The effect of an external electric field on the hydrogen lines

Epstein and Schwarzschild have given a detailed theory of the effect of an external homogeneous electric field on the hydrogen spectrum, discovered by Stark.

Their basis was the general theory of conditionally periodic systems which allow of separation of variables.

This problem may solved simply through the considerations about perturbed periodic systems, developed in Section 2.

Consider an electron of mass m and charge −e. This rotates around a positive nucleus of infinite mass and of charge Ne.

It is subject to a homogeneous electric field of intensity F.

We shall neglect the small effect of the Relativity modifications.

Using rectangular coordinates, and taking the nucleus as origin and the z-axis parallel to the external field, we get for the potential of the system relative to the external field, omitting an arbitrary constant, Ω = eF z.

Calculating now the mean value of Ω over a period σ of the undisturbed motion, we see at once, from considerations of symmetry, that this mean value ψ will depend only on the component of the external electric force in the direction of the major axis of the orbit. We have therefore

…

where:

• ϕ is the angle between the z-axis and the major axis, taken in the direction from the nucleus to the aphelium, and
• r is the length of the radius-vector from the nucleus to the electron, and ϑ the angle between this radius-vector and the major axis. By means of the well known equations for a Keplerian motion

…

where:

• 2α is the major axis, ε the eccentricity and u the so called eccentric anomaly, this gives

…

Thus, ψ is equal to the potential energy relative to the external field, which the system would possess, if the electron was placed at a point, situated on the major axis of the ellipse and dividing the distance 2εa between the foci in the ratio 3 : 1. This point may be denoted as the “electrical centre” of the orbit.

From the approximate constancy of ψ during the motion, proved in § 2, it follows therefore in the first place that, with neglect of small quantities of the same order of magnitude as the ratio between the external force and the attraction from the nucleus, the electrical centre will during the perturbations of the orbit remain in a fixed plane perpendicular to the direction of the external force.

From the considerations in § 2 it follows further, that the total energy in the stationary states of the system in the presence of the field, with neglect of small quantities proportional to F, will be equal to En + ψ, where En is the energy of the hydrogen atom in its undisturbed stationary state.

Since both ε and cos ϕ are numerically smaller than one, we obtain therefore at once from (74) a lower and an upper limit for the possible variations of the energy in the stationary states, due to the field. Introducing from (41) the values of En and αn, and neglecting, here as well as in the following calculations in this section, the small correction due to the finite mass of the nucleus—not only in the expression for the additional energy but, for the sake of brevity, also in the main term—we get for these limits

which formula coincides with the expression previously deduced by the writer by applying the condition I = nh to the two (physically not realisable) limiting cases, corresponding to z = 1 and cos ϕ = ±1, in which the orbit remains periodic in the presence of the field.

1. In order to obtain further information as to the values of the energy in the stationary states in the presence of the field, it is necessary to consider more closely the variation of the orbit during the perturbations. Since the external forces possess axial symmetry, the problem of the stationary states might be treated by means of the procedure indicated in § 2 on page 107. In the present special case, however, the stationary states of the atom may be very simply determined, due to the fact that the secular perturbations are simply periodic independent of the initial shape and position of the orbit, so that we are concerned with a degenerate case of a perturbed periodic system. This property of the perturbations follows already from some calculations given by Schwarzschild

2. See N. Bohr, Phil. Mag. XXVII, p. 506 (1914) and XXX, p. 394 (1915). Compare also E. Warburg, Verh. d. D. Phys. Ges. XV, p. 1259 (1913), where it was pointed out, for the first time, that the effect of an electric field on the hydrogen lines to be expected on the quantum theory was of the same order of magnitude as the effect observed by Stark.

3. in a previous attempt to explain the Stark effect of the hydrogen lines, without the help of the quantum theory, by means of a direct consideration of the harmonic vibrations into which the motion may be resolved, according to the analytical theory of conditionally periodic systems. Starting from the above result, that the electrical centre moves in a plane perpendicular to the direction of the external field, the periodicity of the perturbations may also be proved in the following way, by means of a simple consideration of the variation of the angular momentum of the electron round the nucleus, due to the effect of the external electric force.

Using again rectangular coordinates with the nucleus at the origin and the z-axis parallel to the direction of the electric force, and calling the coordinates of the electrical centre ξ, η, ζ, we have

1. K. Schwarzschild, Verh. d. D. Phys. Ges. XVI, p. 20 (1914).

according to formula (74)

Denoting the components parallel to the x y and z-axis of the angular momentum of the electron round the nucleus, considered as a vector, by Px, Py and Pz, we have next

Since the angular momentum is perpendicular to the plane of the orbit, we have further ξPx + ηPy + ζPz = 0. (3∗)

Now we have for the mean values of the rates of variation of Px and Py with the time

From this we get, differentiating (1∗) and (2∗) with respect to the time, and remembering that α and ω remain constant during the perturbations,

where

On the other hand we have, differentiating (3∗ ) and introducing (4∗),

which together with (5∗) gives

from which we get, by means of (4∗),

the solution of which is ξ = A cos 2π(vt + a), η = B cos 2π(vt + b), (7∗) where A, a, B and b are constants, and where, introducing (6∗), we have

During the perturbations the electrical centre will thus perform slow harmonic vibrations perpendicular to the direction of the electric force, with a frequency which is proportional to the intensity of the electric field, but, for a given value of F, quite independent of the initial shape of the orbit and its position relative to the direction of the field.

For the value of this frequency in the multitude of states of the perturbed system, for which the mean value of the inner energy is equal to the energy En in a stationary state of the undisturbed system corresponding to a given value of n, we get from the above calculation, introducing for α and ω the values of αn and ωn given by (41),

Now from the periodic motion of the electrical centre we may conclude that, in the presence of the field, the system will be able to emit or absorb a radiation of frequency vF , and that accordingly the possible values of the additional energy of the system in the presence of the field will be given directly by Planck’s fundamental formula (9), holding for the possible values of the total energy of a linear harmonic vibrator, if in this formula ω is replaced by the above frequency vF.

Since further a circular orbit, perpendicular to the direction of the electric force, will not undergo secular perturbations during a slow establishment of the field, and therefore must be included among the stationary states of the perturbed system, we get for the total energy of the atom in the presence of the field\

where:

• n is an entire number which in the present case may be taken positive as well as negative.

From a comparison between (75) and (77), we see that the presence of the external field imposes the restriction on the motion of the atom in the stationary states, that the plane in which the electrical centre of the orbit moves must have a distance from the nucleus

equal to an entire multiple of the n th part of its maximum distance 3 2 αn.

The result, contained in formula (77), is in agreement with the expression for the total energy in the stationary states, deduced by Epstein and Schwarzschild by means of the general theory of conditionally periodic systems based on the conditions (22).

The treatment of these authors rests on the fact, that, as mentioned in Part I, the equations of motion for the electron in the present problem may be solved by means of separation of variables in parabolic coordinates (compare page 36).

Taking for q1 and q2 the parameters of the two paraboloids of revolution, which pass through the instantaneous position of the electron and which have their foci at the nucleus and their axes parallel to the direction of the field, and for q3 the angular distance between the plane through the electron and the axis of the system and a fixed plane through this axis, the momenta p1, p2, p3 will during the motion depend on the corresponding q’s only, and the stationary states will be fixed by three conditions of the type (22). With neglect of small quantities proportional to higher powers of F, the final formula for the total energy, obtained by Epstein in this way, is given by:

where:
- n1, n2, n3 are the positive entire numbers which occur as factors to Planck’s constant on the right sides of the
mentioned three conditions. 

As regards the possible values of the total energy of the hydrogen atom in the presence of the electric field, it will be
seen that (78) coincides with (77) if we put n1 + n2 + n3 = n
and n1 − n2 = n. At the same time it will be observed, however, that the motion in the stationary states, as fixed by the
procedure followed by Epstein, is more restricted than was
necessary in order to secure the right relation between the
additional energy and the frequency of the secular perturbations. Thus, in addition to the condition which fixes the
plane in which the electrical centre moves, Epstein’s theory
involves the further condition, that the angular momentum
of the electron round the axis of the perturbed system is
equal to an entire multiple of h/2π; which multiple is seen
to be even or uneven, according as n+n is an even or an uneven number respectively. This circumstance is intimately
connected with the fact that, although the perturbed system under consideration is degenerate if we look apart from
small quantities proportional to the square of the intensity
of the external force, the degenerate character of the system
does not reveal itself from the point of view of the theory
of stationary states based on the conditions (22), because
the system under consideration allows of separation of variables only in one set of positional coordinates. On the other
1
) P. Epstein. Ann. d. Phys. L, p. 508 (1916).
146
hand, this degenerate character of the system has been emphasised by Schwarzschild1
) on the basis of the theory
of stationary states based on the introduction of angle variables, in which the periodicity properties of the motion play
an essential part. In a later discussion of this point Epstein2
) calls attention to the fact that, if small quantities
proportional to the square of the electric force are taken into
account, the system appears no more as degenerate; and he
finds therein a justification of the fixation of the stationary
states by means of (22). From the point of view of perturbed
systems, this would mean that the motion in the stationary
states of the system in question, as fixed by (22), would certainly be stable for infinitely small disturbances, but that
we should expect finite deviations from the motion in these
states, already if the system was exposed to a second perturbing field, the intensity of which was only of the same
order as the product of the external electric force with the
ratio between this force and the attraction from the nucleus.


## The influence of the relativity modifications

The stability of the motion in the stationary states, as determined by (22), is actually often much higher.

The external force has to be stronger to cause finite deviations from this motion. 

This is because the external force is of the same order as the product of the attraction from the nucleus with the square of the ratio of the electron's speed and the light's speed.

The resulting formula (78) needs to explain 2 things:

1. The effect on the motion of the electron due to the small modifications in the laws of mechanics claimed by Relativity
2. The effect of possible forces which might act on the electron, corresponding to the reaction from the radiation in ordinary electrodynamics. 


<!-- ) K. Schwarzschild, Ber. Akad. Berlin, 1916, p. 548.
2
) P. Epstein, Ann. d. Phys. LI, p. 168 (1916). -->



<!-- To this point we shall come back at the end of this section, when considering the simultaneous perturbing influence on the motion of the election in the hydrogen atom, due to the relativity modifications and an external electric field. -->




If, however, for the moment we exclude all stationary states for which the angular momentum round the axis of the system
would be equal to zero (n3 = 0), the total angular momentum of the electron round the nucleus will during the perturbations always remain larger than or equal to h/2π, just as in the stationary states considered in the theory of the fine
structure; and, according to the considerations on page 128,
we shall therefore expect that the effect of the neglect of
possible “radiation” forces will be small compared with the
effect of the relativity modifications. On the other hand, if
the intensity of the electric field is of the same order of magnitude as that applied in Stark’s experiments, the effect of
these modifications must again be expected to be very small
compared with the total effect of the electric force on the hydrogen lines, since the perturbing effect of this force on the
Keplerian motion of the electron will be very large compared
with the corresponding effects of the relativity modifications.
148
If, on the contrary, we would consider a state of the atom
for which n2 was equal to zero, the orbit would be plane and
would during the perturbations assume shapes, for which the
total angular momentum round the nucleus was very small,
and in which the electron during the revolution would pass
within a very short distance from the nucleus. In such a
state the effect of the relativity modifications on the motion of the electron would be considerable, but quite apart
from this a rough calculation shows that the amount of energy, which, on ordinary electrodynamics, would be emitted
during the intervals in which the angular momentum during
the perturbations of the orbit remains small, is so large that
it would hardly seem justifiable to calculate the motion and
the energy in these states by neglecting all forces corresponding to the radiation forces in ordinary electrodynamics. We
need not, however, enter more closely on these difficulties,
because, on the general considerations in Part I about the
a-priori probability of the different stationary states, we are
forced to conclude that, for any value of the external electric field, no state which would correspond to n3 = 0 will
be physically possible; since any such state might be transformed continuously, and without passing through a degenerate system, into a state which obviously cannot represent
a physically realisable stationary state (compare page 49).
In fact, if we imagine that an external central field of force,
varying as the inverse cube of the distance from the nucleus,
is slowly established, it would be possible to compensate the
secular effect of the relativity modifications and to obtain
149
orbits in which the electron would pass within any given,
however small, distance from the nucleus. As regards the
other stationary states fixed by (22), which correspond to
n3 > 1, we shall according to the considerations in Part I
expect that their a-priori probabilities are all equal.1
)
1
) By a simple enumeration it follows from this result, that the total
number of different stationary states of the hydrogen atom, subject to
a small homogeneous electric field, which corresponds to a stationary
state of the undisturbed atom, characterised by a given value of n in the
condition I = nh, is equal to n(n + 1). This expression is directly obtained, if we remember that n = n1+n2+n3 and if we count each state,
characterised by a given combination of the positive integers n1, n2, n3,
as double, corresponding to the two possible opposite directions of rotation of the electron round the axis of the field. With reference to
the necessary stability for a small variation of the external conditions
of the statistical distribution of the values of the energy among a large
number of atoms in temperature equilibrium (see Note on page 82), it
will be seen that the expression n(n+ 1) may be taken as a measure for
the relative value of the a-priori probability of the different stationary
states of the undisturbed hydrogen atom, corresponding to different values of n. The problem of the determination of this a-priori probability
has been discussed by K. Herzfeld (Ann. d. Phys. LI, p. 261 (1916))
who, by an examination of the volumes of the different extensions in
the phase space which might be considered as belonging to the different
stationary states of the hydrogen atom, has arrived at an expression
for the a-priori probability of these states which differs from the above.
From the point of view, as regards the principles of the quantum theory, taken in the present paper, a consideration of this kind, however,
does not, as explained in Part I on page 47, afford a rational means
of determining the a-priori probability of the stationary states of an
atomic system.
150
As regards the comparison between the theory and the
experiments, it will be remembered that Stark found that
every hydrogen line in the presence of an electric field was
split up in a number of polarised components, in a way different for the different lines. When viewed parallel to the direction of the field, there appeared a number of components
polarised parallel to the field and a number of components
polarised perpendicular to the field; when viewed in the direction of the field, only the latter components appeared,
but without showing characteristic polarisation. Apart from
the marked symmetry of the resolution of every line, the
distances between successive components and their relative
intensities varied in an apparently irregular way from component to component. As pointed out by Epstein and
Schwarzschild, however, it is possible by means of (78),
in connection with relation (1), to account in a convincing
way for Stark’s measurements as regards the frequencies
of the components. Especially a closer examination of these
measurements showed that all the differences between the
frequencies of the components were equal to entire multipla
of a certain quantity, which was the same for all lines in the
spectrum and, within the limits of experimental errors, equal
to the theoretical value 3hF
8π
2Nem
. On the other hand, the
theories of Epstein and Schwarzschild gave no direct information as regards the question of the polarisation and intensity of the different components. Comparing formula (78)
with Stark’s observations, Epstein pointed out, however,
151
that the polarisation of the different components observed
could apparently be accounted for by the rule: that a transition between two stationary states gives rise to a component
polarised parallel to the field, if n3 remains unchanged or
is changed by an even number of units; while a component,
corresponding to a transition in which n2 is changed by an
uneven number of units, is polarised perpendicular to the
field. This result may be simply interpreted on the basis of
the general formal relation between the quantum theory of
line spectra and the ordinary theory of radiation. In fact, it
was shown in Part I that, for a conditionally periodic system possessing an axis of symmetry, we shall expect only
two types of transitions to be possible. In transitions of
the first type n3 remains unchanged, and the emitted radiation is polarised parallel to the axis of symmetry, while
the transitions of the second type, in which n3 varies by
one unit, give rise to a radiation of circular polarisation in a
plane perpendicular to this axis (see page 64). In order to
show that this agrees with the empirical rule of Epstein,
it may be noted in the first place that, for any component
which might be ascribed to a certain transition in which
n3 changes by a given entire number of units, there exists
always another transition which will give rise to a radiation
of the same frequency but in which n3 remains unchanged or
changes by one unit, according to whether the given number is even or uneven. Next it will be seen that, in case of
the effect of an electric field on the hydrogen spectrum, we
cannot detect by means of direct observations the circular
152
polarisation of the radiation corresponding to transitions of
the second type; because, for each transition giving rise to a
radiation of circular polarisation in one direction, there will
exist another transition giving rise to a radiation which possesses the same frequency but is polarised in the opposite
direction. Besides on the problem of the polarisations of the
different components into which the hydrogen lines are split
up in the presence of the electric field, the general considerations in Part I allow also to throw light on the question of the
relative intensities of these components, by considering the
harmonic vibrations into which the motion of the electron
in the stationary states can be resolved. Compared with the
problem of the relative intensities of the components of the
fine structure of the hydrogen lines, the present problem is
simpler in that respect, that the stationary states may be
assumed to be a-priori equally probable. Since the different
components, into which a given hydrogen line is split up in
the electric field, correspond to transitions between pairs of
states which for all components have very nearly the same
values for the total energy, these states may therefore be
expected to be of approximately equal occurrence in the luminous gas. According to the considerations in Part I, we
shall consequently assume that for a given hydrogen line the
relative intensities of the different Stark effect components,
corresponding to transitions between different pairs of stationary states characterised by n1 = n
0
1
, n2 = n
0
2
, n3 = n
0
3
and n1 = n
00
1
, n2 = n
00
2
, n3 = n
00
3
respectively, will be intimately connected with the intensities of the radiations of
153
frequency (n
0
1 − n
00
1
)ω1 + (n
0
2 − n
00
2
)ω2 + (n
0
3 − n
00
3
)ω3, which on
ordinary electrodynamics would be emitted by the atom in
the two states involved in the transition in question; ω1, ω2,
and ω3 being the fundamental frequencies entering in the expression (31) for the displacement of the electron. In order
to test how far such a connection is actually brought out by
the observations, it is necessary to determine the numerical values of the amplitudes of the harmonic vibrations into
which the motion of the electron can be resolved. The examination of this problem has been undertaken by Mr. H. A.
Kramers, who has deduced complete expressions for these
amplitudes, by means of which it was found possible, for each
of the hydrogen lines Hα, Hβ, Hγ and Hδ, to account in a
convincing way for the apparently capricious laws which govern the intensities of the components observed by Stark.
1
)
This agreement offered at the same time a direct experimen1
) Note added during the proof. In recent papers H. Nyquist (Phys.
Rev. X, p. 226 (1917)) and J. Stark (Ann. d. Physik, LVI, p. 569
(1918)) have published measurements on the effect of an electric field
on certain lines of the helium spectrum which is given by (35), if in (40)
we put N = 2. As will be seen from (78), the differences between the
frequencies of the components into which these lines are split up will,
for the same intensity of the external electric field, be smaller than
for the hydrogen lines. In conformity with this it was not possible,
with the experimental arrangement used by the authors mentioned, to
observe separately the numerous components to be expected on the
theory, but only to obtain certain rough features of the resolution of
the lines in question. For the interpretation of these observations a
detailed consideration of the relative intensities to be expected for the
different theoretical components is therefore essential; and, as it will be
154
tal support for the conclusions mentioned above: that there
exist no stationary states corresponding to n3 = 0, while the
stationary states corresponding to other values of n2 are apriori equally probable; and that transitions can only take
place between pairs of stationary states for which n3 is the
same or differs by one unit. A general discussion of these
problems will be given by Kramers in the paper, mentioned
on page 136 in the last section, in which also the problem of
the intensity of the fine structure components is treated in
detail.
In the former section and in the present we have seen, how
the problems of the influence of the relativity modifications
on the lines of the hydrogen spectrum and of the influence of
an external electric field on this spectrum can be treated, by
regarding the motion of the electron as a perturbed periodic
motion, and by fixing the stationary states on the basis of the
relation between the energy and the frequencies of the secular
perturbations. As it was done originally by Sommerfeld
and Epstein, both these problems can also be treated by
means of the theory of the stationary states of conditionally
periodic systems which allow of separation of variables in a
fixed set of positional coordinates. If, however, we consider
the problem of the simultaneous influence on the hydrogen
spectrum of the relativity modifications and a homogeneous
shown in Kramers’ paper, it is possible, on the basis of the calculation
of the amplitudes of the harmonic vibrations into which the motion
of the electron in the stationary states can be resolved, to account
satisfactorily for Nyquist’s and Stark’s results.
155
electric field of any given intensity, there does not exist a
set of coordinates for which a separation of variables can be
obtained. On the other hand it is possible, also in this case,
to apply the general considerations about perturbed periodic
systems developed in the preceding. In fact, with reference
to the treatment given in § 3 of the problem of the fine structure of the hydrogen lines, it will be seen that the deviations
of the orbit of the electron from a Keplerian ellipse in the
problem under consideration will be the same as the secular
perturbations produced on a Keplerian motion by the simultaneous influence of an external homogeneous field of force
and an external central force proportional to the inverse cube
of the distance from the nucleus. Since these two fields together form a perturbing field possessing axial symmetry, it
follows therefore that the secular perturbations, when the
relativity modifications are taken into account, will be conditionally periodic and that the problem of the stationary
states may be treated by means of the method mentioned
in § 2 on page 107. In this way we obtain in the first place
the result, that, for any value of the intensity of the external
electric field, we must expect that the hydrogen lines will be
split up in a number of sharp components. Next, since for
any value of this intensity different from zero the system will
be non-degenerate, it follows from the conditions (61), that
we must assume that the angular momentum round the axis
of the field is always equal to an entire multiple of h/2π; in
consistence with the assumption of the validity of the analogous condition involved in the fixation of the stationary
156
states by means of the method of separation of variables,
when applied to an explanation of the Stark effect with
neglect of the relativity modifications (compare page 145).
On the basis of the conditions (61) it is possible to predict
in detail, how the fine structure of the hydrogen lines will
be influenced by an increasing electric field until, for a sufficiently large intensity of this field, the phenomenon develops
gradually into the ordinary Stark effect. The problem of
this transmutation will be treated in a later paper by Mr.
H. A. Kramers,
1
) who has kindly drawn my attention to
this interesting application of the method of perturbations,
and has thereby given a valuable impetus to the detailed
elaboration of this method as regard the treatment of more
complicate problems.