And now for something completely different! I’ve decided to write another series of posts in conjunction with my discussion of Hilbert spaces. As many of you know, I did quite a bit of undergraduate work in physics, and I thought it would be fun to write about some of the things I learned from those courses. There were two branches of physics that I took a particular interest in: electrodynamics (for those of you who saw my senior thesis defense, my research could be considered “classical” electrodynamics) and quantum mechanics. As I hope you will see, the traditional mathematical formulation of quantum mechanics is closely related to the Hilbert spaces I’ve been discussing in my last few posts, and so it seems natural to write about some of the physical applications of abstract Hilbert space theory.
Since many of our readers may not have had basic physics for some time (if ever!) I’ll start with a brief review of some basic physical concepts like energy and momentum and the “classical” equations that describe such systems. I’ll then give a very short description of where these equations break down and why we need quantum mechanics. I’ll discuss the qualitative differences between classical mechanics and quantum mechanics and how we have to change some our intuition about “classical” quantities when we look at the world on a quantum scale. After that, I’ll be able to write down some of the mathematical formalism of quantum mechanics. I’ll describe the equations that govern quantum systems, and then (hopefully) apply these laws to derive the solutions to the quantum harmonic oscillator. Finally, if I make it this far, I’ll show how the quantum harmonic oscillator equations can be applied to the vacuum, and when this is done we’ll see one of the biggest problems in physics: vacuum fluctuations. Anyways, this post will mostly focus on a review of classical physics. Let’s get started.
Imagine you are standing on top of a building and you drop a ball. As time passes, the gravitational field of the earth exerts a force on the ball, causing it to accelerate in the “vertical” direction. If the building is not too tall (so that we neglect air resistance), the ball will continue to accelerate until it comes in contact with ground, where there will be an interaction. The ball and the ground will exchange momentum and the ball will change its trajectory.
The above situation is an example of what would be called a classical physical system and can be analyzed using the laws of classical mechanics. For example, we might try to write down a function that describes the location of the ball at any point in time. For example, since the ball only moves vertically, we might consider some function whose value at the time is the height of the ball relative to the ground. I’ll talk about find this function later. For now, I’ll try to give a bit more precise definition to some of the terms I mentioned above. I won’t try to rigorously define the concepts of position, time, or force; the reader’s intuition should suffice for these. In what follows, unless noted otherwise, the variable will always denote time and will always denote the position function of an object (in some chosen frame of reference, i.e. we choose some coordinate system for ). Boldface variables will denote vectors, while non-boldface variables denote scalars.
Some basic physical quantities:
- The velocity vector of an object is defined to be the vector . It is the (vector) rate of change of an object’s position with respect to time and is also fairly intuitive from one’s every day life.
- The acceleration vector is defined as . It measures the rate of change of the velocity vector.
- The momentum vector is defined as (where is the mass of the object). Momentum is conserved in physical interactions in the sense that if two objects interact (I won’t rigorously what I mean by this either), then the sum of the initial momentum vectors of the objects equals the sum of the final momentum vectors of the objects. Note that momentum is proportional to both velocity and mass.
- The kinetic energy of the object is given by . Energy is also conserved in all closed physical systems, which is a useful fact for performing analysis of a system. Energy, however, can “change forms”. Often we consider both kinetic energy and so called potential energy of a system. In the above problem, there is “potential” gravitational energy given by the initial height of the ball.
- Newton’s Laws, (in particular, the second law), give equations of motion that govern a physical system under the influence of a force . The law states that . This second order differential equation can, in theory, be solved to give the position function mentioned above.
Note that all of these quantities are “continuous”, in the sense that there are no restrictions on possible values for them (except for the obvious ones: an object won’t have negative kinetic energy, etc…). The ball could have energy 0, , or any value in between. We’ll see that this is NOT the case in quantum mechanics. In fact, in the quantum world, the idea of a “position function” won’t make sense! For now, I’ll show one example of how Newton’s laws can be applied to another typical classical physics problem and discuss the solution.
Example: The Harmonic Oscillator Suppose a particle is subject to a (one dimensional) force in the direction given by , where is a (positive) proportionality constant. In other words, the further the object moves from the origin, the stronger the “restoring” force pulls it back. Suppose that the object is initially placed at a location at time . We aim to find the position function that describes the -position of the particle at any time . According to Newton’s second law, the position function can be obtained by solving the initial value problem
I’ll leave the details of this problem to the reader (ask Nicki, she likes these things!), but it turns out the solution is given by
So the particle oscillates back and forth about the origin with a frequency . The velocity and accelerate also oscillate (take derivatives), and so do the momentum and energy. One might ask other questions like, “where is one most likely to find the particle.” Without giving a rigorous argument, the answer is “at the turning points”. This is where the velocity of the particle is lowest, so the particle intuitively should spend most of its time away from the origin. We might ask for the total energy of the system. It is clear the potential energy in this problem comes from the restoring force, which is zero when , which happens (first) at . The velocity at this time is , so , and so the potential energy at the turning points (when ) is . It’s not terribly hard to extend this argument to show that, in fact, at any point , we have (where denotes the potential energy) and that . This idea of a “potential” function will be useful in quantum mechanics, where we will lose the notion of a force.
The harmonic oscillator is an important system in classical physics that shows up in many different places, and I could spend two or three posts discussing its importance. However, the main reason I presented it here is because, later, when we have the tools to describe quantum systems, we will be able to solve the similar quantum harmonic oscillator system, and you may be surprised at some of the differences. I think this about wraps up my discussion of classical (Newtonian) mechanics. There are other formulations of classical mechanics, but they won’t be incredibly useful for us, and the ideas that we do need from these formalisms will be explained later in the context of quantum mechanics. Next time I’ll try to explain where classical mechanics breaks down and why we need to introduce quantum mechanics. What I ask of you is this: if you have any questions about anything from this post or any other classical physical ideas, please ask about them in the comments and I’ll do my best to answer your questions. I want to make sure everyone has decent understanding of physical quantities in the classical sense, so when we pass to the strange world of quantum mechanics, at least we’ll have some intuition for the physical quantities we want to describe mathematically.