[NOTES/QM-06001] Why Start with Postulates?

The inadequacy of classical theory and efforts to explain the observed physical phenomena led to a major revision of classical concepts and of the athematical structure. Some of the earliest points of departure from the classical theory were

i) discrete nature of physical processes, as in Planck’s hypothesis and

(ii) quantisation of dynamical variables}, for example, angular momentum in Bohr’s theory.

The wave particle duality had far reaching consequences and has changed the way we understand and do physics. Here we will briefly highlight the important points that are best seen to emerge from implications of wave particle duality on thought experiments.

In a double slit interference experiment for electrons, because of the dual nature we would expect to see an interference pattern. However, when the intensity of the incident beam is reduced to single electron at a time, we are inclined to conclude that one would see a spot on the screen, and not an interference pattern. Still an interference pattern appears  when the experiment is repeated with a large number of times. This can be understood only by associating a {\it probability amplitude} of electron reaching a point on the screen. It is accepted that the a prediction about exact location of the spot for in experiment with single electron is impossible and indeterminacy  has entered in an essential way.

On similar lines, an analysis of the thought experiments on photon polarization in the limit of low intensities involving single photon  leads to the {\it indeterminacy} of polarization state of a photon. This in turn  introduces  indeterminacyin outcomes of polarization measurements. Again we are led to {\it probabilities}  being associated with the out come of measurements of physical quantities. The indeterminacy is  seen to  arise due the  state of the photon being superposition of two different states of polarization. It has to emphasized that the suggestion here is that indeterminacy is of a fundamental nature and not due to some  limitation of of the quantum theory or of the measuring apparatus.

An analysis of  Heisenberg microscope thought experiments to {\it measure position and momentum of particles  simultaneously} leads to the Heisenberg uncertainty principle. It asserts that the position and momentum of a particle cannot be measured simultaneously. This is forced  due to the observed dual nature of electrons and  photons.

Thus we are naturally led to ask the following, and many more, questions.

• If we have to incorporate the superposition principle of quantum states, how are the states and dynamical variables are to be described in quantum theory?
• If some dynamical variables are quantized, how are we to compute the allowed values of dynamical variables?
• Acceptance of indeterminacy leads us to ask what is the scheme to compute the probabilities of different possible outcomes of an experiment?
• The uncertainty principle  is an out come of Heisenberg microscope thought experiment and a consequence of wave particle duality. This analysis gives us no clue about which other pairs of dynamical variables cannot be measured simultaneously. In fact we do not have a precise mathematical formulation of the uncertainty principle; it is not clear ``How is the uncertainty in a dynamical variable even defined precisely?''

There are several different ways of formulating quantization applying to physical systems. The postulates  will in general  appear different in different scheme. Remember that the postulates, to be described below, give us one of several possible schemes, of 'doing' quantum mechanics.

 Learn from Masters:
We strongly recommend that the reader should go through a few pages of a terse but an illuminating discussion of superposition principle, indeterminacy, simultaneous measurement, and related issues from page 5, the very first section of the first chapter of Landau Lifshitz. A 'teaser' from of this part of the book, reproduced below, should serve as an inducement for going to the original book.