#import "@preview/physica:0.9.2": *

#import "/book.typ": book-page
#show: book-page.with(title: "Hello, typst")

// = The Principle of Superposition

= The need for a quantum theory

#smallcaps[Classical] Mechanics has been developed continuously from the time of Newton and applied to an ever-widening range of dynamical systems,
including the electromagnetic field in interaction with matter.
//
The underlying ideas and the laws governing their application form a simple and elegant scheme,
which one would be inclined to think could not be seriously modified without having all its attractive features spoilt.
//
Nevertheless it has been found possible to set up a new scheme,
called quantum mechanics,
which is more suitable for the description of phenomena on the atomic scale and which is in some respects more elegant and satisfying than classical scheme.
//
This possibility is due to the changes which the new scheme involves being of a very profound character and not clashing with the features of the classical theory that make it so attractive,
as a result of which all these features can be incorporated in the new scheme.

The necessity for a departure from classical mechanics is clearly shown by experimental results.
//
In the first place the forces known in classical electrodynamics are inadequate for the explanation of the remarkable stability of atoms and molecules,
which is necessary in order that materials may have any definite physical and chemical properties at all.
//
The introduction of new hypothetical forces will not save the situation,
since there exist general principles of classical mechanics,
holding for all kinds of forces,
leading to results in direct disagreement with observation.
//
For example,
if an atomic system has its equilibrium disturbed in any way and is then left alone,
it will be set in oscillation and the oscillations will get impressed on surrounding electromagnetic field,
so that their frequencies may be observed with a spectroscope.
//
Now whatever the laws of force governing the equilibrium,
one would expect to be able to include the various frequencies in scheme comprising certain fundamental frequencies and their harmonics.
//
This is not observed to be the case.
//
Instead,
there is observed a new and unexpected connexion between the frequencies,
called Ritz's Combination Law of Spectroscopy,
according to which all the frequencies can be expressed as differences between certain terms,
the number of terms being much less than the number of frequencies.
//
This law is quite unintelligible from the classical standpoint.

One might try to get over the difficulty without departing from classical mechanics by assuming each of the spectroscopically observed frequencies to be a fundamental frequency with its own degree of freedom,
the laws of force being such that the harmonic vibrations do not occur.
//
Such a theory will not do,
however, even apart from the fact that it would give no explanation of Combination Law,
since it would immediately bring one into conflict with the experimental evidence on specific heats.
//
Classical statistical mechanics enables one to establish a general connexion between the total number of degrees of freedom of an assembly of vibrating systems and its specific heat.
//
If one assumes all the spectroscopic frequencies of an atom to correspond to different degrees of freedom,
one would get a specific heat for any kind of matter very much greater than the observed value.
//
In fact the observed specific heats at ordinary temperatures are given fairly well by a theory that takes into account merely the motion of each atom as a whole and assigns no internal motion to it at all.

This leads us to a new clash between classical mechanics and the results of experiment.
//
There must certainly be some internal motion in an atom to account for its spectrum,
but the internal degrees of freedom,
for some classically inexplicable reason,
do not contribute to the specific heat.
//
A similar clash is found in connexion with the energy of oscillation of the electromagnetic field in a vacuum.
//
Classical mechanics requires the specific heat corresponding to this energy to be infinite,
but it is observed to be quite finite.
//
A general conclusion from experimental results is that oscillations of high frequency do not contribute their classical quota to the specific heat.


As another illustration of the failure of classical mechanics we may consider the behaviour of light.
//
We have, on the one had, the phenomena of interference and diffraction,
which can be explained only on the basis of a wave theory;
on the other, phenomena such as photo-electric emission and scattering by free electrons,
which show that light is composed of small particles.
//
These particles,
which are called photons,
have each a definite energy and momentum,
depending on the frequency of light,
and appear to have just as real an existence as electrons,
or any other particles known in physics.
//
A fraction of photon is never observed.


Experiment have shown that this anomalous behaviour is not peculiar to light,
but is quite general.
//
All material particles have wave properties,
which can be exhibited under suitable conditions.
//
We have here a very striking and general example of the breakdown of classical mechanics---not merely an inaccuracy in its laws of motion but #underline[an inadequacy of its concepts to supply us with description of atomic events.]


The necessity to depart from classical ideas when one wishes to account for the ultimate structure of matter may be seen,
not only from experimentally established facts,
but also from general philosophical grounds.
//
In a classical explanation of the constitution of matter,
one would assume it to be made up of a large number of small constituent parts and one would postulate laws for the behaviour of these parts,
from which the laws of the matter in bulk could be deduced.
//
This would not complete the explanation,
however,
since the question of the structure and stability of the constituent parts is left untouched.
//
To go into this question,
it becomes necessary to postulate that each constituent part is itself made up of smaller parts,
in terms of which its behaviour is to be explained.
//
There is clearly no end to this procedure,
sot that one can never arrive at the ultimate structure of matter on these lines,
//
So long as #underline[big] and #underline[small] are merely relative concepts,
it is no help to explain the big in terms of the small.
//
It is therefore necessary to modify classical ideas in such a way as to give an absolute meaning to size.


At this stage it becomes important to remember that science is concerned only with observable things and that we can observe an object only by letting it interact with some outside influence.
//
An act of observation is thus necessarily accompanied by some disturbance of the object observed.
//
We may define an object to be 
big when the disturbance accompanying our observation of it may be neglected,
and small when the disturbance cannot be neglected.
//
This definition is close agreement with common meanings of big and small.

It is usually assumed that,
by being careful,
we may cut down the disturbance accompanying our observation to any desired extent.
//
The concepts of big and small are then purely relative and refer to the gentleness of our means of observation as well as to the object being described.
//
In order to give an absolute meaning to size,
such as is required for any theory of the ultimate structure of matter,
we have to assume that #underline[there is a limit to the fineness of our powers of observation and the smallness of the accompanying disturbance---a limit which is inherent in the nature of things and can never be surpassed by improved technique or increased skill on the part of the observer.]
//
If the object under observation is such that the unavoidable limiting disturbance is negligible,
then the object is big in the absolute sense and we may apply classical mechanics to it.
//
If, on the other hand,
the limiting disturbance is not negligible, 
then the object is small in the absolute sense and we require a new theory for dealing with it.


A consequence of the preceding discussion is that we must revise our ideas of causality.
//
Causality applies only to a system which is left undisturbed.
//
If a system is small,
we cannot observe it without producing a serious disturbance and hence we cannot expect to find any causal connexion between the results of our observations.
//
Causality will still be assumed to apply to undisturbed systems and the equations which will be set up to describe an undisturbed system will be differential equations expressing a causal connexion between conditions at one time and conditions at a later time.
//
These equations will be in close correspondence with the equations of classical mechanics,
but they will be connected only indirectly with the results of observations.
//
There is an unavoidable indeterminacy in the calculation of observational results,
the theory enabling us to calculate in general only the probability of our obtaining a particular result when wen make an observation.


