22. Quantum Mechanics and Richard Feynman
Wave-Particle Duality, Probabilities and Wave Functions
Last weekend, I picked up a book on quantum mechanics, Quantum Man: Richard Feynman’s Life in Science, from an Upper West Side bookstore. Admittedly, I initially intended to go to the Natural History Museum, but stumbled into this store after the musuem’s line was around the block… and I wasn’t waiting in that line, especially in the rain. With luck, I recovered the day by finding this book. The notes below are from one of the chapters.
Wave-Particle Duality
Quantum mechanics says that particles appear to be in many different places at once, while also doing many different things at the same time in each place.
Also, all particles behave in some sense like waves and all waves behave in some sense like particles. This gives rise to the wave-particle duality theory, which not necessarily contradicts classical physics, but extends the description of how particles and waves behave.
For instance, classical physics describes particles like electrons and photons as distinct entities that are either particle-like or wave-like, depending on the experiment. Light was considered more wave-like in the context of interference and diffraction, while particles were treated as discrete entities when discussing their motion.
For clarity, the difference between a particle and a wave is that a particle is located at a specific point, while the wave is spread out over some region of space.
Now, wave functions don’t describe the particle itself, but the probability of finding the particle at any given place in space at a specific time.
If the wave function, and thus the probability of finding a particle, is nonzero at many different places, then the particle acts like it is in many different places at any one time!
This is what’s known in quantum mechanics as superposition. Put differently, it describes the ability of quantum systems (i.e., particles) to exist in many different states simultaneously.
The description of this quantum state is represented by the wave function. The quantum state represents a range of possible outcomes or properties of the system. So, superposition is the combination of these possible outcomes / states at one time.
Wave Functions
Moving into a more detailed discussions on wave functions - given the wave function at any one time, you can calculate the wave function of the particle at a later time.
This is no different than the deterministic nature of Newton’s Laws of Motion that describe, say, a baseball’s motion over time. Or, Maxwell’s famous equations describing how electromagnetic waves evolve over time.
However, the difference is that the quantity in quantum mechanics that evolves in time in a deterministic way is not directly observable.
Rather, the quantity represents a set of probabilities for making certain observations, in this case for measuring the particle to be at a certain place in time.
It is the square of the wave function, not the wave function itself, that gives the probabilities of finding a particle at some time.
We’ll look at why it’s the square, which is interesting…
As you know, probabilities are generally positive. You rarely, if ever, see a negative probability for an event or occurrence. Rather, you would say it has a zero probability of occurring. So probabilities are generally >= 0.
And… the square of a quantity is also always greater than or equal to zero (less accurately, the square is positive).
But, this implies that the wave function can be positive or negative. For instance, (-1/2) and (1/2) both return (1/4) when squared.
Here’s an example.
Suppose P1 represents the value of the wave function that corresponds to finding particle A at position X.
P2 represents the value of the wave function that corresponds to finding particle B at position X.
So, the probability of finding either particle A or particle B at position X is now:
(P1 + P2)^2
Suppose P1 = (1/2) and P2 = (-1/2)
The probability of finding particle A at position X is given by:
(1/2)^2 = (1/4)
The probability of finding particle B at position X is given by:
(-1/2)^2 = (1/4)
But, here’s where quantum mechanics gets interesting!
If there are two particles, the probability of finding either particle at position X is:
( (1/2) + (-1/2) )^2 = 0
Think about what this equation shows conceptually. What does the equation mean in reality?
Interference
We know that waves can interfere with each other. Think about noise cancelling headphones in my last post. This is the product of two waves collapsing as they interfere with one another.
Imagine dropping two pebbles simultaneously into a pond. You will have two identical water waves that move outward in circles. When the peak of one wave meets the trough of the other wave at a specific point in the pond, the waves interfere destructively and cancel out. The amplitudes (or height of the wave) are equal in magnitude but oppositive in direction, so they collapse, like the noise cancelling headphones example. This results a flat water surface at this interference point.
Destructive wave interference can happen with other waves, like sounds waves. Concert halls are designed specifically to eliminate dead spots arising from waves meeting and canceling out.
Quantum mechanics, with probabilities being determined by squares, tells us that the particles too can interfere with each other. This makes sense from what we discussed at the beginning around the wave-particle duality theory - if waves are subject to interference, and particles behave like waves, then particles may be subject to interference.
So, if there are two particles in a box, the probability of finding either particle at a given location can end up being less than the probability of finding one of them if there is a single particle in the box.
The math supporting this is above when we looked at the P1 and P2 example! When you observe P1 or P2 individually, the probability is 1/4. But when you observe both, the two particles interfere, and reduce the probability to zero in that example!
The waves functions interfere and collapse, in this case of looking for the two particles. And this makes sense because it’s the mathematical representation of the noise cancelling headphones.
In the headphone example, we assumed we had two opposing waves with the same amplitude (wave height) that cancel out to zero.
In the particle example, the same amplitude can be represented by the probability amplitude of (1/2) and (-1/2), which cancel out!
More on Probabilities, But From a Path Perspective
Let’s look at probabilities from a classical lens.
Suppose a car wants to travel from point A to point C but pass through point B. The probability of choosing some route from A to B is given by P(ab). The probability of choosing some route from B to C is given by P(bc).
The probability of traveling from A to C along a route that goes through point B is given by the product of the two probabilities.
P(abc) = P(ab) x P(bc)
For instance, suppose there is a 50% chance the car takes the route from A to B and a 50% chance of taking the route from B to C. That results in a 25% chance of taking the route A —> B —> C
Now, suppose we send four cars out from point A to point B. Two cars will make it to point B. And, only one of those two cars will make it to point C, yielding the 25% probability of taking the path from A to C through B.
Now, say we don’t care which point B the car passes through between A and C.
P(ac), the probability of traveling A to C, is the sum of the probabilities P(abc) of choosing to go through any point B between A and C.
This makes sense classically because if we go from A to C, and B represents the totality of different points we can cross through, then we have to go through one of these points during the journey.
Now imagine instead of a car we are referring to light rays. Using the principle of least time, also known as Fermat’s principle, we can determine that going through one of the routes, that of least time, represents 100% probability this path will be taken.
Fermat’s principle says that light, when traveling between two points, will follow the path that takes the least time to traverse.
Meaning, in our multiple point B example (pictured above), that one path will have a 100% probability while all others have 0% probabilities.
That’s how classical thinking describes probabilities in this context, but isn’t how quantum mechanics works.
In quantum mechanics, probabilities are determined by the squares of the probability amplitude to go from A to C.
So the probability of going from A to C is not given by the sum of the probabilities to go from A to C via any definitive point B.
Why? Because in quantum mechanics, it is the separate probability amplitudes for each route that multiply and not the probabilities themselves.
The probability amplitude to go from A to C through a definite point B is given by multiplying the probability amplitude to go from A to B times the probability amplitude to go from B to C.
But the actual probability is the square of the sum of these products.
The Effect of Measurement
If you don’t measure which of the two points, say B1 and B2, a particle traverses as it travels through one of the points between A and C, then the probability of arriving at point C is the sum of the squares of the probability amplitudes for the two paths allowed.
However, if we measure which point, B1 or B2, the particle traverses in between A and C, then the probability is the square of the probability for a single point! Which, as we saw mathematically above, can be different from the probability of both when not observed.
The results are different if you don’t measure the particle between the beginning and end compared to what happens if you do take a measurement.
Quantum mechanics allows for all possible paths, with all values of B to be chosen at the same time.
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Until next time.
John Galt