A CAUSAL INTERPRETATION OF A QUANTUM HARMONIC OSCILLATOR

Classical quantum mechanics (known as the Copenhagen Interpretation) contends that one can only assign a probability to a particle, or put another way, you can't say exactly where and when it exists at any one time.

"Not so!" cry a small school of thought who contend that you can know where and when a particle exists in time - adherents to this other school fall into the camp of Bohm's Hidden Variable Theory (A Causal Interpretation of Quantum Mechanics) which develops a deterministic rather than a probabalistic approach to Quantum Mechanics.

It was into this quasi-religious debate I stepped, when I did my B.Sc thesis back in 1990-1991. Initially, I was open minded as to the 2 sides arguments, but the more I got involved in the Causal Intepretation, the more I became convinced that it was viable. In fact, I think from a purely philisophical point of view, a "Quantum Potential" has a lot going for it, and on a more esoteric level, could tie in on a scientific level with Eastern concepts such as "Chi" as well as possibly providing a scientific mechanism for some paranormal phenomenon as well as consciousness.

"You're crazy" the rationalists cry. Yes, maybe, but then again, nature has a way of producing phenomenon which make the above statements seem rational. Black holes and Superstring theory are equally bizarre, and would have been thought completely impractical not long ago. Yet I don't exactly see a huge chorus of disapproval in the physics community to their existence now.

So rather than philosophising, off I went to work on this, working through the mathematics involved . The result is summarised in the following abstract from the thesis:

The Causal Interpretation is illustrated by means of a common example, in this case, a two level quantum harmonic oscillator, to which time dependent perturbation theory is applied. In this example, the particle trajectories are demonstrated firstly by analytical methods and then by graphical display; both by means of a computer software model.

Translation into English ?

It means that if you have a particle which is fairly inactive, and you whack it. It jumps up to a higher energy state. After a certain time, it drops back down to the position where it was originally. OK, so far, so good ?

Now, if you whack the particle at a constant rate, and with a constant amount of energy, every time it drops back down to its original position, you get what could be called oscillatory motion. i.e. the particle oscillates between the two positions or states. This is called a harmonic oscillator.

According to classical quantum mechanics, the particle will only be in one of the two positions, anything else isn't allowed.

So, armed with what I had learnt, and was learning on my Quantum Mechanics part of my Physics degree, I then proceeded to look at the possibility that the particle could be in any number of positions between the upper and lower energy states. Developing a model for this was interesting, since applying Hidden Variable Theory, a Quantum Potential term dropped out very nicely. The model was then coded up to provide a computer simulation that could model particle trajectories. The model was very processor intensive, requiring hours of dedicated processing time to generate an output of the particle trajectories, but it provided valuable feedback for the calculations in the model. My conclusions:

"maybe there was something to this Hidden Variable theory" since the hidden variable term in the equation "dropped out" nicely.

Mind you, I still wouldn't want to incorporate it into quantum gravity! The maths would be "interesting" to say the least!