
The Schr¨odinger form of Quantum Mechanics permits one to deﬁne in a natural way the frequency of occurrence of a state with help of the intensity of the associated eigenfunction. This understanding carries over to the theory of scattering where the transition probability is determined through the asymptotic cases of aperiodic solutions.
1 Introduction
Collision processes have not only supplied convincing experimental proof for the basic assumptions of Quantum Theory, but also seem suitable to explain more about the physical meaning of the formal laws of the so-called “Quantum Mechanics”. On the one hand it seems that Quantum Mechanics always gives the correct values for stationary states and the correct values for radiative transitions. However, with regards to the physical interpretation of the formulas, opinions are divided. The authors of the matrix formulation of Quantum Mechanics1 are of the opinion that an exact description of the processes in space and time are principally impossible. They are content therefore with a list of relations between physical observables. Only in the classical limit can these physical observables be interpretted as posessing the characteristics of [classical?] motion.
Schr¨odinger2 on the other hand, appears to ascribe to the wave (which he, according to de Broglie’s process, regards as the carrier of atomic processes) the same kind of characteristics as those which a light wave possesses. He tries to construct wave groups
∗See the preliminary publication, ZS. f. Phys. 37, 863, 1926. 1W. Heisenberg, ZS. f. Phys. 33, 879, 1925; M Born and P Jordan, ibid. 34, 858, 1925. See also P.A.M. Dirac, Proc, Roy, Soc, 109, 642, 1925; 110, 561, 1926. 2E. Schr¨odinger, Ann. d. Phys. 79, 361, 489, 734, 1926. Vlg. besonders die zweite Mitteilung, S. 499. Ferner Naturw. 14, 664, 1926.
1

[wave packets?] which have relatively small dimensions in all directions and should, as it seems [to Born], directly represent moving corpuscles.
Neither of these two views seem satisfactory to me. I would like to attempt here a third interpretation and test its applicability to collision processes. I thereby pin my hopes on a comment of Einstein’s regarding the relationship between the wave ﬁeld and light quanta. He says roughly that the waves may only be seen as guiding [showing?] the way for corpuscular light quanta, and he spoke in the same sense of a “ghost ﬁeld”. This determines the probability that one light quantum, which is the carrier of energy and momentum, chooses a particular [deﬁnite?] path. The ﬁeld itself, however, does not have energy or momentum.
If one wants to bring Einstein’s thoughts into direct relation with Quantum Mechanics, one is better oﬀ waiting until the electromagnetic ﬁeld has been included into the formalism. With regards to the complete analogy between a light quantum and the electron, one should not forget to formulate the laws of electron motion in a similar manner. And here it is obvious to regard the de Broglie-Schr¨odinger waves as a “ghost ﬁeld”, or even better as a guiding ﬁeld.
I would like to pursue the association further: The guiding ﬁeld, represented by a scalar function ψ of the co¨ordinates of all participating particles and time, evolves according to Schr¨odinger’s diﬀerential equation. Energy and momentum, however, are transferred as if corpuscles (electrons) are literally ﬂying around [behaving classically?]. The paths of these corpuscles are only determined insofar as energy and momentum conservation restrict them. Apart from that, we only have a probability for the choice of a certain path. This probability is given by the value of the distribution of the function ψ. One could summarise approximately, somewhat paradoxically: The movement of particles follows a probability law, the probability itself however evolves in accordance with the law of causality3.
If one views overall the three steps of the development of Quantum Mechanics, one sees that the ﬁrst, namely periodic processes, is wholly unsuitable for verifying the usefulness of [the proposed] picture. The second step, namely aperiodic stationary processes, is somewhat more productive. It is this subject which will occupy us in this article. Really the third step should be seen as the most signiﬁcant one ie: non-stationary events. These processes must show whether the interference of damped probability waves is suﬃcient to explain phenomena which apparently point to a coupling independent of space and time.
A more precise description is only possible if grounded in a mathematical development4. It is to this development that we now turn; later we will return to the hypothesis itself.
3I deﬁne the law of causality thus: that the complete knowledge of a state at a certain instant determines the distribution of the state for all later times.
4Herr Prof. N. Wiener of Cambridge, Mass. has helped me with the mathematical aspects of this work in a most friendly way. For this, I would like to express my thanks and acknowledge that without him, I would have been unable to reach my goal.
2

2 Deﬁnition of Weights and Frequencies of occurrence for Periodic Systems

We begin with a completely formal view of a stationary non-degenerate state. This system may be characterised by Schr¨odinger’s diﬀerential equation:

[H − W, ψ] = 0 .

(1)

The eigenfunctions are normalised to unity5:

ψn(q)ψm∗ (q)dq = δnm .

(2)

Any arbitrary function ψ(q) may be expanded in terms of eigenfunctions

ψ(q) = cnψn(q) .

(3)

n

So far we have only directed our attention towards the eigenfuctions ψn and eigenvalues Wn. the picture developed in the introduction suggested that equation 3 should be related to the probability that in a collection of equal, uncoupled atoms, a state should occur with a certain frequency [multiplicity?]. The completeness relation

|ψ(q)|2 dq = |cn|2

(4)

n

suggests regarding this integral as representing the number of atoms. We have for a single normalised eigenfunction a value of 1. (Or a-priori, the state has weight 1.) |cn|2 represents the frequency of occurrence of a state n and the entire number of atoms is
assembled additively out of these pieces.

In order to justify this interpretation, we consider for example the motion of a point mass in 3-dimensional space under the inﬂuence of a potential energy U (x, y, z). Then the diﬀerential equation reads

2µ

∆ψ + (W − U ) ψ = 0 .

(5)

2

One substitutes here for W and ψ, an eigenvalue Wn and eigenfunction ψn, multiply the equation by ψm∗ and integrate over all space. Thus one obtains

d3q

ψm∗ ∆ψn

+

2µ
2

(Wn

−

U)

ψm∗ ψn

= 0.

According to Green’s Theorem and with the orthogonality relation (2) we have

2

δmnWn =

d3q 2µ

(∇ψn).(∇ψm∗ ) + U ψnψm∗

.

(6)

5For simplicity I set the Dichtigkeitsfunktion equal to 1. (Sorry, I cannot translate this - can anyone help?)

3

Every energy level can be understood as a volume integral of the energy density. One can construct now the appropriate integral for any arbitrary function:

2

W = d3q |∇ψ|2 + U |ψ|2 .

(7)

2µ

So one obtains through the use of (3) the expression

W = |cn|2Wn .

(8)

n

According to our interpretation of |cn|2, the right-hand side of (8) is the average value of the total energy of the atomic system. This average value can be represented as a volume integral of energy density of the function ψ. As long as we dwell on periodic processes, there is nothing else substantial to be said in favour of our Ansatz.

3 Aperiodic Systems

We therefore consider aperiodic processes and regard ﬁrst for the sake of simplicity the

case of linear motion with constant velocity along the x-axis. Here the diﬀerential equation

reads:

d2ψ dx2

+

k2ψ

=

0;

k2

=

2µ 2W

.

(9)

Its eigenvalues W are all positive and it has eigenfuctions:

ψ = ce±ikx .

In order to deﬁne weights and frequencies of occurrence, one must normalise all the eigenfunctions. The integral form of equation (2) fails. (The integral diverges.) It is therefore obvious to use the average value instead:

1 lim

+a
|ψ(k, x)|2 dx =

c2 lim

+a
e−ikxeikxdx = 1 .

a→∞ 2a −a

a→∞ 2a −a

(10)

It follows that c = 1 and one has as normalised eigenfunctions:

ψ(k, x) = e±ikx .

(11)

Any function of x can be constructed out of these. It is still necessary to select a calibration

for the k-scale. It is necessary to determine on which portion [of the x-axis?] the weight

1 will fall. For this, one may view free motion as the limiting case of periodic motion;

namely the normal modes of a ﬁnite piece of the x-axis. It is generally known that the

∆k

1

number of normal modes per length ∆x and per interval ∆k is given by = ∆

2π

λ

where λ is the wavelength. As one expects,

ψ(x) =

∞

k

c(k)ψ(k, x)d

1 =

∞
c(k)eikxdk

(12)

−∞

2π

2π −∞

4

with

c(−k) = c∗(k) .

(13)

Then |c(k)|2 can be viewed as the measure of frequency of occurrence for the interval dk
. For a collection of atoms whose distribution is given by c(k), the number of atoms is 2π represented as in equation (4), by the integral

∞
|ψ(x)|2 dx =

1

∞

∞

2

dx

c(k)eikxdk .

(14)

−∞

(2π)2 −∞

−∞

Consider the case where only a small interval k1 ≤ k ≤ k2 is occupied. Then

∞
c(k)eikxdk = c¯

k2
eikxdk =

c¯

eik2x − eik1x

,

−∞

k1

ix

where c¯ represents the average value. Therefore one has:

∞
|ψ(x)|2 dx

=

|c¯|2

∞ dx eik2x − eik1x e−ik2x − e−ik1x

−∞

4π2 −∞ x2

|c¯|2 =

∞ dx sin

k2 − k1 x

4π2 −∞ x2

2

|c¯|2 = 2π (k2 − k1) .

Now the momentum of the translatory motion, belonging to the eigenfunctions (11), is

given by de Broglie’s equation

h

p = = k.

(15)

λ

It is perhaps not superﬂuous to remark that p can be understood as a “matrix”. One must therefore deﬁne the matrix in the regime of a continuous spectrum through average values rather than through integrals.

p(k, k ) = =

1 lim i a→∞ 2a
1 lim i a→∞ 2a

+a ψ∗(k, x) ∂ψ(k , x) dx

−a

∂x

+a
e−ikxik e−ik xdx .

−a

k for k = k ,

⇒ p(k, k ) =

(16)

0 for k = k .

∆p Finally, through , one sets ∆k = k2 − k1, and so,

∞ |ψ(x)|2 dx = |c¯|2 ∆p .

(17)

−∞

h

With this, one has the result that a box of dimensions ∆x = 1 and ∆p = h has a weight of 1, which is in agreement with the multiply veriﬁed ansatz of Sackur and Tetrode6. The

6A. Sacker, Ann. d. Phys. 36, 958, 1911; 40, 67, 1913; H. Tetrode, Phys. ZS. 14, 212, 1913; Ann. d. Phys. 38, 434, 1912.

5

second result is that |c(k)|2 is the frequency of occurrence of a motion whose momentum is p = k.
Now we turn our attention to accelerated motion. Here we can naturally deﬁne a certain distribution of events in an analogous manner. This is however not a rational [approach?] with regards to collision processes. With these processes, one has asymptotic, straight-line motion before and after a collision. The particles exist therefore in practically free states for long periods of time (as compared with the duration of the impact itself) before and after the collision. One comes therefore to the following understanding (which is in accordance with problems raised by experiments): If, for asymptotic motion before impact, the distribution function |c(k)|2 is known, can one obtain from it the distribution function after impact?
Naturally, we are referring here to a stationary beam of particles. Mathematically, our task amounts to the following: The stationary oscillation ﬁeld ψ must be divided into incoming and outgoing waves. These are asymptotic plane waves. One can represent both of these through Fourier integrals of the form (12) by choosing the coeﬃcient function c(k) for the incoming wave arbitrarily. Then it shall be shown that c(k) for the outgoing wave is completely determined. c(k) yields a distribution into which the given collection of particles is transformed by the impacts. To gain a clearer picture, we must ﬁrst consider the 1-dimensional case.

4 Asymptotic Behaviour of Eigenfunctions of the Continuous Spectrum for One Degree of Freedom

The Schr¨odinger diﬀerential equation reads:

d2ψ 2µ

+ (W − U (x)) ψ = 0 ,

(18)

dx2

2

where U (x) represents the potential energy. For expediency, we set

2µ W

=

k2

and

2µ U (x) = V (x) .

(19)

2

2

Then we have

d2ψ + k2ψ = V ψ .

(20)

dx2

We examine the asymptotic behaviour of the solution at inﬁnity. To have simple relations, 1
we assume that V (x) falls oﬀ faster than as x → ∞, ie: x2

K

|V (x)| < ,

(21)

x2

where K is a positive number7. We now determine ψ(x) through an itterative process.

We start with

u0(x) = eikx .

(22)

7With this assumption, the cases of pure Coulomb and dipole ﬁelds are excluded.

6

Then u1(x), u2(x), . . . are solutions of the approximation equation

d2un dx2

+

k2un

=

V

un−1

,

which vanish as x → ∞.

Then

1∞ un(x) = k x un−1(ξ)V (ξ) sin k(ξ − x)dξ ,

as one can directly verify. One has

1∞ un(x) ≤ k x |un−1(ξ)| . |V (ξ)| dξ .

We show now that

1 Kn |un−1(ξ)| ≤ n! kx .

For n = 0, this equation is true since, in agreement with (22), |u0(x)| ≤ 1. We assume now that it is correct for n − 1:

1

K n−1

|un−1(ξ)| ≤ (n − 1)! kξ

;

it then follows

11 |un−1(x)| ≤ k (n − 1)!

K n−1 K

∞
ξ−n+1ξ−2dξ =

1

k

x

n!

Kn ,
kx

as has been stated. Therefore the series

∞

ψ(x) = un(x)

(23)

n=0

converges uniformly for any ﬁnite interval of x. ψ(x) lends itself to being diﬀerentiated term by term arbitrarily often, and is thus, as can clearly be seen, the desired solution of our diﬀerential equation.

However, since all terms u1, . . . , un vanish as x → ∞, the function ψ asymptotically approaches u0(x) = eikx as x → ∞. In a similar way, one sees that another solution exists. This solution asymptotically approaches e−ikr as x → ∞. Since the general solution has
only two constants, the solution for x → ∞ must have the form

ψ+ = aeikx + be−ikx.

(24)

Here the degeneracy of the system becomes obvious. To each energy W , there are associacted two values of k (±k) and two linearly independent solutions.

It follows similarly that the general solution for x → −∞ must have the same form:

ψ− = Aeikx + Be−ikx.

(25)

7

Therefore the amplitutes A and B are deﬁnite functions of a and b.

We separate the solution into incoming and outgoing waves and we include a time factor eikvt (kv = 2πν = W ) and set

a = cieiϕit, A = Coeiφot, b = coe−iϕot, B = Cie−iφit.

(26)

Then

ψ+(x) = cieik(x+vt+ϕi) + coe−ik(x−vt+ϕo), ψ−(x) = Coeik(x+vt+φo) + Cie−ik(x−vt+φi).

(27)

The respective parts are distinguished with indices i and o. i denotes the incoming wave

and o denotes the outgoing wave. We are interested in the case where only one wave

arrives at x = +∞. In this case Ci = 0. Moreover, one can arbitrarily set ϕi = 0. Then

one obtains:

ψ+(x) = cieik(x+vt) + coe−ik(x−vt+ϕo), ψ−(x) = Coeik(x+vt+φo).

(28)

It has been seen that through the integration, ψ− is determined by ψ+, ie: A and B are

particular functions of a and b. In the case Ci = 0, B = 0 also; in addition we have two

equations of the form

A = A(a, b), (29)
0 = B(a, b).

From these two, one can express b in terms of a and thus express A in terms of a alone. This means however that the constants of the reﬂected and transmitted waves can be calculated from the amplitude of the incident wave. One can now see that a relation exists between the intensities of the three waves. The simplest way to obtain this relationship is through energy conservation.

5 The Principle of Conservation of Energy

To derive this principle, we must return to the form of the Schr¨odinger equation which does not presuppose purely periodic oscillations. The wave equation takes on the form:

∂2ψ 1 ∂2ψ

∂x2 − v2 ∂t2 = 0

(30)

Here v is the wave velocity. One arrives to the Schr¨odinger equation with the help of de Broglie8:
hν = W = µ u2 + U, 2 v = λν,
h = p = µu.
λ

8We disregard Special Relativity and calculate with Classical Mechanics

8

Then we have

1 h2 =

1

µ2u2 =

=

µ 2

u2.2µ

=

2µ (W − U ).

(31)

v2 λ2 h2ν2 W 2

W2

W2

One now seeks a solution whose time dependence is give by the factor e2πiνt = eiW t/ . So

one obtains:

d2ψ 2µ

+ (W − U ) = 0.

dx2

2

∂ψ We consider however the general form (30) and multiply the equation by :
∂t

∂2ψ ∂ψ ∂ ∂ψ ∂ψ

∂ψ ∂2ψ ∂ ∂ψ ∂ψ

∂ 1 ∂ψ 2

∂x2 ∂t = ∂x

∂x ∂t

−

=

∂x ∂x∂t ∂x

∂x ∂t

− ∂t 2

∂x

.

When v depends only on x, we obtain

∂ ∂ψ ∂ψ

∂ 1 ∂ψ 2 1 ∂ψ 2

− ∂x ∂x ∂t ∂t 2 ∂x

+ 2v2 ∂t

= 0.

(32)

Integrating over all space, one obtains:

∂ψ ∂ψ ∞ ∂ ∞ 1 ∂ψ 2 1 ∂ψ 2

− ∂x ∂t −∞ ∂t −∞ 2

∂x + v2 ∂t

dx = 0

(33)

As was show in section 2, the volume integral has to be interpretted as the total energy available in space. However, the expression does not interest us because we are concerned with the in and out-ﬂow of energy. This is represented by the surface term in equation (33). The time average of the second term vanishes for a periodic process. One obtains by use of the notations introduced in equations (24) and (25)

∂ψ− ∂ψ− ∂ψ+ ∂ψ+

=

.

(34)

∂x ∂t ∂x ∂t

This equation demonstrates that the incoming and outgoing energies are equal. By inserting here the real parts of (27) we obtain

Co2 − Ci2 = c2i − c2o,

(35)

or in the case Ci = 0 (as in equation (28)):

c2i = c2o + Co2.

(36)

This means however that for any wave front of a given k, the incoming intensity is split up into both left and right scattered waves. Or in the language of corpuscular theory: If a particle of given energy strikes an atom, it is either reﬂected or transmitted. The sum of probabilities for both these events is 1.

The principle of conservation of energy implies therefore the conservation of number of particles. The reason for this lies in the degeneracy of the system. There is more than one path associated with the same energy and these paths are set in relation with each other.

9

6 Generalisation to Three Degrees of Freedom. Motion With Constant Velocity

We consider now the motion of particles in space under the inﬂuence of a potential energy U (x, y.z). Similarly to (30), one has the diﬀerential equation

1 ∂2ψ

∆ψ −

= 0,

(37)

v2 ∂t2

where v (in approximation with Classical Mechanics) is given again by equation (31). Here the conservation principle reads:

∇.

∂ψ ∇ψ

∂1 −

(∇ψ)2 + 1

∂2ψ

= 0,

(38)

∂t

∂t 2

v2 ∂t2

or integrating over all space,

∂ψ ∂ψ ∂ ds −

1 (∇ψ)2 + 1 ∂ψ 2 d3r = 0,

(39)

∞ ∂t ∂n

∂t 2

v2 ∂t

where ds is an inﬁnite closed surface with outer normal nˆ. For periodic processes it follows

that the time average

∂ψ ∂ψ

ds = 0.

(40)

∞ ∂t ∂n

In this case the diﬀerential equation reads

∆ψ + (k2 − V )ψ = o,

(41)

where

k2

=

2µ W,

2µ V (r) = U (r).

(42)

2

2

The diﬀerential equation for unaccelerated motion is given by

∆ψ + k2ψ = 0,

(43)

with the solution:

ψ = eik.r.

(44)

Here r is the vector (x, y, z) and the vector k satisﬁes the equation

2

k = kx2 + ky2 + kz2 ≡ k2.

(45)

It is equal, up to a factor, to the momentum vector:

p = k.

(46)

10

h The de Broglie wavelength is given by = p = |p| = k. The solution (44) should be
λ seen as normalised in the sense of an average (see (10)).
The general solution of (43) is

ψ(r) = u0(r) = c(sˆ)eik(r.sˆ)dω; c(sˆ) = c∗(sˆ),

(47)

where sˆ is a unit vector and dω is an element of solid angle. This represents unaccelerated
[inertial] motion in all possible directions with the same energy. According to our principle, |c(sˆ)|2 computes the number of particles ﬂying in direction sˆ per unit solid angle.

We want to deduce an asymptotic representation for u0 which will clearly show the behaviour of u0 at inﬁnity. Although it is very simple to obtain this result, we want to obtain it here by means of a more general method; one which can be transferred later to handle more complicated cases. We consider here a new orthogonal co¨ordinate system introduced with the help of the orthogonal transformation:

x = a11X + a12Y + a13Z, X = a11x + a21y + a31z, 

 

y = a21X + a22Y + a23Z, Y = a12x + a22y + a32z,

(48)

 z = a31X + a32Y + a33Z, Z = a13x + a23y + a33z 

At the same time we introduce a new unit vector Sˆ in place of sˆ with the aid of the same orthogonal transformation. Then the volume angle element dω becomes dΩ and

r.sˆ = R.Sˆ.

(49)

Now we choose the new co¨ordinate system in particular so that

X = 0, Y = 0,

(50)

and so

Z = r = x2 + y2 + z2.

(51)

Our integral becomes

u0(x, y, z) = u0(a13Z, a23Z, a33Z) = dΩc(a11Sx + a12Sy + a13Sz, . . .)eikZSz .

Now we introduce polar co¨ordinates for Sˆ:

Sx = sin ϑ cos ϕ, Sy = sin ϑ sin ϕ, Sz = cos ϑ,

(52)

and setting cos ϑ = µ;

2π

1

u0 = dϕ dµc

0

−1

1 − µ2(a11 cos ϕ + a12 sin ϕ) + µa13, . . . eikZµ.

11

Through partial integration

1 u0 = ikZ

2π
dϕ c(a13, a23, a33)eikZ − c(−a13, −a23, −a33)e−ikZ
0

1 2π d

−

dϕ c

ikZ 0 dµ

1 − µ2(a11 cos ϕ + a12 sin ϕ) + µa13, . . . eikZµdµ.

Repeated application of the same process shows that the second term vanishes as Z−2. One

xx

inserts

now Z

= r,

a13

=

Z

=

, . . ., r

and

so

one

obtains

the

asymptotic

representation

u∞ 0 (x, y, z)

=

2π ikr

c

xyz ,,
rrr

eikr − c

xyz − ,− ,−
rrr

e−ikr ,

(53)

or in Euler notation with c = |c|eikγ:

u∞ 0 (r)

=

4π k

c

xyz ,,
rrr

sin k

r+γ

x r

,

y r

,

z r

r

.

(54)

This means that u0 behaves asymptotically like a spherical wave whose amplitude and phase depend on direction. The intensity as a function of sˆ determines the rate of incoming

particles passing through a solid angle element dω in the direction sˆ.

7 Elastic Collisions

We turn our attention now to the integration of the general equation (41).

∆ψ + (k2 − V )ψ = 0.

(55)

This represents the physical case where a single electron collides with an unexcited atom. As in section 4, we determined ψ through an itterative procedure. This served as a starting point for the function u0 (equation (47)). So then we can calculate u1, u2, . . . one after another from the approximation equation

∆unn + k2un = V un−1 = Fn−1.

(56)

Green’s Theorem yields the solution which is equivalent to an outgoing wave with time

factor eikvt:

1 un(r) = − 4π

Fn−1(r

e−ik|r−r )

|
d3r .

|r − r |

(57)

The convergence of the procedure can be proven based on the assumption that V falls

oﬀ9 as

1 .

We do not focus on this however.

Instead we assume that a solution may be

r2

represented by a series

∞

ψ(r) = un(r).

n=0

9Herewith, the case of ions is excluded. Concerning this, one must take as a starting point for our approximation procedure, a hyperbolic path rather than linear motion. See here also the the soon to be appearing publication of J.R. Oppenheimer, Proc. Cambridge Phil. Soc. 26 July 1926.

12

We examine the asymptotic behaviour of un(r). We write explicitly:

1 un(r) = − 4π

Fn−1(x , y , z )

√

e−ik (x−x )2+(y−y )2+(z−z )2

dx dy dz .

(x − x )2 + (y − y )2 + (z − z )2

As in section 6, we rotate the co¨ordinate system and apply the same rotation to the integration variables. We have then

√

1

e−ik X 2+Y 2+Z 2

un(x,y,z)

=

un(a13Z,

a23Z,

a33 Z )

=

− 4π

Fn−1(X ,Y ,Z )

dX dY dZ . X 2 +Y 2 +(Z −Z )2

(58)

Therefore

Fn−1 (X , Y , Z ) = Fn−1 (a11X + a12Y + a13Z , . . . , . . .) .

(59)

Now introduce polar co¨ordinates

X = ρ sin ϑ cos ϕ; Y = ρ sin ϑ cos ϕ; Z = ρ cos ϑ.

Then

1

un

=

− 4π

2π
dϕ
0

∞
ρ2dρ
0

π
sin ϑdϑFn−1(ρ sin ϑ cos ϕ, . . .)
0

√ e−ik ρ2+Z2−2ρZ cos ϑ
. ρ2 + Z2 − 2ρZ cos ϑ

Instead of ϑ, we introduce the integration variable µ through

ρ2 + Z2 − 2ρZ cos ϑ = Zµ,

Z sin ϑdϑ = µdµ.

ρ

The limits of integration become ρ
ϑ=0 : µ= −1 ; Z

ρ ϑ=π : µ= +1
Z

and cos ϑ and sin ϑ become special functions c(ρ, Z, µ) and s(ρ, Z, µ). c(ρ, Z, µ) and s(ρ, Z, µ) take on values c = 1, s = 0 in the lower limit and c = −1, s = 0 in the upper limit. Thus one obtains

1

un

=

− 4π

2π
dϕ
0

∞
ρdρ
0

ρ Z

+1

Fn−1(ρs

cos

ϕ,

ρs

sin

ϕ,

ρc)e−ikµZ dµ.

|

ρ Z

−1|

Through partial integration we obtain the asymptotic representation as in section 6:

u∞ n

=

1 4π

2π
dϕ
0

∞

1

ρdρ

0

ikZ

Fn−1(0, 0, ρ)e−ik(Z+ρ) − Fn−1(0, 0, −ρ)e−ik|Z−ρ| .

Using equation (59)

ρx ρy ρz

Fn−1(0, 0, ρ) = Fn−1(a13ρ, a23ρ, a33ρ) = Fn−1

,, rrr

,

13

ρx ρy ρz

Fn−1(0, 0, −ρ) = Fn−1(−a13ρ, −a23ρ, −a33ρ) = Fn−1

− ,− ,− rrr

.

So we obtain:

u∞ n

=

e−ikr 2ikr

∞
ρdρFn−1
0

ρx , . . . e−ikρ− e−ikr

r

2ikr

r
ρdρFn−1
0

ρx −,

.

.

. eikρ−

eikr

r

2ikr

∞
ρdρFn−1
r

− ρx , . . . e−ikρ. r

Here the last integral vanishes as r → ∞. In explaining why this term vanishes, we use

as

a

starting

point

|V

|

≤

ar−2

and

so,

because

|u0|

≤

br−1,

we

have

|Fn−1|

≤

A ,
r3

and

thus

∞
ρdρFn−1
r

ρx − ,...
r

e−ikρ

≤A

∞ dρ A =.
r ρ2 r

With this, we ﬁnally obtain

u∞ n

=

e−ikr 2ikr

∞
ρdρ Fn−1
0

ρx ,...
r

e−ikρ − Fn−1

ρx − ,...
r

e−ikρ .

(60)

This expression can be brought into a more transparent form. To that end, we introduce the Fourier co-eﬃcients of the function Fn−1:

1 fn−1(k) = (2π)3

Fn−1 (r)e−ik.r d3 r

=

1 (2π)3

∞
r2dr
0

dωFn−1(rsˆ)e−ir(k.sˆ).

(61)

By the procedure already used, we determine the asymptotic value and obtain:

fn∞−1(k)

=

1 4π2ik

∞
rdr Fn−1
0

rkx , . . . k

eikr − Fn−1

− rkx , . . . k

e−ikr .

Therefore

fn∞−1(−ksˆ)

=

1 4π2ik

∞
ρdρ Fn−1
0

ρx ,...
r

e−ikρ − Fn−1

ρx − ,...
r

eikρ .

(62)

Inserting equation (60), we ﬁnally obtain

u∞ n (r)

=

2π2fn∞−1

(−ksˆ)

e−ikr r

.

(63)

Comparing this with equations (47) and (54) we can see tha an observer standing at inﬁnity will recognise the scattered radiation as a plane wave whose amplitude

k 2π2 2π

fn∞−1(−ksˆ)

= kπ

fn∞−1(−ksˆ)

depends on the direction sˆ. Therefore the probablility that an electron with direction sˆ is scattered into a solid angle element dω, is given by

∞

2

Φdω = π2k2 fn∞(−ksˆ) dω.

(64)

n=0

14

The complete solution has the asymptotic form:

ψ∞ = u∞ 0 +

∞

u∞ n

=

2π k

∞
|c(sˆ)|eik(r+δ) + kπ fn∞(−ksˆ)e−ikr

.

n=1

n=1

One includes here a time factor eikvt and so one easily obtains “conservation of number of particles” from equation (40).

To ﬁrst approximation one has

Φdω = π2k2 |f0∞(−ksˆ)|2 dω,

(65)

where one either strictly calculates f0 from the formula

1 f0(k) = (2π)2

F0 (r)e−ik.r d3 r,

(66)

or one immediately makes use of the expression given by (62):

f0∞(−ksˆ)

=

1 4π2ik

∞
ρdρ F0 (ρsˆ) eikρ − F0 (−ρsˆ) e−ikρ .
0

(67)

8 Inelastic Electron Collisions

An atom (or a molecule, but we prefer to always speak of an “atom”) is represented by the Hamiltonian function Ha(p, q)10. If the Schr¨odinger diﬀerential equation has been solved, the eigenvalues Wna and eigenfunctions ψna(q) identically satisfy the equations

[Ha − Wna, ψna] = 0.

(68)

An electron collides with an atom. The Hamiltonian function of the free electron is

Hε = 1 2µ

p2x + p2y + p2z

.

Its eigenvalues W ε are all positive and the eigenfuctions are

e±kr (11)

where

k2

=

2µ
2

W

ε.

(69)

The general solution which corresponds to the incoming wave is

ψkε =

c0(sˆ)eik(r.sˆ)dω. r.sˆ>0

It satisﬁes the diﬀerential equation

[Hε − W ε, ψkε] = 0 or ∆ψkε + k2ψkε = 0.
10We write in abbreviated form p, q in place of p1, p2, . . . , pn, q1, q2, . . . , qn. 11This must be a misprint in the original paper. It should read e±ikr.

(70) (71)

15

The potential energy of the atom-electron interaction is

U (q; x, y, z).

(72)

The interaction between the two particles leads to the Hamiltonian function H = H0 + λH(1),

where

H0 = Ha + Hε and λH(1) = U.

The undisturbed system has the solution

Wn0k = Wna + W ε,

ψn0k = ψnaψkε.

We solve the Schr¨odinger diﬀerential equation of the perturbed system

[H − W, ψ]

through the Ansatz

ψ = ψ0 + λψ(1) + . . .

The one obtains the approximation equations

H0 − Wn0k, ψn(1k) = −U ψn0k,
H0 − Wn0k, ψn(2k) = −U ψn(1k), ......................... ...............

whose left hand sides are in agreement with each other. We write explicitly:

Ha, ψn(1k) + Hε, ψn(1k) − Wn0kψn(1k) = −U ψn0k,

or
2
Ha, ψn(1k) − 2µ ∆ψn(1k) − W (0)ψn(1k) = −U ψn0k. We seek to solve this equation through the Ansatz:

ψn(1k) =

u(n1m) (r)ψma ,

m

ie: through the expansion in eigenfuctions of the unperturbed atom whose coeﬃcients are still undetermined functions of the position vector r of the electron.

Now, according to (68),

Ha, ψn(1k) =

u(n1m) (r) [Ha, ψma ]

m

=

u(n1m) (r)Wma ψma .

m

16

The right hand side of the equation can be expanded in the same manner:

U ψn0k = ψkεU ψna = ψkε Unmψma .
m

The co-eﬃcients Unm make up a matrix which represents the potential energy. We insert this expression into the diﬀerential equation and obtain:

2

ψma u(n1m) (r)Wma − 2µ ∆u(n1m) (r) − u(n1m) (r) (Wma + W ε) = −

ψma Unmψkε.

m

m

By equating co-eﬃcients of ψma one obtains a diﬀerential equation for u(n1m) (r). We multiply 2µ
this equation by − , and using the abbreviations
2

2µ

2µ

V = U,
2

Vnm = 2 Unm,

(73)

kn2m

=

2µ
2

(Wna

− Wma

+ W ε)

=

2µ
2

(hνnam

+ W ε) ,

(74)

we have

∆u(n1m) + kn2mu(n1m) = Vnmψkε.

(75)

In this way we have related the problem to the earlier treatment of inelastic collisions,

since all following approximations lead to the same wave equation. The diﬀerence, how-

ever between this case and the earlier one is as follows: Any transition (n → m) of an

atom corresponds to a distinct [unique?] diﬀerential equation whose right hand side is

determined by the corresponding matrix element of the potential energy. Furthermore,

the k value of the incoming wave is always replaced by another k value, knm, whose energy

corresponds to

2
Wnεm = 2µ kn2m = hνnam + W ε.

(76)

Already the fundamental qualitative laws of electron scattering follow: The energy of an

electron after a collision is not generally equal to its energy before, but the energy of

the atom diﬀers by an energy step of hνnam. For any collision process, a corresponding

probability function

Φnm = π2kn2m |f0∞(−knmsˆ)|2

(77)

may be computed with the aid of equations (66) and (67).

9 Physical Conclusions
First we see that our formulas correctly describe the qualitative behaviour or atoms in collisions, therefore also the fact of “energy limits [levels?]”. These energy limits [levels?] are always seen as the cornerstone of Quantum Mechanics and the grossest violation of Classical Mechanics. We arrange the energy levels of the atom from smallest to largest:
W0a < W1a < W2a < . . .
17

The index 0 labels the ground state and so

hνnam = Wna − Wma > 0 for n > m.
We consider ﬁrst the case where the atom starts in its ground state. Then νma 0 > 0 for all m and it follows from equation (76) that

W0εm = W ε − hνma 0.
If W ε < hν1a0, W0εm would become negative for m > 0, which is impossible. Thus m = 0 and
W0ε0 = W ε. Elastic reﬂection takes place with [yield?] Φ00. Let W ε increase until
hν1a0 < W ε < hν2a0.
Now W0εm is only positive for m = 0 and m = 1. Therefore one either has elastic reﬂection with yield Φ00, or resonance excitation with yield Φ01. If we increase W ε further until
hν2a0 < W ε < hν3a0,

then we have three cases: Elastic reﬂection with yield Φ00, excitation of the ﬁrst quantum jump with yield Φ01, or excitation of the second quantum jump with yield Φ02. One can continue in the same way.

Now consider the case where the atom starts in its second quantum state (n = 1). Then ν1a0 > 0 and ν1am < 0 for m = 2, 3, . . . One has therefore

W1ε0 = W ε + hν1a0, W1ε1 = W ε, W1εm = W ε − hν1am,

m = 2, 3, . . .

If W ε < hν2a1 then W1εm is negative for m = 2, 3, . . . Therefore either a collision of the second type with energy gain of the electron of hν1a0 and yield Φ10 or elastic reﬂection with yield Φ11 exists. If
hν2a1 < W ε < hν3a1,
then the state n = 2 contributes to these processes with yield Φ12. And so it goes on . . .

In the general case, if the atom starts in the nth state, for

W ε < hνna+1,n

there are only collisions of the second type. Here the atom can drop into states 0, 1, . . . , n− 1 and transfers energies hνna0, hνna0, . . . , hνna,n−1 to the electron with yields Φn0, Φn1, . . . , Φn,n−1 and elastic reﬂection Φnn. Increase W ε above hνna+1,n so that
hνna+1,n < W ε < hνma +1,n,

18

then excitations contribute with yields Φn,n+1, Φn,n+2, . . . , Φn,m.

The next task would be to discuss the formula for the yields (77). However we will have
to be satisﬁed here with a very preliminary [provisional?] and probably quite controversial 1
picture. We assume that the potential U can be expanded in a power series in . For a r
neutral atom we have to ﬁrst approximation the dipole equation:

e

U (x, y, z) = r3 Br,

(78)

where B(q) is the electrical moment of the atom. We represent this with a matrix Bnm.

Then according to (73),

2µc

r

Vnm = 2 Bnm r3 .

(79)

Naturally this ansatz can only be correct for electrons which pass by the atom at a considerable distance. Our view is therefore limited to such electrons where12 r > r0. Therefore from equation (67)

f0∞(−knmsˆ)

=

1 4π2iknm

∞
ρdρ Fnm (ρsˆ) e−iknmρ − Fnm (−ρsˆ) eiknmρ .
0

We assume that the incoming electrons constitute a parallel bundle corresponding to a

plane wave. Then

Fnm(ρsˆ)

=

Vnmeikρzˆ

=

2µe
2

(Bnm, sˆ)

eikρzˆ ρ2 .

Now

iπknmf0∞(−knmsˆ)

=

µe π2

(Bnm, sˆ) A,

(80)

where, with zˆ = cos ϑ;

∞ dρ

A=
r0

ρ cos [ρ(k cos ϑ − knm)] ,

(81)

or

A = Ci (r0[k cos ϑ − knm]) ,

(82)

where Ci(x) is the integral cosine13.

Therefore from equation (77), the yield function becomes

Φnm =

µe π2

2 |Bnm, sˆ|2 A2.

(83)

Finally, one averages over all positions of the atom and so the products of pairs of com-

ponents of Bnm all vanish. The averages of squares of components of Bnm become equal

to

1 3

|Pnm|2

where

P

denotes

the

magnitudes

of

the

electrical

moments.

So

one

obtains:

Φnm

=

4µ2e2 32

|Pnm|2

A.

(84)

12The exclusion of central impacts means that we must preliminarily give up the (ability to?) interpret an extremely interesting group of phenomena, namely the transparency of atoms to slow electrons (Ramsauer Eﬀect).
13S.E. Jahnke and F. Emde, Funktionentafeln, Leipzig 1909, S. 19.

19

We want to brieﬂy discuss this expression for the yield function.

First one sees that in our approximation, the yield is proportional to |Pnm|2, ie: the

yield is proportional to the co-eﬃcients of the transition probabilities bnm of the Einstein

radiation theory. These co-eﬃcients correspond to the processes of absorption and stim-

ulated

emission

(but

no

the

probabilities

for

spontaneous

emission

anm

=

8πhνn3m c3

bnm)14.

The yield for elastic reﬂection is proportional to |Pnn|2, a magnitude which is optically unfeasible. In general, the diagonal elements of the matrix Pnm become zero15, except in a few cases where the linear Stark eﬀect exists (such as the hydrogen atom). Herr Pauli has informed me that he can even deduce that the diagonal elements of quadrupole and higher multipole moments vanish for the s-wave states of the alkali metals and for the ground states of the noble gases and alkali-earth metals. This result represents an exact expression for the spherical symmetry of the eﬀective [scattering?] region of the atom. Our approximation is insuﬃcient for the calculation of elastic reﬂections. For this, one must take the approximation a step further. It should soon be possible to test our theory against large quantities of data (Lenard and others) of mean free paths of electons in unexcited gases. Without exact calculation one can see that terms of fourth order in Pnm will determine the yield. These terms are much smaller than |Pnm|2. Thereafter we can see that the cross-section of atoms for slow electrons (which is of the same order of magnitude as that calculated using the kinetic theory of gases) is far smaller than the cross-section for fast electrons (which are capable of exciting the atom)16.

The dependence of the yield on direction is determined by the function A2 from (82). This apparently corresponds to diﬀraction.

W. Elsasser17 drew this conclusion from the de Broglie theory about a year ago. In
taking seriously the wave picture, he concluded that slow electrons must scatter oﬀ an
atom in such a way that their distribution after scattering should correspond approximately to the intensity pattern of light diﬀracting around a small sphere18. He showed the connection between the observations of Ramsauer19 about the mean free path of elec-

14S.J.H. van Vleck, Phys. Rev. 23, 330, 1924; Journ. Opt. Soc. Amer. 9, 27, 1924. M. Born and P.
Jordan, ZS. f. Phys. 33, 479, 1925. 15In the case of the harmonic oscillator for example, they are zero, in the case of the anharmonic
oscillator, they exist. 16This may be found in the literature in the recently published book by J. Franck and P. Jordan,
“Anregung von Quantenspru¨ngen durch St¨osse” (Berlin, J. Springer, 1926). 17W. Elsasser, Die Naturwiss. 13, 711, 1925. The relations between orders of magnitude upon which
Elsasser’s considerations are based, are based on de Broglie’s formula for the wavelength:

λ = 2π = √ h

k

2µW

For 300V radiation, one has roughtly λ = 7 × 10−9cm, in other words, waves of atomic dimensions. 18S.K. Schwartzchild, Sitzungsber. d. Kgl. Bayer. Akad. d. Wiss., S. 293, 1901; G. Mie, Ann d. Phys.
25, 377, 1908; P. Debye, Ann. d. Phys. 30, 57, 1909. 19C. Ramsauer, Ann. d. Phys. 64, 513, 1921; 66, 546, 1921; 72, 345, 1923. For further literature see
“Ergebnisse der exakten Naturwissenschaften”, 3. Bd. (Berlin, J. Springer, 1924), Artikel R. Minkowski

20

trons and the experiments of Davisson and Kunsman20 about the angular distribution of electrons being reﬂected oﬀ a platinum plate. In the mean time, the correctness of these considerations has been proven by the experiments of Dymond21. He directly observed the occurance of interference maxima of electrons reﬂected oﬀ helium. A proof of our formalism from the data will follow later.
10 Final Remarks
Based on the previous considerations, I would like to express the opinion that Quantum Mechanics not only permits the formulation and solution of the problem of stationary states, but also the problem of transition processes. The Schr¨odinger formulation seems to do justice to these problems in the simplest way. Furthermore, it allows us to retain our conventional [ordinary?] view of space and time, in which events take place in a wholly normal manner. However, the proposed theory is not in accordance with the consequences of the causal determinism of single events. I have especially stressed this point about indeterminism in my preliminary publication, since it seems to me to be in agreement with the practical experience of experimenters. It is however natural for anyone who is dissatisﬁed with the above interpretation, to freely assume the existence of further parameters which may be introduced into the theory and which will determine single events. In Classical Mechanics, this appears as the “phase” of motion. For example, the co¨ordinates of a particle at a certain instant. At ﬁrst it seemed to me improbable that one could include physical quantities corresponding to these phases into the new theory. However, Herr Frenkel as informed me that it may be possible after all. In any case, this possibility would not change the practical indeterminism of collision processes, since one cannot give the values of these phases. In addition, this possibility must lead to the same formulas which appear in the suggested “phaseless” theory.
I would like to believe that the laws of motion for light quanta allow themselves to be treated in a completely analogous manner22. Now, immediately with the basic problem of free radiation, we no longer have a periodic process, but rather a decay process. This means an initial value problem rather than a boundary value problem for the for the coupled wave equation of Schr¨odinger’s ψ and the electromagnetic ﬁeld. Understanding the laws of this coupling is probably one of the most urgent problems. This is, as I know, being worked on in more than one place23. When these laws have been formulated, it will perhaps become possible to develop: a rational theory of lifetimes of states, the transition probabilities of radiative processes and the damping and spectral line widths.
and H. Sponer, S. 67. 20Davisson and Kunsman, Phys. Rev. 22, 243, 1923. 21Dymond, Nature. 22The diﬃculties one has encountered so far with the introduction of “ghost ﬁelds” into optics seems
to me to be based partially on the implicit assumption that the wave centre and emitting particle must be in the same place. This is however already not the case with the Compton eﬀect and will probably never be the case in general.
23See for example the recently appearing publication by O. Klein, ZS. f. Phys. 37, 895, 1926.
21

11 Translators’ Notes and Acknowledgments
I would like to thank Andreas Zech and Michael Sindel, without whose valued assistance, this translation would have been impossible.
The purpose of this translation is to analyse the language and analogy used by Born. Hence some of the translation is rather literal and if there is some uncertainty in his language use, I have put alternatives in square parenthesis. If I am still unsure of the best translation, the alternative is accompanied by a question mark. Otherwise the alternatives may have a more elaborative purpose. There are occasions where the exact meaning is not important and in these places I have tended to be loose in translation. I have also taken the liberty of updating some, but not all of the antiquated notation.
22