## Schrodinger’s Zoo

I have been enjoying reading Richard Muller’s “Now: The Physics of Time” – Muller is an extremely imaginative experimental physicist and his writings on the “arrow of time” are quite a nice compendium of the various proposed solutions. Even though none of those solutions is to my liking, they are certainly worth a read.

Meanwhile, his chapter on the famous Schrödinger’s cat experiment is interesting though I think I can come up with a far better explanation. Some of these ideas have come out of discussions with two colleagues at Rutgers (though I would hate to have them labeled as the reason for any errors), so thanks to them for being open to discussing these oft-ignored subjects.

The “cat” experiment is simple. An unfortunate kitty is stuffed into a strongly built box, where a radioactive atom is also placed. The atom has a chance, but only a chance, to decay and when it does (through, let’s say, beta decay) produces an electron. The electron triggers a cathode-ray amplifier that sets off a bomb and “poof!”😪 – poor kitty is chasing mice in a different universe!

The kerfuffle appears to be that if we describe the system using standard quantum mechanics, Muller writes that it appears that we should describe the system (in the future) as

$|\Psi> = \frac{|cat \: alive>+|cat \: dead>}{\sqrt{2}}$

To be an accurate depiction of what he means, the initial state of the system is

$|\Psi(0)> = |cat \: alive, undecayed \: atom>$

and due to the passage of time and the possibility of decay of the atom, we should write the state in the future (at time $t$) as

$|\Psi(t)>=|cat \: alive, undecayed \: atom > (1 - p(t)) + \\ \: \: \: \: \: \: |cat \: dead, decayed \: atom> p(t)$

Here, $p(t)$ is the probability that the atom has decayed in the time $t$. Clearly, $p(0)=0$ since the atom starts out undecayed and $p(\infty)=1$, since the atom will eventually decay. A “superposition” of states is much more than “this or that”. The system truly has to be thought of as being in both states and cannot be even conceived of as being in one of these states at that instant. Thinking it as being in “one” or the “other” leads to logical and experimental contradictions, due to the presence of “interference” between the two states.

To be honest, if someone asked you if cats were intelligent, you would probably say “Yes” and they certainly seem quite capable of understanding that they are alive, or facing imminent demise in a rapidly exploding fireball. Stating that the system is in a state where the cat is either alive or dead and worse, in a superposition of these states, is truly silly. Einstein found this galling, asking in eloquent terms, “Do you believe that the moon is not there when you are not looking at it?”.

Having the state of the system “collapse” into one of several alternatives is picturesquely referred to as “wave-function collapse”. Many physicist – hours have been spent thinking about how wave-functions must collapse (physicist hours are like man hours, except they can be charged to the NSF and other funding organizations). Solutions range from the GHZ mechanism, which says that wave-functions can spontaneously collapse (like stock prices can randomly jump) and this is modeled by a stochastic jump process (just as with stock prices). There is also the QB-ism approach (of David Mermin and others) that suggests that wave-function descriptions of nature are in your head and that they represent YOUR knowledge and beliefs about the state of the system. While that seems like an acceptable way to handle the “cat” experiment above, it begs the question why we all seem to agree on the Hamiltonian and state variables for systems in general. Could we manufacture, from our sense impressions, an alternative-reality explanation of the same situation that agrees in detail with the “real” description. This is one that a lot of Trump supporters would certainly embrace! But it doesn’t give one a warm and fuzzy feeling that there is “a reality”out there to explain.

Other solutions rely on the consciousness of the observer and how his/her consciousness collapses wave-functions. To which, I would ask (paraphrasing Aharonov, a famous physicist), “If you believe that you collapse wave-functions yourself, is this an inherited ability? Did your parents collapse wave-functions too?” If you believe that only you, the observer, has the right to decide when a “wave-function has collapsed”, it seems to give little meaning to my experience and the experience of millions of others. At that rate, the Taj Mahal only exists the moment you yourself see it. And America didn’t exist till Columbus saw it, unless you were Native American of course!

The key, in my opinion, to understanding all this is to recognize (admit, rather), that we, as observers, are just another cog in the quantum universe. We participate in measurements, we are part of wave-functions and a wave-function is supposed to “collapse” to varying degrees based on how many interactions the system it represents has had with the surroundings. So, if the only thing that happened was that a radioactive atom decayed, that’s one atom that emitted an electron, an anti-neutrino and created a proton out of a neutron, little, if anything in the universe was altered. This piffle of a disturbance isn’t big enough to collapse anything, let alone a wave-function. If the electron subsequently collided with a couple of other atoms on its way to the anode of the amplifier, that’s piffle too, The “state” of the system, which includes the original atom, the electron, the anti-neutrino, the proton, the couple of other atoms is still in a superposition of several quantum states, including states where the original atom hasn’t even decayed.

The fun begins when the disturbance is colossal enough to affect a macroscopic number of atomic-sized particles. Now, there is one state (in the superposition of quantum states) where nothing happened and no other particles were affected. There are a ${\bf multitude}$ of other states where a colossal number of other particles were affected. At some point, we reach no-return – the number of “un-doings” we would need to do to “un-do” the effect of the (potential) decay on the (colossal number of) other particles is so large and the energy expended to reverse ALL the after-effects would be so humongous that there is no comparison between the two states. The wave-function has collapsed. Think of it this way – if the bomb exploded and the compartment where the cat was sitting was wrecked, the amount of effort you would need to put in to “un-do” the bomb would be macroscopically large. This is what we would refer to as wave-function collapse. Of course, if you were an observer the size of our galaxy and to you this energy was a piddlingly small amount, you might say – it is still too small! However, this energy is much much larger (by colossal amounts) than the energy you would need to expend to not do anything if the decay were ${\bf not}$ to happen.

This explanation seems to capture the idea of “macroscopic effect” as being the way to understand wave-function collapse. It also allows one to quantitatively think of this problem and relate to it the way one relates to, say, statistical mechanics.

So, if you think one little kitty-cat is “piffle”, consider a zoo of animals in a colossal compartment, along with their keepers – Schrodinger’s Zoo, to be precise. Now, consider the effect of the detonation on this macroscopic collection of particles. It should be clear that even if you, as an observer, are far from the scene, a macroscopic number of variables were affected and you should consider the wave-function to have collapsed – even if you didn’t know about it at the time, since you were clearly not measuring things properly. No one said that physics needs to explain incorrect measurements, did we?

I’d like to thank Scott Thomas and Thomas Banks, both of Rutgers and both incomparably good physicists, for educating me through useful discussions. Tom, in particular, has a forthcoming book on Quantum Mechanics that has an extensive discussion of these matters.

## Why do chocolate wrappers stick to things

Here’s something I saw while lazily surfing the net this morning. Someone throws a candy wrapper towards the floor and it sticks to the curtain or a book cover. How long will it stick?

First, the reason this happens is because of static electricity – and this is why this rarely happens in humid climates (except inside the confines of an air-conditioned room, which is usually much drier). Why static electricity – when you eagerly unwrap a piece of chocolate, some charges jump in or out. Usually slow moving positive ions stay where they are – the electrons jump and which direction they jump in is a simple question with a complicated answer.

Clearly the answer to how long it will stick depends on how dry the air is – the mechanism for leaking the charge back between the wrapper to the curtain to the ground depends on the resistance to current flow offered by the air between the wrapper and the curtain and the capacitance of the wrapper-curtain capacitor.

Since I finished teaching “RC” circuits to a whole lot of undergraduates last month, I thought it would be fun to carry out a small calculation.

What does the system look like – let’s idealize the wrapper/curtain capacitor to the below picture.

The effective area of contact is of area $A$ and separated by a distance $d$ from the curtain. This is a parallel plate capacitor, whose capacitance is (from your dim memory of high school physics) $= \frac{ \epsilon_0 A}{d}$ where $\epsilon_0$ is the permittivity of the vacuum and I have used a dielctric constant of 1 for air. Using Meter-Kilogram-Seconds-SI units, if you make the entirely reasonable assumption that $d = 1 \: mm$ and $A = 1 \; mm^2$, this translates to a capacitance of $\approx 10^{-14}$ farads. The resistivity of air (between the plates of this parallel plate capacitor) is $\rho \approx 10^{16} \Omega -m$ and again the resistance is $R = \rho \frac{d}{A}$, which approximates to $10^{19} \Omega$. The value of $R \times C \approx 10^5 secs \approx 1 \: \: day$ which is an enormously long time constant – post one day, chances are the force keeping it up (the electrical attraction) is not big enough to withstand gravity and it is likely to fall. So a candy wrapper could remain stuck for a day at least – I would bet on more since the contact area might actually be much more than $1 mm^2$ given lots of folds in the wrapper and the curtain, as well as all the sticky saliva on the wrapper from licking the candy off….reminds me I need to get back to finishing the candy!