A quantum version of Zeno's paradox

The quantum Zeno effect describes the situation where an unstable particle, say a radioactive atom, won’t decay if it is observed continuously. More generally, it says that if you repeatedly interact with a quantum system through measurement then you can effectively freeze its quantum state.

The phenomenon is named after a Greek philosopher of ancient times, Zeno of Elea. Zeno is known best for a set of paradoxes (we know of 9 of them) that he posed as arguments  against Aristotle’s concept of motion. Here we are interested in the arrow or fletcher’s paradox.

If you observe an arrow flying through the air at some particular instant in time, then it would have a definite position, meaning it isn’t moving at that specific moment. However, you can think of the arrow’s motion as happening one moment at a time. This says that motion must be impossible since it is made up of this long sequence of motionless moments. 

Of course, as far as we can tell, the world is not static and objects in it are not forever motionless. What’s lacking with Zeno’s assertion is the mathematical notion of continuity. Motion is possible because time doesn’t flow like a series of separate frames in a film but more like the seamless current of a steady stream.

Spins, magnets, and quantum mechanics

Quantum mechanics is often described as an area of physics that deal with energy and matter at the atomic scales, where different weird, unusual stuff happen. To some extent, it is true that quantum objects behave in ways that seem counter to our everyday, common-sense intuition. Unfortunately, focusing on these particular aspects of quantum theory might give the impression that it is mysterious, mystical, and difficult to understand. And that is simply not true. Things like superposition require a little getting used but, for the most part, quantum mechanics works in ways you expect and that naturally make sense. 

I hope to demonstrate this by discussing an experiment with atoms and magnets that is explained in fairly simple terms using quantum mechanics.  This is largely how I was  introduced to the subject many years ago. But to start, we will need to go over some basic ideas regarding magnets and magnetic fields.

Most of us are familiar with magnets from their ability to attract iron and similar metals but in physics, a magnet is just any material that produces a magnetic field. A magnet influences its surroundings through the magnetic field it creates and reacts to the magnetic fields it experiences from other magnetic objects.

A nice thing about magnets is we can understand how they work without having to be very precise. I'm sure many of you have played around with a bar magnet before, which is often found bent into a horseshoe shape, to create a region of particularly strong magnetic field in between the ends labeled north pole and south pole. You probably also know about and maybe experienced first-hand how opposite poles attract and similar ones repel, and how they attract or repel more when two magnets are brought closer together.  
 
Something less familiar is what determines the force at which a magnet attracts and repels objects. The strength of a magnet is measured by a property called magnetic moment, which is responsible for a magnet's tendency to align with magnetic fields.

Orthogonal states and quantum contextuality

In this post, we use the idea of orthogonal quantum states to describe a fascinating result in quantum mechanics known as the Kochen-Specker theorem. To begin, we review a bit of necessary mathematics.

Recall our usual example of a qubit represented by the spin of an electron. Typically, write the state |E) of such an electron as a superposition of the spin pointing up, which we write as the state |u), and spin pointing down, which we write as the state |d):

|E) = a |u) + b |d),

where a and b are numbers such that |a|^2 is the probability of measuring the spin as up and |b|^2 is the probability of measuring the spin as down.

We've seen before that the set of all possible states for a qubit corresponds to all possible directions the spin can point to, which can be described by using points on a sphere. However, we may choose to write our qubit states using any pair of polar opposite points, and normally we would choose the directions corresponding to north and south pole, which are labeled as the spin states |u) and |d), respectively.