Thought experiments with waves

During the first three decades of the twentieth century when quantum theory was being developed several classical concepts had to be revised. A list of major changes that took places is as follows.

1. Discontinuous nature of physical process such as emission of radiation in Bohr model absorption of radiation in photoelectric effect.
2. Quantization of physical observable quantities, for example angular momentum and energy in Bohr model. In quantum description, the dynamical variables are quantized, in general, they can take only some discrete values.
3. Wave particle duality was an important change in concepts that brought in major changes in the way we think of physical system. The fact that in quantum world both matter and radiation have dual nature had far reaching consequences. However, It must be remembered that the two natures are complimentary and do not manifest themselves in any single experiment ( Bohr complimentary principle).

In addition to the above mentioned changes, the quantum theory brought many new concepts and forced revision of several classical ideas. We recapitulate some important classical concepts which underwent a complete revision after the quantum revolution.

• The classical theories are deterministic, once initial state is specified the motion of the system is deterministic; out come of any measurement can be predicted. The quantum theory is probabilistic; only probabilities of different possible outcomes of experiments can be predicted by the theory.
• In classical theory we associate a well defined trajectory with motion of particles. Waves are not localized and one cannot associate definite trajectories with waves. Properties of particles and waves are incompatible properties.
• In classical mechanics the states of a physical system are described by generalized coordinates and momenta. This changes completely in the quantum theory.
• In classical theories there is no restriction on simultaneous measurement of a pair of variables. Unlike classical theories, a generalised coordinate and canonical conjugate momentum can not be measured to arbitrary accuracy simultaneously. In general two arbitrary dynamical variables cannot be measured simultaneously.
• The classical motion of particle is confined to regions where the total energy is greater than the potential energy. A particle cannot cross a region where the potential energy is higher than the kinetic energy. A quantum particle can tunnel through a barrier, as is the case in alpha decay.
• In quantum world all particles and radiation have dual nature. However, The two natures are complimentary and do not manifest themselves in any single experiment. ( Bohr complimentarity principle)
• In quantum description identical particles cannot be distinguished, they loose their identity.

Our understanding of classical concepts requires a major shift, or even a complete change. In addition many new concepts are brought in by the quantum theory. In addition entire mathematical framework needed for description of quantum phenomena changes. While the mathematics prerequisite for classical mechanics for solution of problems is differential equations and partial differential equations, quantum mechanics brings in Hilbert spaces and probability theory in an essential way. Also note that the kind of questions that are meaningful for a classical system, do not all remain valid questions in quantum mechanics. For example For a classical point particle we may ask for its position and momentum at different times but not for a quantum particle ( which is also a wave ). There are a whole host of new physically meaningful questions that are not asked in the classical physics.\par At this stage you need to remember that above ideas from classical formulations will need a change. Why change is needed, what is the replacement , if any, and all such questions will be dealt with at a suitable stage later, The transition from classical to quantum mechanics is conventionally made by following a route known as canonical quantization. This is the route we will follow. Later you might learn that there are several routes (~10) to quantization. You will understand all this more clearly as you move on and learn the subject. So, by the time we reach the end of this course, you will have understood basics of one of the most common approach to quantum mechanics . Several revisits will be needed for you to gain better and fuller appreciation of the subject. . Before closing, I leave you with a quote from the quantum mechanics book by Landau and Lifshitz p3. 
``Thus quantum mechanics occupies a very unusual place among physical theories; it contains classical mechanics as a limiting case, yet at the same time it requires this limiting case for its own formalism.''

 Think about the statement Landau and Lifshitz make and about `intriguing' relation between the classical and quantum theories.

Ref :  Landau L. D. and E. M. Lifshitz,{\it Quantum Mechanics} Pergamon Press