The Wavefunction

Now that we are right in the heart of quantum mechanics, it is time we reviewed all that has happened so far. The start of the 20th century saw a surge in the number of experiments conducted by physicists, all in the same field of trying to figure out a way to model the seemingly unpredictable behaviour of electrons and similar particles.

In the double-slit experiment, thought experiments were performed about directing a single electron at the set up. By physical measurement, it is a simple task (atleast in theory) to ascertain which slit the electron went through. But that gives us no real information, if you think of it. Well we know for sure where the electron is, now, but that tells us nothing about where it tends to go, in general. This was quite a revelation at that time, considering the fact that a large number of scientists were breaking their heads over the particle (or wave-like) nature of the electron.

One thing was clear: the electron was and is a particle. There are no two ways about it.

But this particle behaves as if its properties (speed, energy, momentum and the like) were determined by a wave-like probability distribution.



Wait, what?

A wave-like probability? Does that mean that interference can take place? - Yes! Why not?

We must note that although these properties seem very counter-intuitive, they are the results of experimental study and we need to craft a model to fit the experiments best.

Since interference can take place, it is no surprise that the likelihood of finding an electron at one place is influenced by the probability distribution of another electron in the same system. So we need a parameter that can depict the probability distribution of a particle, and also give us information about its energy.

Welcome the wavefunction - ψ.

Αs of now, it will suffice to say that psi is such that the square of its modulus gives us the probability of finding the electron somewhere. A wavefunction needs to obey a few properties for it to be of use to us. We will be looking at these properties later.

The Rutherfordian Model and a peek into the quantum system

We have seen the Thomson Model and the reasons for its failure. To continue with the chronological scheme of things, I will give here some more insight on the Rutherford Model of the atom that followed the rejection of the Plum-Pudding Model.

We have heard quite a bit about the Rutherford Model of a dense positively charged nucleus surrounded by particle-like electrons that revolve around it like a system of planets under a gravitational field. We have also read about the fact that given such a crude model, the electron will inevitably collapse into the nucleus.

Let's see why.

Consider a stationary charge. Now let the charge experience a sudden acceleration which causes it to move in a particular direction. Initially, the electric field lines from the charge emanate as they should, from a stationary particle. Once the charge is accelerated, distant observers must be able to get the news of this surge, not instantly, but at the speed of information transfer(the speed of light). To enable this, the charge sends out a transverse field line, which travels at the speed of light. This, hence gives rise to the perpendicular component of the electric field which is 1/r dependent.







This phenomenon is responsible for the Larmor formula for radiation. In essence, the information transfer that accompanies the acceleration of the charge causes it to lose energy. As the particle moves, it radiates energy (in the form of electromagnetic radiation, due to the transverse field lines) as:

Now what do we do with this?

We know that the total energy of a particle of charge q, revolving around a central nucleus is:

Substituting this value of energy in the Larmor equation and solving for the time rate of change of radius of the particle, we get:

(This equation was obtained by taking 'a' as the centripetal acceleration = 

 
)


Integrating both sides with respect to the radius, we get the time when the particle falls into the nucleus as:

Putting in the values of final radius as 1 fm and initial radius as 1 angstrom, we get the time taken to fall to be around ten-billionth of a second. 

Well that number is way smaller than the average lifetime of most of the atoms we see around us. Hence, the Rutherfordian Model, though elegant and intuitive, is not quite so much correct.

But that wasn't the end of the problems for physicists back then. As we know now, the puzzling question of the dual nature of light kept popping up, with the photoelectric effect and the ground-breaking Young's double slit experiment, each standing testament to one side of the case.

This was the right time for a new model of thinking, and as usual, physics didn't disappoint.

The Pre-Quantum World

The Pre-Quantum World

Let's review the infamous 'Thomson Plum-Pudding Model' first. When I tried doing some study on the model, I was amazed. Blown away, to be exact - not by the fact that the model is (now) deemed to be absurd, (its lack of scientific backing notwithstanding) but by the issue of how Thomson could come up with a solution to how the electrons could distance themselves so as to minimize the interaction between themselves.

On further exploration, I came across this article on 'The Thomson Problem'. The question goes as follows:

"Given N electrons on the surface of the unit sphere that exhibit interactions that follow Coulomb's Law, find the configuration of minimum potential energy."

This problem, apparently, is a subset of a larger question on arranging objects that interact with each other according to the inverse-square law. Now, let us, for a moment forget about the solution to such a problem. Intuitively, symmetrical structures like a line that connects the electrons (in the case of a two electron system), whose length is the diameter of the pudding or a regular tetrahedron (for 4 electrons) can solve the issue. 

Note: We are dealing with an environment that has rid itself of the surrounding positive charge.

But here is one question that crept in. As we increase the number of electrons, the rigidity of the structure of energy equilibrium increases, i.e., the degrees of freedom of the electrons go down drastically. So we are dealing with a situation where even the most electro-negative elements would be unwilling to give in to an excitation of the electron, simply because the rigidity of the structure leads to a collapse of the equilibrium state, on disruption. An enormous amount of energy would hence be required for excitation, to take into account the re-establishment of equilibrium of the remaining electrons.

The Thomson Model is a failure today, but that was just some food for thought on a topological problem.

A Good Day to Begin

A Good Day to Begin

"The scientific mind does not so much provide the right answers as ask the right questions."

                                                                                                          - Claude Levi-Strauss

We have seen enough drudgery in our school and college lives to yearn for something different; something that can allow us to explore the things that actually go on out there in the scientific world. I believe that this blog is a product of such a drive to learn and catalog the wonderful facts about the science of the objects that surround us.

I started this page to serve as a record of my journey in the world of Quantum Physics. I have just begun, and am already fascinated by the mind-bending stuff that lies there. I am sure that there are multitudes of young people like me who wish to read up on the Science that interests them, and I hope this page will serve as a portal for all of us to discuss and thrash out ideas.

Let's begin!