Friday, January 2, 2009

QM!

After a long break, and with Quantum Chemistry looming in the near future of the coming semester, let us go into a short (and I do mean ‘short’) discourse on the structure of Quantum Mechanics.

So, what is the principal difference between the so-called Classical Physics and Quantum Physics? Is it… that quantities are quantized and no longer continuous? Well actually no, even in the classical realm things are quantized, just that in such small portions that they seem continuous to the human senses! So what is the main difference that makes quantum mechanical treatment ‘quantum’?

First off, you should be acquainted with the term ‘observable’, which in general refers to any dynamical quantity that can be observed in real life; that is, properties such as momentum, position, weight, mass, energy etc. By the word ‘observe’, we mean to say that the property belongs to a system, and the system possesses that property or quantity and what we measure depends on the state of the system.

In all Physics, we are concerned with two types of quantities: ‘parameters’ and observables. So what is a ‘parameter’? Time, for instance, is a parameter – you can measure time no doubt, but time doesn’t belong to a system. A system evolves with time, but it doesn’t possess time per se, so as to speak. With that, time is a parameter, and not an observable. That is to say, the measure of time, doesn’t depend on the state of any system, and evolves very much by itself. In that case, time isn’t an observable!

Alright, so with the definition of an observable laid out, we can now state the fundamental difference:

‘In classical physics, observables are represented by functions
but
in quantum physics, observables are represented by operators.’

So what is this difference? Now… a function by itself, means something. For instance, if I tell you the position of say, my Chemistry professor depends on time as such:


I’m pretty sure you’d be tracing out a parabola in your head. Or maybe not. But the point is, in classical physics, the observable takes on a pretty much, well, for lack of a better word, ‘observable’ form. But if I tell you now, that say, the momentum of my Chemistry professor depends on an operator as such:


Then, what exactly does this mean? The derivative isn’t operating on anything at all! How can I say that the momentum is equal to this derivative – the derivative of what? So it seems like, hey, quantum observables aren’t that easy to observe, huh?

Therein lies the first magic of quantum physics that all science students should be aware of: without any measurement, you can’t say anything about any system. Which makes sense: if you don’t allow the operator to ‘operate’ (i.e. measure) on a system, then no information can be obtained, because an operator needs to operate on a system before it can extract any information!

So that’s why in quantum physics, there needs to be something called the ‘wavefunction’ – essentially, this wavefunction contains all information about a system. If you want to find out something about this system, easy! Just use the appropriate operator on the wavefunction, and it’ll give you a numerical value about just the thing you want to know about the system.

I guess you should be aware of the Hamiltonian; the Hamiltonian is the ‘total energy operator’, meaning, when you apply the Hamiltonian to a wavefunction, you obtain the total energy of the system. And of course, we see this most often in the much-celebrated Schrodinger wave equation:


That is, if I wish to determine the total energy of a system, I just apply the Hamiltonian to the wavefunction of the system, and bam! I get the energy ‘E’!

Alright, I’m tired, and it’s 2:37 am in the morning, and I think I’ll be taking some of these thoughts to bed. Haha.

1 comment:

quirK said...

A thoroughly excellent primer on QM. I think somewhere in the far future when your career is established, you can seriously consider publishing your own textbook.