Jetée

originally posted May 16, 2008

author's comments:
I'm sure that many of you are familiar with Schrödinger’s cat but I still feel compelled to
give a short description here. However, please note that I will only be explaining enough to
make sense of the comic and that many details will be omitted.
The idea of “Schrödinger’s cat (paradox)” was put forth by the Austrian physicist Erwin
Schrödinger in 1935 as a thought experiment to illustrate the absurdity of (what has come
to be known as) the Copenhagen interpretation of quantum mechanics when applied to the
“real” world of common sense and macroscopic objects.

Imagine a box that is so perfectly sealed that no physical influence can get in or out.

Now imagine that a cat is inside the box along with a device that can kill the cat when
triggered by some “quantum event”. That is the setting for Schrödinger's cat. In
Schrödinger's original version, the quantum event was the decay of a radioactive atom.
Schrödinger asserted that the Copenhagen interpretation implies that the cat remains in a
"superposition" of states: (both alive and dead) until the box is opened.
I used a bit of jargon in the previous paragraphs so let's backtrack a little with a miniscience
lesson. First of all what is quantum mechanics? To put it simply, quantum
mechanics is the theoretical framework that describes the universe at the “smallest” scales:
atoms, electrons, protons, quarks, etc. At such small scales, the “rules” are very different
from the rules of the macroscopic world which is described by classical mechanics. Take,
for example, an ordinary baseball. According to classical mechanics, the baseball has a
definite trajectory (position and momentum) at any given time and we can theoretically
predict the position of the baseball at a later time if we know its trajectory at an earlier time.

This seems to conform to our common sense notion about how everyday objects that we
see around us should behave. However, when we are dealing with small objects (e.g.
electrons), quantum physics tells us that such common sense no longer applies. At the
quantum level, we must describe an object by its state vector.
Suppose, for example, that we wanted to know the position of such a microscopic object.

According to quantum physics, the object has no definite position until it is measured. In
fact, before its position is measured, we can think of the object as having a probability of
being in any possible position available to it (this idea, by the way, is one of the central
tenets of the Copenhagen interpretation). This probability distribution is described by the
state vector which, by convention, is represented by the Greek letter Ψ (psi). In this case,
Ψ describes the object's possible positions.

As you may have surmised from the name, state vectors are examples of mathematical
objects called vectors. To be more precise, they are vectors in a complex vector space
called a Hilbert space, but we won't get into that here. The important point is that different
vectors can be added together to give another vector. So for example, if Ψ and Χ are two
different vectors, then Ψ + Χ would be another vector. Vectors can also be multiplied with a
(complex) number to give another vector so if Ψ is a vector, then cΨ would be another
vector (where c is a number). Physicists have adopted a notation for these state vectors
(called bra-ket notation) in which each vector is denoted by a symbol in angled brackets can
be written as ΙΨ> + ΙΧ> and the multiplication of a vector by a number can be written
as c ΙΨ>.

Now let's see this bra-ket notation in action for a simple example. Suppose we have a
microscopic particle whose state vector for position is Ψ and it is expressed as the
weighted sum of two other vectors

ΙΨ> = c ΙA> + d ΙB>.

What this expression says is that the particle can be in two possible positions, A or B.
Before measuring the position, the particle cannot be thought of as occupying any of the
two positions. We can only say that it has a probability of being in either of the two
positions. Physicists would say that the particle is in a quantum superposition of the two
positions. The numbers c and d are called probability amplitudes. The square of the
probability amplitudes (actually the squared moduli) gives the probability of finding the
particle in that position after measurement. In this case, if we measured the particle's
position, the probability of finding it in position A would be |c|2 and the probability of finding
it in position B would be |d|2. The amplitudes are usually “adjusted” so that their squares
sum to 1 but that's another detail which I won't get into here. The process by which the
state vector representing the superpostion of different states reduces to a single state is
referred to as state vector reduction (it can sometimes be referred to as wavefunction
collapse). The idea of the reduction of the state vector is another one of the central
aspects of the Copenhagen interpretation of quantum physics.

Now let's apply what we've learned so far to poor Schrödinger's cat. We know that the
cat can be in two possible states: live or dead. Let ΙΨ> be the state vector and suppose
that the probability of finding a dead cat is ½ and that the probability of finding a live cat is
½. Then one possible way to express the state vector is

ΙΨ> = (1/√2) Ι live cat> + (1/√2) Ι dead cat>.

So there you have it. Now you understand the equation in the first panel of the comic;
but what about the equation in the last panel? The symbol є is generally used by
mathematicians to represent an infinitesimal quantity. Hence, the equation in the last panel
expresses the idea that there was an infinitesimal chance that an angry monkey could
have magically appeared in the box.

Abstruse Goose

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+ bonus: "Schrödinger’s Miscalculation - Part 2"

__________________
As human beings, our greatness lies not so much in being able to remake the world (that is the myth of the Atomic Age) as in being able to remake ourselves.

—Mohandas K. Gandhi