In view of the increasing number students facing problems with concepts of Quantum Mechanics, I have decided to start a series of articles where I'll try to present these concepts in the simplest manner possible without compromising with the accuracy and subtle details. I will try to demonstrate how the concept of wavefunctions (or wavepackets) enters into picture and what do we actually mean by the Wave - Particle Duality (this is the starting point of all the confusion and is completely unnecessary). I'll discuss the meaning of abstract quantities like spin quantum number, angular momentum quantum number, what do they actuaaly mean and where do they come from, etc.
Mathematical Nature Of Physics
Quantum Mechanics is considered as one of the most fascinating subjects in the field of academics often because of its abstruse concepts which people find difficult to reconcile. This difficulty is mostly because of the misconstrued concepts and premeditated thoughts about the working of nature and its representation in physics. Therefore, before going into the subject, it is necessary for one to know the basic nature of physics, what it is and how should one proceed to understand it.
Physics is essentially a mathematical theory of nature. By this I mean that everything you encounter in physics has a mathematical origin constructed in order to suffice explanation to a problem or a set of problems at hand. We use mathematics as a tool, more like a language, to explain the observed physical processes in nature or in lab. Why is this preference given to mathematics over the languages we use in our daily lives? Wouldn’t it be a lot easier to use our regular language with which we are already familiar?
The answer to these questions lies in the construction of mathematics. Mathematics is based on some definite self-evident postulates also called axioms and developed thereafter by logic, inductive and deductive reasoning. This logical development leaves is no room for ambiguity caused by human intuition, emotions and premeditated thoughts, something that our regular languages fail to do as it was developed for the purpose of expressing our emotions. This kind of unambiguous environment is necessary to work with complicated problems where our intuitive capabilities fail to produce meaningful conclusions in accord with the observed results.
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| An atom : A classical picture (this is wrong) |
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| An atom : A quantum picture (we will discuss in detail where do they come from) |
Another advantage of mathematics, over other languages, is that it is possible to develope methods accomodating a large array of logical arguments which can be used to solve multifarious and apparently complicated problems which our brains may not (and usually are not) be able to handle intuitively. For instance, we find it easy to calculate the area of a square or a rectangle - product of sides, but what about the area of a circle. It is not possible to calculate it intuitively and our ancestors struggled to solve it, but can be readily obtained with ease using calculus. As a matter of fact, surface areas and volumes for most of the symmetrical figures can be obtained easily with calculus, all we are required to do is proceed with the logic and already existing methods of the calculus. The same laws of calculus are used in classical mechanics in finding the trajectory of an object in a force field, the same laws are used in dealing with the dynamics of the charged particles (evaluating potentials, fields, etc. in electrodynamics), in solving statistical problems and the same laws are used in analysing quantum mechanical systems. Thus, being conversant with the laws of calculus helps one in solving and analysing a wide variety of physical problems. This shows the universality of the laws of mathematics (because they are built on logic) something that our languages lack.
Now, let me point out some generally misunderstood concepts that should be avoided in physics. There are many terminologies in physics which we use quite regularly in our daily lives. Indeed so regularly that their regular interpretation interferes with their intrepretation in physics which causes a lot of un-necessary confusion.
Let us take the example of energy. The term energy is used so widely that people tend to think that it is something real, just as real as this very palpable computer screen. Whereas, from the physics point of view, energy is a well defined and abstract mathematical quantity, it is a number, which is found to remain constant before and after a physical process (A detailed account of this is given in Feynman’s lecture series part 1) and thats it. Its just an abstract mathematical quantity. It is not something real or something which we can feel or perceive like objects around us. The expressions for various kinds of energies like kinetic or potential, etc. of a system come while mathematically constructing the Lagrangian of that system using symmetry arguments. There are reasons why we cannot have cube of velocity in the kinetic energy term or square of height in potential energy term, and they all are mathematical. Take the example of charge. Like energy, it is also an abstract mathematical quantity. It is a fundamental mathematical property of a particle and nobody knows what it really is or what causes it. The same story follows for quantities like momentum, angular momentum, etc. These are abstract mathematical quatities which are found to remain constant for a system under certain conditions. All these quantities are a part of mathematical models of physical phenomena which are required to explain the observed behaviour of physical systems. I will elaborate on the actual meaning of these quantities and how should they be understood in later articles.
The moral of the story is that one should not always try to look for a real world analogy of things we encounter in physics and must avoid trying to seek physical interpretation of each and every thing. This habit later manifests itself in the form of confusion that people face when they are not able to make a physical sense of the quantities that they deal with in physics (like stress tensors, crystal momentum, generalised co-ordinates, etc.) and this habit is the root cause for the confusion that is prevalent, in case of quantum mechanics, among students as well as teachers. It is obligatory for one to be conversant with these subtleties before proceeding with quantum mechanics.
In the next part, I plan to give a brief historical outline of development of quantum physics. This will give the reader an idea of what made classical physics obsolete and necessitated the development of a very different theory - the Quantum Theory.
I welcome any critical comments or queries.
(Images are borrowed from http://www.forgottenplanet.com/studyguide/chem210/printguide.html)
2 comments:
Can u gie some more clarification to the fact that we need not look for the physical interpretation of each and every physical quantity.I mean yes, I too had a tough time getting convinced that conserved things like energy, momentum,charge and even structures like langrangians and hamiltonian such objects of composite nature have any physical meaning or not. I mean that how do we know that a mathematically defined thing has a physical meaning or not.For example fields were introduced as completely mathematical objects but later they were manifested as physical things for the theories of gravitation and electromagnetism have effects that do give the status of physical objects to fields.My question is that when do we know whether a mathematically introduced quantity can be interpreted physically and when not.Please guide.
The following is the link to the article which is a continuation of the present post:
http://khuntersscience.blogspot.com/2012/05/quantum-mechanics-2-mathematical-naute.html
I think I have addressed these questions in the above article but let me know if you still have queries or if you find the explanation ambiguous. When we say "...give status of physical objects to field" we mean that the fields, which are mathematical models, are able able to explain/simulate all the physical phenomena observed so far successfully. There is no way, absolutely no way, to imagine what these fields are or look like "physically." In all of the physical sciences, by physical status we mean a successful association of the observed physical phenomena to some mathematical model. But its only a mathematical model and will always remain as one. if tomorrow somebody observes a phenomena which can't be accounted for by the present theory/model, we'll have to either modify the present theory or completely change the way we perceive fundamental concepts so that we can accommodate the new observation along with all others. This way we end up with a more general theory and with new mathematical quantities which can be successfully associated with physical objects. Its an everlasting process and its always the this association between physical world and mathematics that we refer to as physics and physical interpretation.
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