Let's get the idea of Schrodinger's cat right: It's a funky aspect of human perspectives. The math that goes with it is really easy, contrary to popular opinion, and entertaining, too!
(This article is about Schrodinger’s cat. If you thought it was about LOLCATS… well, it could be.)
When you’re studying the world through the lens of quantum mechanics, you’ll have a lot of freedom in defining what you want to study. This is because you’re essentially probing the nature of nature, you’re stirring through reality’s constituents, and you’re manipulating (on paper, at least) the apparatus to see how the experiment called consciousness is affected by it.
You can look at how making some changes alters a system’s freedoms, modifies the way it moves through space and in time, becomes more or less, energetic, etc. For each of these viewpoints, there’s a framework – a set of rules and operations – that makes it easier to study it.
It’s like you’re making a speech to an audience. Depending on who’s listening to you, you modify your language, your tone and style, and your body language so your message gets across – whatever that message may be. If you’re talking to kids, you try to emphasise your message in terms of their future. If you’re talking to job-seekers, you emphasise your message in terms of creation of jobs, increase of social security, etc.
Similarly, while studying quantum mechanics, you have the liberty of addressing your problem from different angles. In fact, to make it simpler for you, physics itself offers what are called degrees of freedom. A degree of freedom (DoF) is simply an avenue to effect change by doing work in the universe. A universe that contains a long stick can remain so; if the stick starts to move forward and backward in space-time, we have a universe with a moving stick; if the stick starts to revolve about its central axis, we have a universe with a revolving. So, there you go: Two degrees of freedom for a universe that contains a stick!
Each DoF is an “angle” for you to approach a problem with. As I briefly mentioned earlier, studying how the energy of a system varies is one such angle, and the framework that goes with this model is called the Hamiltonian formulation of dynamics. In classical (i.e., Newtonian) mechanics, solving the Hamiltonian for a system yields the total energy of the system – be it an electron, or a system of electrons in, say, a helium atom.
To wit: A helium atom consists of two electrons, e1 and e2. They have masses m1 and m2, have momentums p1 and p2, Z is the atomic number, e is the value of the elementary charge, and r1 and r2 are the position vectors of the electrons with respect to the nucleus (don’t hyperventilate yet). For this system, the Hamiltonian is
We can simplify this one step further: All electrons have the same charge (e) and the same mass (m). So, the Hamiltonian becomes
Because of the charge-and-mass uniformity, taking e1 and putting it in e2’s place and taking e2 and putting it in e1’s should still yield the same Hamiltonian. This is because p1 and p2 are measured in terms of m, and r1 and r2 are measured in terms of e. Since m and e for both electrons are fixed, interchanging the electrons shouldn’t affect how we’re measuring p and r!
In other words, H(1,2) = H(2,1).
This formulation is only valid whenever the principle of indistinguishability holds: One electron is exactly like another electron as long as you’re treating them in terms of their mass and charge. Thus, the Hamiltonian for a two-proton system will also be compatible with H(1,2) = H(2,1). However, switching two electrons when you’re studying them in a framework that involves their spin will break the H(x, y) equivalence.
In fact, when you’re treating particles as being distinguishable, here’s the more general Hamiltonian:
Solving this Hamiltonian gives the position and velocity of the two electrons for all possible initial conditions (except if you’re getting them started at the speed of light).
All this trouble has been taken to figure out the classical energy of a system in terms of two electrons. Let’s go quantum, and let me dispel the notion that it gets harder: It doesn’t, not if you’ve a little imagination to spare.
First, a disaster strikes…
There once was a man named Werner Heisenberg, and he was a buzzkill. While he was pottering around the world of particles, he noticed that they existed in a world in which a small change of energy meant everything in the system would change. Because humans study most of the world around them by bouncing off light, sound, heat, or other forms of radiation, and letting the brain interpret the splashback, Heisenberg realised it’s inevitable that any observation made on the quantum world changed it in such a way that it rendered our measurement useless.
“Thou shalt not simultaneously know the position and momentum of a particle,” quoth Heisenberg, and ‘twas called his uncertainty principle hence.
So, what was once a very determinable system – we could say that, given r1 and r2, we know p1 and p2 – has now become a probabilistic system: From “We know it’s there”, it’s become “We know it may be there”. Consequently, we throw out p1, p2, r1 and r2, and bring in a freak named ψ (psi – yes, Heisenberg will be thrilled to know it’s all Gangnam-style now).
ψ is called the wave function. It is a function in space and time - a time-traveller, if you will - that describes the chances of finding a particle in a given region in space at a given time. As a solution of the Schrodinger equation, it's described thus:
(Here, ∂/∂t measures how one parameter, such as ψ, changes with time; i is a complex number and h/2π is the Dirac’s constant.)
Even in the case of the quantum-Hamiltonian, interchanging the two electrons results in the same value for Hψ. Here’s how.
Interchanging 1 and 2,
Since H(1,2) = H(2,1),
Hence, proved! That was simple enough, wasn’t it?
Enter: A cat out of a box (i.e., cattus ex machina)
Now, we have two indistinguishable states, ψ(1,2) and ψ(2,1). They are both probabilistic, which means the total energy of a two-electron system is based on the momentum of the two electrons and where they may probably be at some time (given by H(1,2)ψ(1,2)).
Also, since ψ(1,2) and ψ(2,1) are interchangeable, it’s probable that they could both be true – but at different times.
To resolve this “but at different times” issue, we create what’s called a linear combination in mathematics which, in English, reads: “At any time, the wave function ψ(1,2) has equal chances of being a combination of ψ(1,2) and ψ(2,1) and a combination of only ψ(1,2) and not ψ(2,1).”
Since the probability of a certainly-happening event is 1, the probability of an event that may happen half the time should be ½, or 0.5. So, as an equation:
ψ(1,2) = EITHER [ψ(1,2) + ψ(2,1)] OR [ψ(1,2) – ψ(2,1)]
ψ(1,2) = 0.5[ψ(1,2) + ψ(2,1)] + 0.5[ψ(1,2) – ψ(2,1)]
Here, [ψ(1,2) + ψ(2,1)] is called the symmetric state and [ψ(1,2) – ψ(2,1)] is called the antisymmetric state. Therefore, every wave function describing a system – such as a cat in a closed box with a bowl of poison – is a superposition of a symmetric state – the cat is alive – as well as an antisymmetric state – the cat is dead. And how do you know which state it is in at some point of time?
You open the box and make a measurement. This distorts the energy of the system and collapses the wave function, trapping ∂/∂t to, say, 2 pm, and solving the Hamiltonian to give you either a dead cat or a live cat!
And this, my dear reader, is the Schrodinger’s cat formulation.