At long last, the previously mentioned Quantum Mechanics post!!! Quantum Mechanics is essential to our understanding of the world, and was born around the turn of the (19th-20th) century. It’s rather counter-intuitive from the point of view of our everyday experience so things like thought experiments must be used to gain intuition about what’s going on. WARNING: although quantum mechanics is a very successful theory experimentally, there are many interpretations of what is actually happening. Indeed, there is an entire field of physics (which I am NOT an expert in) devoted to understanding quantum mechanics.
However, I think it is fair to say (please comment if you disagree) that for all currently observable phenomena, different interpretations of quantum mechanics are indistinguishable. Therefore, I’ll present the point of view I think is easiest to understand. We’ve got to start with `classical’ physics, that is all of physics before quantum mechanics. This is the physics of dropping balls and crashing cars and predicting when trains will arrive. Our intuition about how this works should be pretty good, as it describes the macroscopic world in which we live.
The most succinct way I know to formulate classical physics is in terms of the ‘Least Action Principle': Given a starting configuration of a system and an ending configuration, I want to know how the system proceeds from the starting configuration to the ending configuration. For example, if a baseball (from a really boring American sport) is in my hand now and through my office window 2 seconds later, what path did it take to get there? For each possible path, like going on a straight line between my hand and the window or taking a detour around the moon before hitting the window, I can assign a number, called the ‘action’. The path going around the moon has waaaaaaay more action than the straight line path. In fact, the path that the baseball takes is the one with the least action. End of story. For a given starting and ending point, it always takes this least action path.
Of course, I can also turn this around and specify the initial position of the baseball (my hand) and the direction and magnitude of its initial velocity (toward the window and super fast). Once I do that, the path that the baseball takes is fixed. Always. If I throw a million baseballs through a million office windows (yeah, it’s that kind of day at work), all starting from my hand and ending at the window 2 seconds later, each one will follow the same least action path.
This calls for an equation! There’s a general principle in physics that things behave smoothly, except when they don’t. When I’m assigning an action to each path, two paths which are almost identical will have almost identical actions. As I move toward the least action path, the action should decrease in a more-or-less smooth fashion, with the minimum at the the least action path (duh). If I move past this least action (or ‘classical’) path, the action will increase again (also duh). So a cartoon plot of what’s going on might look like this:
Fig.1: As the path is smoothly varied toward the classical path, the action decreases. The tangent line at the classical path has zero slope
This plot is just a fancy way to say that the least action path is the one which has the minimum action. Because of that, the tangent to the curve at the least action path is flat, i.e. it has zero slope. We can use this fact to encapsulate all of classical physics in a single tiny equation:
This equation says that the path the system takes is the one for which the action has a flat tangent, i.e. the slope of the tangent of the action at the classical path is zero. This also means that as I change a tiny bit from the classical path the action does not change that much. The bottom of the well in the above plot is the only place where this is true as any other point on the curve does not have a flat tangent.
Finally! We can stop talking about boring old classical physics! The main point of the preceding discussion was that classical physics is deterministic, i.e. if I specify the start and end points, or the start point and the initial velocity, I can predict with certainty what the system will do at every moment of the future. When we are doing a classical physics homework problem (booo!) the `answer’ is x(t), the state of the system at all time, after specifying the start and end points, or the start point and initial velocity.
Quantum mechanics is way different. If I were doing a quantum mechanics homework problem (<cough> nerd!) the answer would be instead be a set of numbers for each time, usually written
Each number is the probability of finding the system in state x at time t. In our example, given the initial and final positions of the baseball (my hand and the shattered window), at each time I have no idea where the baseball is, I only have a probability for each point in space. In fact, at any given time, there is a non-zero (but very small) probability that the baseball may be on the moon! Of course I do know that the baseball has to be somewhere at each time so I can write
which just says that the sum of all probabilities at a certain time has to be one, i.e. that I will always find the baseball somewhere in the universe. The thing on the right side of Eq. (2) is a bit strange. The angle brackets inside the absolute value sign are called a probability amplitude and represent a complex number. What the hell is a complex number, and why am I using it to describe something in the real world? A complex number is just a number with two parts, a real part and an imaginary part. A complex number can be specified two ways, either by giving the real and imaginary parts, or by giving the distance from zero and the angle with the real axis. See the figure. Do it.
Fig. 2: A complex number has real and imaginary parts. It can also be specified by it's length (distance from 0) and the angle it makes with the Real axis.
Formally, I can write the two representations of the same complex number as
where x,y, r, and φ are given in the figure and i is the square root of -1 (crazy).
So, the problem of calculating the probability density ρ(x,t) has been reduced to calculating the complex probability amplitude. Then we just take the length of that complex number and square it to get the probability density. There are many ways to calculate this amplitude but I think the easiest one to understand was given by Richard Feynman (in his P.H.D thesis!) which expresses the amplitude as a sum over all paths (or histories) that the system could possibly take:
This is by far the most complicated equation we’ve encountered so far. The left side is the probability amplitude for finding the system in the specified initial and final states at the initial and final times, respectively. Remember, if we wanted the actual probability, we would have to take the length of this complex number and square it. The right side of Eq. 5 is a sum over all possible paths that start in the initial state and end in the final state. Each path gets weighted by a ‘phase factor’ in the sum. This is the exponential factor, which is just a complex number (look at Eq. 4) with length one. This equation also contains a new fundamental constant of Nature: Planck’s constant (h).
It is the small numerical value of this constant (h = 6.62606957(29)×10−34J·s) that makes quantum mechanical effects so far removed from our everyday world. For comparison, the action (which has units of [energy] [time]) of a 0.14kg baseball traveling at 75mph ( roughly 33.5m/s, way slower than during my teenage baseball years) for 0.1 seconds is 78.55 J·s, or about 35 orders of magnitude (factors of 10) bigger than h!
The smallness of this constant may also be used to predict Eq. 1, the defining equation of classical physics. When summing up all the phase factors corresponding to all the possible paths in Eq. (5), we have to add the arrows head-to-tail. If we have an arrow pointing along the Real axis (angle = 0 degrees) and we add to it an arrow pointing the opposite way (angle = 180 degrees), we end up with nothing. In our case, the angle corresponds to the ratio S/h, the action of a particular path divided by Planck’s constant. If we examine a random path in the sum, say the one that goes around the moon before hitting my office window, the action will change quite a lot (compared to h) if we vary the path a tiny bit. This is because this path does not have a flat tangent in Fig. 1. In fact, for every path that we choose in this way, there is a nearby path whose arrow points in exactly the opposite direction, and the effect of the two paths cancel out as we sum them.
The only path which does not have a nearby path to cancel it is the path for which the action does not change very much when we move away from it. This is precisely the condition that we gave earlier for the classical path!!! So for a system (like the baseball and window) where the action is much larger than Planck’s constant, the only path that significantly contributes to the sum in Eq. (5) is the classical path, and we can recover Eq. (1) which determines the condition for the classical path. Furthermore, for this situation the probability that the system does not follow the classical path is certainly not zero, but it is so small that we will never see it happen. Maybe not even once if I throw baseballs at my office window all day everyday for the entire age of the universe. Perhaps I should write a research grant application for such an experiment!
Anyway, now that we’ve seen that classical mechanics is recovered as a limiting case of the Feynman path integral, we can ponder some of the implications. Of course the probabilities calculated from the path integral agree extremely well with observations, so for systems where the action is not so much larger than h we can observe effects of this sum over paths. Some famous manifestations of this are the double-slit and Stern-Gerlach experiments, but there are a ton of others. I suppose the Keanu-Reeves-Bill-and-Ted’s-Excellent-Adventure quasi-philosophical interpretation of what’s happening would be that when deciding the probability for a certain initial and final state, Nature somehow travels all possible paths (assigning each one an arrow) before summing them up to get the probability, causing the `wave particle duality’ that is so peculiar in quantum mechanics. However, this is simply one interpretation.
Of course, there are a lot of things I glossed over in this post, like the problem of measurement in quantum mechanics, some of the paradoxes, and different interpretations (here is an overview). Anyway, all I’ve done is provide a rule for calculating probabilities, which are all that can be measured. Back to throwing baseballs at office windows!!!