Schrödinger's Cormorant

6.10 The Cormorant Particle

Investigating the transition from quantum physics to classical physics

Cormorant has had a thought. “I could be a particle in a box!” His plan is simple: he would be inserted into a pipe within which he is free to move (zero potential energy throughout, near enough, he feels) but which is firmly locked shut at both ends. Then, he suggests, he can just bounce around freely just like in our model system. Nefertiti immediately produces a long, long list of objections – air resistance, inelastic collisons, just for starters. “How about a Gedankenexperiment”? Cormorant reluctantly agrees.

“So, if I’m in my ground state”, Cormorant muses, recalling the probability density distributions we deduced in section 6-7, ”I’m much more likely to be in the middle of the tube than at the ends. That doesn’t seem to fit in with my admittedly limited experience of being trapped in a pipe.” Nefertiti agrees; it doesn’t feel right. Maybe he won’t be in his ground state. A calculation is in order.

Question 6-12

What this exercise highlights is that the difference in scale between an electron and a seabird has profound consequences. For the electron in its molecular box, quantum effects dominate: the widely spaced energy levels leave the electron becalmed in its ground state with a substantial zero point energy and a wavefunction that gives it a much higher probability of being found in the middle of the box than near the edges.

For Cormorant in his pipe, however, two things conspire to make these weird effects all but disappear: his much bigger mass and the much bigger box. Looking at the energy level expression:

we can see these two terms (m and L) combine to make the much bigger denominator which makes the zero point energy negligible and squashes the higher levels so close together that there is effectively a continuum. In all but the lowest few of these accessible states, there are so many individual maxima in the probability density distribution that their separation all but vanishes, meaning that Cormorant can move freely within the box. He is, to all intents and purposes a classical seabird. This is a very important feature of quantum mechanics, first proposed by Niels Bohr, which is called the correspondence principle: at the high quantum numbers that naturally emerge when masses become macroscopically large and dimensions become macroscopically long, the predictions of the quantum treatment become indistinguishable from those of classical physics.

There’s still something that bothers Cormorant, however (well, many things actually, but he’ll focus on this for the moment). The point about the wavefunction is that if a measurement is made it allows the particle to be detected anywhere – except at a node – in the box and where that will be is absolutely not predictable until the measurement is made. “But I will already know where I am in the pipe. So how does that work?” This is actually a hard question which we’ll return to (though not necessarily answer) after we’ve delved a little into the murky world of wave function superposition.

Back

Chapter Guide

Page updated

Google Sites

Report abuse