Saturday, February 14, 2015

How to see Schrödinger’s Cat as half-alive

Schrödinger’s Cat is a thought experiment used to illustrate the weirdness of quantum mechanics. Namely, unlike classical Newtonian mechanics where a particle can only be in one state at a given time, quantum theory says that it can be in two or more states ‘at the same time’. The latter is often referred to as ‘superposition’ of states.
File:Schrodinger cat in box.jpgIf D refers to the state that the cat is dead, and if A is the state that the cat is alive, then classically we can only have either A or D at any given time — we cannot have both states or neither.
In quantum theory, however, we can not only have states A and D but can also have many more `in between' states, such as for example this combination or superposition state:
$\large \Psi = 0.8A + 0.6D$
Notice that the coefficient numbers 0.8 and 0.6 here have their squares adding up to exactly 1:
$\large (0.8)^2 + (0.6)^2 = 0.64 + 0.35 = 1.$
This is because these squares, (0.8)2 and (0.6)2, refer to the probabilities associated to states A and D, respectively. (The fact that this sum is 1 is what a 'normalized' wavefunction means.) Interpretation: there is a 64% chance that this mixed state $\Psi$ will collapse to the state A (cat is alive) and a 36% chance it will collapse to D (cat is dead) — when one proceeds to measure or find out the status of the cat were it to be in a quantum mechanical state described by $\Psi$.
Realistically, it is hard to comprehend that a cat can be so described by such a state. Well, maybe it isn’t that hard if we had some information about the probability of decay of the radioactive substance inside the box.
Nevertheless, I thought to share another related experiment where one could better ‘see’ and appreciate superposition states like $\Psi$ above. The great physicist Richard Feynman did a great job illustrating this with his use of the Stern-Gerlach experiment (which I will tell you about). (See chapters 5 and 6 of Volume III of the Feynman Lectures on Physics.) 
In this experiment we have a magnetic field with north/south poles as shown. Then you have a beam of spin-half particles coming out of a furnace heading toward the magnetic field. The result is that the beam of particles will split into two beams. What essentially happened is that the magnetic field made a ‘measurement’ of the spins and some of them turned into spin-up particles (the upper half of the split beam) and the others into spin-down (the bottom beam). So the incoming beam from the furnace is like the cat being in the superposition state $\Psi$ and magnetic field is the agent that determined — measured! — whether the cat is alive (upper beam) or dead (lower beam). (Often in physics books they use the Dirac `bra-ket' notation $|\uparrow\rangle$ for spin-up state and $\langle\downarrow|$ for spin-down.)

In a way, you can now see the initial beam emanating from the furnace as being in a superposition state.
Ok, so the superposition state of the initial beam has now collapsed the state of each particle into two specific states: spin-up state (upper beam) and the spin-down state (lower beam). Does this mean that these states are no longer superposition states?
Yes and No! They are no longer in superposition if the split beams enter another magnetic field that points in the same direction as the original one. If you pass the upper beam into a second identical magnetic field, it will remain an upper beam — and the same with the lower beam. The magnetic field ‘made a decision’ and it’s going to stick with it! :-) 
That is why we call these states (upper and lower beams) ‘eigenstates’ of the original magnetic field. They are no longer mixed superposition states — the cat is either dead or alive as far as this field is concerned and not in any ‘in between fuzzy’ states.
Ok, that addresses the “Yes” part of the answer. Now for the “No” part.
Let’s suppose we have a different magnetic field, one just like the original one but perpendicular in direction to it. (So it’s like you’ve rotated the original field by 90 degrees; you can rotate by a different angle as well.)
In this case if you pass the original upper beam (that was split by the first magnetic field) into the second perpendicular field, this upper beam will split into two beams! So with respect to the second field the upper beam is now in a superposition state!
Essential Principle: the notion of superposition (in quantum theory) is always with respect to a variable (or observable, or quantity) that is being measured. In our case, that variable is the magnetic field, which in this case measures the spin direction of particles. (And here we have two magnetic fields, hence we have two different variables, observables — non-commuting variables as we say in quantum theory.)
Therefore, what was an eigenstate (collapsed state) for the first magnetic field (namely the the upper beam) is no longer an eigenstate (i.e., is no longer ‘collapsed’) for the second (perpendicular) magnetic field. Thus if a wavefunction is collapsed with respect to one field, it is not necessarily a collapsed wavefunction for the second magnetic field! A collapsed wavefunction for one variable could be a superposition wavefunction of another variable.
The Schrodinger Cat experiment could possibly be better understood not as one single experiment, but as a whole series of many (infinitely many) boxes with cats in them. This view might better relate to the fact that we have beam of particles each of which is being ‘measured’ by the field to determine its spin status (as being up or down). 
Why do you say 'infinitely many'? Because, for example, when we say that a tossed coin lands heads or tails with probability 1/2, what we have to do to get that 1/2 is toss the coin a very large number of times and count the number of heads and divide that by the number of tosses. (If you toss it only 10 times, it's possible heads would land 6 or 4 times instead of exactly 5 - yet the probability would still be 1/2.)
Best wishes, Sam
Reference. R. Feynman, R. Leighton, M. Sands, The Feynman Lectures on Physics, Vol. III (Quantum Mechanics), chapters 5 and 6.

Postscript on determinism. It occurred to me to add a uniquely quantum mechanical feature that is contrary to classical physics thinking.   The Stern-Gerlach experiment is a good place to see this feature.
We noted that when the spin-half particles emerge from the furnace and into the magnetic field, they split into upper and lower beams. Classically, one might think that before entering the field individual particles already had their spins determined as either being spin up or spin down before a measurement takes place (i.e., before entering the magnetic field) — just as one might say that the earth already had a certain velocity as it moves around the sun before we measure it. Quantum theory does not see it that way. In the Copenhagen Interpretation of quantum theory (the predominant view), one cannot say a priori that a particle's spin was already either up or down before entering the field. (Recall, the magnetic field in this case is the agent making the spin direction measurement.) The reason we cannot say this is that if we had rotated the field at an angle (say at right angles to the original), the beams would still split into two, but not the same two beams as before! So we cannot say that the particles were already in either an up or down spin state. That is one of the strange features of quantum theory, but wonderfully illustrated by Stern-Gerlach. 

Mathematics. In vector space theory there is a neat way to illustrate this quantum phenomenon by means of `bases’. For example, the vectors $(1,0)$ and $(0,1)$ form a basis for 2D Euclidean space $\Bbb R^2$. So any vector $(x,y)$ can be expressed (uniquely!) as a superposition of them; thus,
$\large (x,y) = x(1,0) + y(0,1).$
So this would be, to make an analogy, like how the beam of particles, described by (x,y), can be split into to beams — described by the basis vectors $(1,0)$ and $(0,1)$.
However, there are a myriad of other bases. For example, $(2,1)$ and $(1,1)$ also form a basis for $\Bbb R^2$. A general vector $(x,y)$ can still be expressed in terms of a superposition of these two:
$\large (x,y) = a(2,1) + b(1,1).$
for some constants $a$ and $b$ (which are easy to solve in terms of $x, y$). So this other basis could, by analogy, represent a magnetic field that is at an angle with respect to the original — and its associated beams $(2,1)$ and $(1,1)$ (it’s eigenstates!) would be different because of their different directions. As a matter of fact, we can see here that these eigenstates (collapsed states), represented by $(2,1)$ and $(1,1)$, are actual (uncollapsed) superpositions of the former two, namely $(1,0)$ and $(0,1)$. And vice versa!
Analogy: let’s suppose, to take a specific example, that the vector $(5,3)$ represents the particle states coming out of the furnace. Let’s think of the basis vectors $(1,0)$ and $(0,1)$ as representing the spin-up and spin-down beams, respectively, as beam enters the first magnetic field, and let the other basis vectors $(2,1)$ and $(1,1)$ represent the spin-up and spin-down beams as they enter the second perpendicular magnetic field. Then the particle state $(5,3)$ is in fact a superposition in each of these magnetic fields (bases)! This is so because
$\large (5,3) = 5(1,0) + 3(0,1)$
$\large (5,3) = 2(2,1) + 1(1,1).$

It is now quite conceivable that the initial mixed state of particles $(5,3)$ as they exit the furnace can in fact split in any number of ways as they enter any magnetic field! Depending on the orientation of the field. I.e., it’s not as though they were initially all either $(1,0),(0,1)$ or $(2,1),(1,1)$, but $(5,3)$ could be a simultaneous superposition of each pair. (In fact (5,3) is combination (superposition) in an infinite number of bases because you can have an infinite number of directions the field can point to.)
Indeed, it now looks like this strange feature of quantum theory can be described naturally from a mathematical perspective! Vector Space bases furnish a good example!

No comments:

Post a Comment