“ … children at
school should not be introduced to quantum phenomena through any description of
the atom which includes Bohr’s orbits” Mott, Contemporary Physics, 1962
With that qualification from the very reverend Rev
Sir Neville Mott, Nobel Laureate 1977, it seems necessary for me to cite what
Bohr himself said:-
“Every sentence I utter must be
understood not as an affirmation, but as a question.” —
Bohr
It is the possible questions in Bohr’s postulates that we need to be
interested in if not in the model itself.
This series of blogs was actually started with the intention of it being
my beginner’s interpretation of the method behind Bohr’s model for the atom.
Why it was necessary? How it evolved? In the next blog I hope to deal with how it
impacted the development of quantum theory as applied now?
As dwelt with in my previous blog (Part II), quantum theory actually began
with Planck (quantum of action) and was extended by Einstein (light quantum).
The nature of Bohr’s contributions a hundred years ago is directly based on the
work of these two giants. However, it has to be remembered that the impact of
Bohr’s work was so great that the Nobel prize (1922) was awarded to him within
nine years of the publication of his paper. Planck would have to wait 18 years
after his 1900 publication; Einstein would have to wait 20 years. Despite this
success, Bohr’s model has been the only one to have been almost completely
ignored (see left of figure below) in
recent times mainly because of its failure to be generally applicable to atoms
other than hydrogen-like atoms and its failure to be applicable to the simplest
molecule, the hydrogen molecule. But then the Nobel committee has its own ways
and Bohr’s success must be taken as a lesson in the way science is sometimes
recognized.
In the latter part of his life (right of above figure) Bohr worked
mainly on the philosophical aspects of Quantum Mechanics most important of
which perhaps is the “... postulation that
the act of observing something affects the results of that observation” as well as the wave particle duality. There
is much to admire about Bohr and I would love to read Ottavani and Purvis’s
illustrated comic book “Suspended in Language” from which two cartoons have
been taken in the right of the figure above. As a Jew, he escaped from Nazi
Germany in the bomber hold of an allied aircraft. He was also in the secret
Manhattan project where he “...soon
became a security concern, however, since his philosophical and globalist
nature drove him to encourage open sharing of nuclear information with the
Russians.”
Around a hundred years ago, Planck’s evidence for the quantum of action
must have been the immediate scientific quantity seeking an application. Bohr
indeed starts his first paper “On the Constitution of Atoms and Molecules” by
noting the “... inadequacy of the
classical electrodynamics in describing the behaviour of systems of atomic
size...” and that “... it seems
necessary to introduce in the laws in question a quantity foreign to the
classical electrodynamics, i. e. Planck's
constant, or as it often is called the elementary quantum of action … .“
Bohr acknowledges that the “... general
importance of' Planck's theory for the discussion of the behaviour of atomic
systems was originally pointed out by Einstein...”.
The pre-Bohr history on application of Planck’s constant to a model for
the atom seems to be important in understanding the way Bohr applied his quantum
model for the hydrogen atom even though he had many shoulders of others to
stand upon. It nevertheless marks the
origin of the quantum chemical model for the structure of atoms which now
stands in his name,
Early contributions to the development of Bohr’s
model.
Bohr has acknowledged by name the contribution of Arthur Eric
Haas, Bohr writes “The agreement as to the order of magnitude between
values observed for the frequencies and dimensions of the atoms, and values for
these quantities ... was first pointed out by Haas*, in an attempt to explain
the meaning and the value of Planck's constant on the basis of J. J. Thomson's
atom model.”
The remarkable aspect of Haas’s work in 1910 is that it was before Rutherford ’s model.(1911). Haas’s model had a heavy
positively charged nucleus with extranuclear electrons and was based on
alpha-particle scattering.
Haas’s aim, it seems, was to produce a simple
model for a Planck resonator and the one-electron case was the simplest. It
need not have been the hydrogen atom. Haas
actually changed Thomson’s plum pudding model where electrons are fixed in
space and instead modelled a negatively charged electron as revolving around
the surface of the atom with one energy quantum.
The important difference with Haas is that he
considered the spatial dimensions of the atom to be fundamental, instead of the
Planck’s constant. Using the constant of
Balmer’s equation, Haas also correctly derived
the Rydberg constant from the action quantum h, the velocity of light c,
and the fundamental magnitudes of the electron, e and m. He achieved this
relation by a very formal second hypothesis, namely, that the frequency derived
from his quantum rule corresponds with the constant of Balmer’s equation. From this size Haas obtained the value of the
Rydberg constant within a numerical factor of eight. The value of Planck’s
constant could then be expressed it in terms of the mass and radius
of the hydrogen atom.
Arthur Erich Haas is a remarkable figure. Born as a Bohemian,
he stumbled across Thomson’s earlier work while working on his thesis on
history of science and the went on to write a treatise on Physics. Haas later went on to become a founding father of cosmology believing,
like a true Christian --- and other seemingly simple folks --- the age of the
universe to be finite. More interestingly (at least for me) Haas was among the
first to propose that the total energy of the universe is zero having both
positive and negative contributions.
Around this time, Bohr was in Copenhagen Cambridge Manchester
When Bohr joined Rutherford’s group around 1912, phenomena associated
with scattering of charged particles when impacted on matter was of natural
interest, with the focus being naturally on what more information can be
obtained on the constitution of matter from way Rutherford ’s
model of the atom. It is this interest that gave rise to powerful new results.
These included, Geiger and Marsden’s crucial results with alpha particle
scattering that confirmed Rutherford ’s theory;
Van den
Broek’s results on atomic number, Z,
and nuclear electrons; Moseley’s association of frequencies of X-ray radiation
with atomic number; Soddy’s discovery of isotopes of elements and so on. All of
these experiments are the jewels in Rutherford ’s
crown that followed directly from his model for the atom. The most shining
theoretical jewel at that time was Bohr’s planetary model for the hydrogen
atom, which precedes chronologically or is contemporaneous with the discoveries
just mentioned.
It must have been “bliss in that dawn to be alive” in
Rutherford’ laboratory. In Spangenbrg and Moser’s book “Niels Bohr: Atomic
Theorist” we have this description
Each afternoon in the lab, work was
set aside for tea. Rutherford would come in, sit down and talk. The lab group
avidly discussed politics and sports and, of course, work. Ideas always were
exchanged freely at these daily get-togethers. So much was happening in physics
that no one was afraid that someone else would take his idea and publish it
first. There were plenty of vital topics for everyone.
Bohr was little motivated by discussions with Rutherford.
Instead, Bohr’s theoretical/mathematical background drew him to Charles Galton
Darwin, an unusual man who later, at sixty, wrote a scientific treatise on the
evolution in “The Next Million Years”. He was perhaps burdened by the
reputation of his grandfather who had his treatise on the Origin of Species during
the previous Million Years. We cannot really judge the merits of CGDarwin’s book
until a million years have passed since it was written!
The Chapter on Material Conditions in this book, is perhaps
the more easily judged from contemporary results. CGDarwin considered various
environmental and energy issues that is relevant. He reveals his views on, what
seems to be, the empiricism of theoreticians. He writes (about the various ages
of the earth) “…theorists claim to have
given an explanation on astronomical grounds for the recent four ages—but then
if there had been five, might they not have discovered a different but equally
cogent reason for there having been five?” This comment becomes relevant
when one considers, say, the recent debate on the status of Pluto as a planet
from both astronomical and astrological considerations. The more notorious recent
example (perhaps not so recent) for the empiricism of theoreticians must be, of
course, the theories for high-temperature superconductivity! During Darwin’s
time also, there were various attempts to describe atomic structure that must
have hampered his own research, since it could have depended on such knowledge.
Galton Darwin was set by Rutherford
to make a theoretical model for the trajectory of α-particles when traversing matter. Darwin Darwin Darwin
Bohr’s first 1913 paper On the Theory of the Decrease of Velocity of
Moving
Electrified Particles on passing
through Matter in Philosophical Magazine built on CGDarwin’s
work. He stressed the importance of forces “ … by which the electrons are kept in their
positions in the atoms. Under the influence of these forces the electrons will
have a sort of vibratory motion if they are disturbed by an impulse from
outside.” This
seems to be the crucial point in Bohr’s modification of Darwin’s approach.
Thus, as Mott writes, “…quantization
applies to any movement of particles within a confined space, or any periodic
motion, but not to unconfined motion such as that of an electron moving in free
space or deflected by a magnetic field.”
The concept of Planck’s constant or the elementary
quantum of action does not appear in Bohr’s first 2013 paper “On the theory of the decrease of
velocity of moving electrified particles on passing through Matter” although he borrowed concepts
from Planck’s resonator. However, in the first of his 1913 papers “On the Constitution of Atoms and Molecules” Bohr,
like Haas before him, would use Planck’s quantum of action:- “Now the essential point in Planck's theory
of radiation is that the energy radiation from an atomic system does not take
place in the continuous way assumed in the ordinary electrodynamics, but that
it, on the contrary, takes place in distinctly separated emissions, the amount
of energy radiated out from an atomic vibrator of frequency n in a single
emission being equal to thn, where t is an
entire number, and h is a
universal constant”. Such forces “… will materially alter the motion of the electrons during the
collision, and consequently the loss of energy of the particle, if the time of vibration
of the electrons is of the same order of magnitude as the time … the particle takes to travel through a
distance of the same order of magnitude as the shortest distance apart of the
electron from the path of the particle.” The decrease in velocity “… will depend purely on the frequency of the
electrons and the velocity of the particles,”
Bohr acknowledged that such a theory borrows from the
electromagnetic theory of dispersion when the frequency of the light is
replaced “by the different times of
collision of particles of different velocities” .Bohr realized that effects
due to changes in the “frequency” (binding energy) of the “vibratory” electrons
would be more rapid than the loss of velocity of “moving” (free) particles one
could get “some more information about
the internal structure of the atoms.” Bohr then concluded “If we adopt Rutherford 's
conception of the constitution of atoms, we see that the experiments on
absorption of a-rays very strongly suggest, that a hydrogen atom contains only
one electron outside the positively charged nucleus.”
Despite his association with Niels Bohr (see below),
CGDarwin does not mention the word “quantum” in his future million years.
CGDarwin writes that ordinarily “… scientific progress means the discovery of
yet more exact effects produced by exact causes, and … that the
cause-and-effect relation is the sole idea in the scientific method. He then
emphasized that “… a very different new
type of procedure is connected with the principle that the result of a great
number of chances may be far more certain than the result of a few. … This newer type of reasoning is connected
with the principle of probability.” CGDarwin has the
Boltzmann-Planck-Bose models in mind.
Planck’s statistical distribution model for the
scattering of light would help Bohr improve upon Darwin’s model for the
scattering of a-particles by matter from which he came to the very
important conclusion of there being only one electron in the hydrogen atom.
Once this was dealt with, the theoretical problem of the hydrogen atom had no
electron-electron interactions to contend with. Further, this single electron
character of the hydrogen atom allows Bohr to assume “that the orbit in question is circular … for systems containing only a
single electron.” For those few working on the structure of atoms at that
time this would have been a blessing. Bohr would be the first to realize this
and the first to exploit this to account for a structural model for the
one-electron hydrogen atom.
From distinct allowed rotation to the idea of quantized angular momentum
seems to be a logical step. It was Nicholson in 1912 who suggested that the angular momentum assumed values
which were integral multiples of h/2π.
”If, therefore, the constant h of
Planck has, as Sommerfeld has suggested, an atomic significance, it may mean
that the angular momentum of an atom can only rise or fall by discrete amounts
when electrons leave or return. It is readily seen that this view presents less
difficulty to the mind than the more usual interpretation, which is believed to
involve an atomic constitution of the energy itself.”
Nicholson (1912)
Mott, writes in his 1962-Contemporary-Physics article
Bohr-as
every schoolboy knows-made the assumption that the angular momentum L should be
given by L =
nh/2p where n
is an integer. The factor 2p, turning up as it does, often seems
arbitrary, and I do not think it can be explained without quantum mechanics.
Bohr introduced it to obtain agreement with experiment for the energy levels of
hydrogen.
Bohr’s first
paper on the structure of the atom
One of the first lines of the first part of his paper
in which he makes general considerations on the binding of electrons by
positive nuclei, Bohr writes:-
Let us at
first assume that there is no energy radiation. In this case the electron will
describe stationary elliptical orbits.
It is the concept of stationary orbits and not
quantization that seemed to be the most important step. In this Nobel lecture
for “… a formulation of the principles of
the quantum theory that could immediately account for the stability in atomic
structure and the properties of the radiation” he states:-
(I). … there
exist a number of so-called stationary states which, in spite of the
fact that the motion of the particles in these states obeys the laws of
classical mechanics …, possess a peculiar, mechanically unexplainable
stability, of such a sort that every permanent change in the motion of the
system must consist in a complete transition from one stationary state to
another.
(2). … in
contradiction to the classical electromagnetic …, a process of transition
between two stationary states can be accompanied by the emission of
electromagnetic radiation, which will have the same properties as that which
would be sent out according to the classical theory from an electrified
particle executing an harmonic vibration with constant frequency. This
frequency v has, however, no simple relation to the motion of the
particles of the atom, but is given by the relation hv = E’ – E”, where h is Planck’s constant,
and E’ and E” are the values of the energy of the atom in the two
stationary states that form the initial and final state of the radiation
process. Conversely, irradiation of the atom with electromagnetic waves of this
frequency can lead to an absorption process, whereby the atom is transformed
back from the latter stationary state to the former.
Although Bohr agreed with Nicholson that from Planck’s
theory the radiation from an oscillator “is sent out in quanta” such a
radiation cannot be “homogeneous … for, as soon as the emission of radiation is
started, the energy and also the frequency of the system are altered.” This
statement is perhaps a quantum equivalent of the classical argument that as
electrons move about the nucleus it would lose energy and would spiral into the
nucleus.
In order to obtain the binding energy of the hydrogen
atom Bohr had to make two important assumptions. The first of these is that “… the stationary states can discussed by help
of the ordinary mechanics …”. In a classical model, the electron’s
acceleration in the hydrogen atom, is the centripetal acceleration, v2/r , and the
only force (e2/4pe0r2)
acting on the electron is the Coulomb attraction of the proton. From Newton Newton
only hold in calculations of certain mean values of
the motion of the electrons. What is necessary in order to account for
spectroscopic facts is that “in obvious
contrast to the ordinary ideas of electrodynamics” “one need not distinguish between the actual motions and their mean values”.
The second point is that the transition between
different stationary states cannot be treated classically. Here, the transition
leads to emission of a homogeneous radiation
following Planck with hv = E’ – E”.
In order to obtain these energies, one requires a knowledge of the radii, r’ and r”, of the two states between which the transitions take place. Bohr
then used the term of a “permanent” state for the “the one among the stationary
states during the formation of which the greatest amount of energy is emitted.”
This “permanent” state is now referred to as the ground state. The calculation
of the energy required a knowledge of the angular momentum.
Bohr acknowledges Nicholson’s effort in showing “… that the ratios between the wave-length of
different sets of lines of the coronal spectrum can be accounted for with great
accuracy by assuming that the ratio between the energy of the system and the
frequency of rotation of the ring is equal to an entire multiple of Planck's constant.”
Bohr would introduce quantization by stating that “… the angular momentum of the electron round the nucleus in a
stationary state of the system is equal to an entire multiple of a universal
value, independent of the charge on
the nucleus.” It may be argued that such a quantization of angular
momentum is conceptually different from Planck’s quantization of energy.
However, the quantization of angular momentum is equivalent to the quantization
of energy.
The “entire multiple” (integer) by which
the angular momentum is quantized was given the notation, t, by Bohr and it is now known as the principal quantum
number. For the present we continue to use Bohr’s notation, t. The “permanent” or ground state would correspond to t = 1, which has the lowest energy and is mainly
occupied at low temperatures. The radius of the “allowed”
(t =
integer) stationary states was obtained as what is now known as the Bohr
radius. For the hydrogen atom, the Bohr radius, aH, for the ground state (t = 1) is given by aH = (h/2p)2/me2. In general, aH(t) = t2(h/2p)2/me2 From the virial theorem we then obtain the energy E(t=1) for the ground state as E(t=1) = -e2/2aH. In general, Et = -me2/2aH(t) = -2p2me4/2t2h2,
If now we suppose that the radiation in question is
homogeneous, and that the amount of energy emitted is equal to hn, where
n is the frequency of the radiation, we get
Et2 - Et1 = 2p2me4/2h2(1/t22 -1/t12)
or
the frequency, v, of the
“homogeneous” radiation is given as
v = (Et2 - Et1)/h = 2p2me4/2h3(1/t22 -1/t12)
Bohr then concludes
“We see that
this expression accounts for the law connecting lines in the spectrum of
hydrogen. If we put t2 = 2 and let t1 vary, we get the
ordinary Balmer series. If we put t2 = 3, we get the series in the
ultra-red observed by Paschen and previously suspected by Ritz. If we put t2
= 1 and t2 = 4, 5, . . , we get series respectively in the extreme
ultra-violet and the extreme ultra-red, which are not observed, but the
existence of which may be expected.”
Bohr commented also
It may be remarked that the fact,
that it has not been possible to observe more than 12 lines of the Balmer
series in experiments with vacuum tubes, while 33 lines are observed in the
spectra of some celestial bodies, is just what we should expect from the above
theory. According to the equation (for aH(t)) the
diameter of the orbit of the electron in the different stationary states is
proportional to t2. For t = 12 the diameter is equal to 1.6 x 10¯6
cm., or equal to the mean distance between the molecules in a gas at a pressure
of about 7 mm. mercury; for t = 33 the diameter is equal to 1.2 x 10¯5 cm.,
corresponding to the mean distance of the molecules at a pressure of about 0.02
mm. mercury. According to the theory the necessary condition for the appearance
of a great number of lines is therefore a very small density of the gas …
All this would strongly suggest that the more important immediate impact
of Bohr’s contributions has been in accounting for the spectral lines of the
hydrogen atom. He was thought at that time to be a spectroscopist explaining
spectral lines such as the Balmer lines. Even as a spectroscopist, Bohr was not
the first to invoke quantization. In his Nobel lecture Bohr acknowledges Bjerrum’s contributions
on spectra of rotating molecules in which Bjerrum “… emphasized the fact that the effect should not consist of a
continuous widening of the lines such as might be expected from classical
theory, which imposed no restrictions on the molecular rotations, but in
accordance with the quantum theory he predicted that the lines should be split
up into a number of components, corresponding to a sequence of distinct
possibilities of rotation. … and the phenomenon may still be regarded as one of
the most striking evidences of the reality of the quantum theory …” (From
Bohr’s Nobel lecture).
Bohr’s derivation of the formula for the spectral
lines of hydrogen atom is accurate. To date, it is the only quantum mechanical
result that gives accurate values using first principle arguments. Later
developments in wave mechanics through Schrödinger’s wave equation, do not give
such striking agreements from completely ab-initio calculations. The formula is
identical with that obtained by wave mechanics.
Bohr is nevertheless historically associated with the
initiation of a quantized approach to electronic structure of atoms that now
dominates modern theoretical thinking, even if it is now in another
Schrodinger’s-wave-function way. Bohr’s model would have many shortcomings. One
of the earliest was when Rutherford would draw Bohr’s
attention to the Stark effect which showed a splitting of lines in the presence
of an electric field. This effect could not be explained, much as Bohr tried,
using non-elliptical electron orbits that is imposed by Bohr’s use of t
(later known as the principal quantum number, n) alone.
Sommerfield (around 1916) extended Bohr’s model to
include elliptical orbits and magnetic fields he was able to model other
quantum numbers such as the angular momentum quantum number, l (- 0, 1, 2, … (n-1)), and the (2l + 1) magnetic quantum
number, ml (= -l, -(l -1), …(l-1), l), and Pauli added the spin quantum
number, ms to complete the
quantum numbers of the so-called old quantum theory. Such quantum numbers were
mainly designed to explain the spectral lines of atoms, before Schrödinger’s
wave equation (1926) became established as the new quantum mechanics. What is important is that the use of such
quantum numbers does not necessarily require Schrödinger’s wave equation as a
pre-requisite.
The various quantum numbers are a consequence of the
Bohr-Sommerfield model which, with its various quantum numbers, gives some
validity to treating atoms like a microscopic solar system --- each electron
keeping its place through its own unique identification card (its adhaar card, that we Indians like to
complain about) consisting of various quantum numbers.
It was Fermi who first examined this aspect in detail. In the
translation (arXiv:cond-mat/9912229v1
[cond-mat.stat-mech] 14 Dec 1999) of his Italian
paper) of his 1926 paper
On the
Quantization of the Monoatomic Ideal Gas (Rend. Lincei,
3,145-149 (1926)) Fermi writes “It is necessary to
admit that we must add some complements to Sommerfeld’s rules, in the case of
systems, … in which the elements are not distinguishable from each other …like
… atoms heavier than hydrogen … containing more than one electron. The fact
that “…the K ring is already saturated when it contains two electrons, and in
the same way the L ring is saturated when it contains 8 electrons, was
interpreted by Stoner,[ E. C. Stoner, Phil. Mag. 48, 719 (1924)] and even more
precisely by Pauli,[ W. Pauli, Zs. f. Phys. 31, 765 (1925)]. … To realize this
fact, it is sufficient to assume that in the atom there can not be two
electrons with the orbits described by the same quantum numbers; in other words
it is required to admit that an electronic orbit is already ”occupied” when it
contains only one electron.” It was this that gave the Pauli exclusion principle and then the complete
Aufbau (building-up) principle from which the electronic configuration of
atoms could be obtained and the
periodicity of the periodic table could be obtained. It is, however, to Bohr’s
credit that it was he who first introduced the concept of an Aufbau principle based
on two quantum numbers, principal and subordinate.
Bohr’s idea of stationary states as states --- such as
those of a hydrogen atom --- is that in which an electron remained in the same
orbital state for all time, in contrast to that in classical physics which
would require the orbiting electron to radiate away its energy and spiral into
the nucleus. Although Bohr’s original concept is now no longer appropriate, the
notion of a stationary state remains a valid one. The stationary states of such
quantum systems do not change in time in the sense that the probabilities of
outcomes of a measurement of any property of the system is the same no matter
at what time the measurement is made. The cartoon below on self-reference
effect (“tendency of people to effectively recall information about
themselves”) probably
represents one view of this stationary state.
Bohr’s
stationary states really follows Hamilton's principle of classical
physics which means that of all the possible equations of motion of a
mechanical system between two time intervals, the motion will occur along the
curve that gives a stationary value (extreme value of a function where the
derivative is zero) to the action integral which is the integral of the
Lagrangian (difference between kinetic and potential energy) over the time
interval. It is this Hamilton’s principle that was used by Schrödinger in
developing his theory for “undulatory mechanics” (see An undulatory theory on the
mechanics of atoms and molecules” Schrödinger, 1926, Phys. Rev.). The Hamilton
principle comes from the “theory of
propagation of light in a non-homogeneous medium, which... became the starting
point for his (Hamilton’s) famous
theories in pure mechanics.”
The philosophies on quantum mechanics that
arose from Bohr’s model of stationary states follows in the next blog.
