January 2021

Conservation of energy is a somewhat sacred principle in physics, though it can be tricky in certain circumstances, such as an expanding universe. Quantum mechanics is another context in which energy conservation is a subtle thing — so much so that it’s still worth writing papers about, which Jackie Lodman and I recently did. In this blog post I’d like to explain two things:

• In the Many-Worlds formulation of quantum mechanics, the energy of the wave function of the universe is perfectly conserved. It doesn’t “require energy to make new universes,” so that is not a respectable objection to Many-Worlds.
• In any formulation of quantum mechanics, energy doesn’t appear to be conserved as seen by actual observers performing quantum measurements. This is a not-very-hard-to-see aspect of quantum mechanics, which nevertheless hasn’t received a great deal of attention in the literature. It is a phenomenon that should be experimentally observable, although as far as I know it hasn’t yet been; we propose a simple experiment to do so.

The first point here is well-accepted and completely obvious to anyone who understands Many-Worlds. The second is much less well-known, and it’s what Jackie and I wrote about. I’m going to try to make this post accessible to folks who don’t know QM, but sometimes it’s hard to make sense without letting the math be the math.

First let’s think about energy in classical mechanics. You have a system characterized by some quantities like position, momentum, angular momentum, and so on, for each moving part within the system. Given some facts of the external environment (like the presence of gravitational or electric fields), the energy is simply a function of these quantities. You have for example kinetic energy, which depends on the momentum (or equivalently on the velocity), potential energy, which depends on the location of the object, and so on. The total energy is just the sum of all these contributions. If we don’t explicitly put any energy into the system or take any out, the energy should be conserved — i.e. the total energy remains constant over time.

There are two main things you need to know about quantum mechanics. First, the state of a quantum system is no longer specified by things like “position” or “momentum” or “spin.” Those classical notions are now thought of as possible measurement outcomes, not well-defined characteristics of the system. The quantum state — or wave function — is a superposition of various possible measurement outcomes, where “superposition” is a fancy term for “linear combination.”

Consider a spinning particle. By doing experiments to measure its spin along a certain axis, we discover that we only ever get two possible outcomes, which we might call “spin-up” or “” and “spin-down” or “.” But before we’ve made the measurement, the system can be in some superposition of both possibilities. We would write , the wave function of the spin, as

   

where and are numerical coefficients, the “amplitudes” corresponding to spin-up and spin-down, respectively. (They will generally be complex numbers, but we don’t have to worry about that.)

The second thing you have to know about quantum mechanics is that measuring the system changes its wave function. When we have a spin in a superposition of this type, we can’t predict with certainty what outcome we will see. All we can predict is the probability, which is given by the amplitude squared. And once that measurement is made, the wave function “collapses” into a state that is purely what is observed. So we have

   

At least, that’s what we teach our students — Many-Worlds has a slightly more careful story to tell, as we’ll see.

We can now ask about energy, but the concept of energy in quantum mechanics is a bit different from what we are used to in classical mechanics. …

Energy Conservation and Non-Conservation in Quantum MechanicsRead More »