I want to make a few very fundamental comparisons between classical and quantum mechanics. I’ll be assuming a lot of background in this particular post to prevent it from getting uncontrollably long, but am planning on writing a series on quantum mechanics at some point.
Let’s assume that the universe consists of N simple point particles (where N is an ungodly large number), each interacting with each other in complicated ways according to their relative positions. These positions are written as x1, x2, …, xN.
The classical description for this simple universe makes each position a function of time, and gives the following set of N equations of motion, one for each particle:
Fk(x1, x2, …, xN) = mk · ∂t2xk
Each force function Fk will be a horribly messy nonlinear function of the positions of all the particles in the universe. These functions encode the details of all of the interactions taking place between the particles.
Analytically solving this equation is completely hopeless – It’s a set of N separate equations, each one a highly nonlinear second order differential equation. You couldn’t solve any of them on their own, and on top of that, they are tightly entangled together, making it impossible to solve any one without also solving all the others.
So if you thought that Newton’s equation F = ma was simple, think again!
Compare this to how quantum mechanics describes our universe. The state of the universe is described by a function Ψ(x1, x2, …, xN, t). This function changes over time according to the Schrödinger equation:
∂tΨ = -i·H[Ψ]
H is a differential operator that is a complicated function of all of the positions of all the particles in the universe. It encodes the information about particle interactions in the same way that the force functions did in classical mechanics.
I claim that Schrodinger’s equation is infinitely easier to solve than Newton’s equation. In fact, I will by the end of this post write out the exact solution to the wave function of the entire universe.
At first glance, you can notice a few features of the equation that make it look potentially simpler than the classical equation. For one, there’s only one single equation, instead of N entangled equations.
Also, the equation is only first order in time derivatives, while Newton’s equation is second order in time derivatives. This is extremely important. The move from a first order differential equation to a second order differential equation is a huge deal. For one thing, there’s a simple general solution to all first order linear differential equations, and nothing close for second order linear differential equations.
Unfortunately… Schrodinger’s equation, just like Newton’s, is highly highly nonlinear, because of the presence of H. If we can’t find a way to simplify this immensely complex operator, then we’re probably stuck.
But quantum mechanics hands us exactly what we need: two magical facts about the universe that allow us to turn Schrodinger’s equation into a linear first-order differential equation.
First: It guarantees us that there exist a set of functions φE(x1, x2, …, xN) such that:
H[φE] = E · φE
E is an ordinary real number, and its physical meaning is the energy of the entire universe. The set of values of E is the set of allowed energies for the universe. And the functions φE(x1, x2, …, xN) are the wave functions that correspond to each allowed energy.
Second: it tells us that no matter what complicated state our universe is in, we can express it as a weighted sum over these functions:
Ψ = ∑ aE · φE
With these two facts, we’re basically omniscient.
Since Ψ is a sum of all the different functions φE, if we want to know how Ψ changes with time, we can just see how each φE changes with time.
How does each φE change with time? We just use the Schrodinger equation:
∂tφE = -i · H[φE]
= -iE · φE
And we end up with a first order linear differential equation. We can write down the solution right away:
φE(x1, x2, …, xN, t) = φE(x1, x2, …, xN) · e-iEt
And just like that, we can write down the wave function of the entire universe:
Ψ(x1, x2, …, xN, t) = ∑ aE · φE(x1, x2, …, xN, t)
= ∑ aE · φE(x1, x2, …, xN) · e-iEt
Hand me the initial conditions of the universe, and I can hand you back its exact and complete future according to quantum mechanics.
Okay, I cheated a little bit. You might have guessed that writing out the exact wave function of the entire universe is not actually doable in a short blog post. The problem can’t be that simple.
But at the same time, everything I said above is actually true, and the final equation I presented really is the correct wave function of the universe. So if the problem must be more complex, where is the complexity hidden away?
The answer is that the complexity is hidden away in the first “magical fact” about allowed energy states.
H[φE] = E · φE
This equation is a highly non-linear and in general second-order differential equation. If we actually wanted to expand out Ψ in terms of the different functions φE, we’d have to solve this equation.
So there is no free lunch here. But what’s interesting is where the complexity moves when switching from classical mechanics to quantum mechanics.
In classical mechanics, virtually zero effort goes into formalizing the space of states, or talking about what configurations of the universe are allowable. All of the hardness of the problem of solving the laws of physics is packed into the dynamics. That is, it is easy to specify an initial condition of the universe. But describing how that initial condition evolves forward in time is virtually impossible.
By contrast, in quantum mechanics, solving the equation of motion is trivially easy. And all of the complexity has moved to defining the system. If somebody hands you the allowed energy levels and energy functions of the universe at a given moment of time, you can solve the future of the rest of the universe immediately. But actually finding the allowed energy levels and corresponding wave functions is virtually impossible.
Let’s get to the strangest (and my favorite) part of this.
If quantum mechanics is an accurate description of the world, then the following must be true:
Ψ(x1, x2, …, xN, 0) = ∑ aE · φE(x1, x2, …, xN)
Ψ(x1, x2, …, xN, t) = ∑ aE · φE(x1, x2, …, xN) · e-iEt
This equation has two especially interesting features. First, each term in the sum can be broken down separately into a function of position and a function of time.
And second, the temporal component of each term is an imaginary exponential – a phase factor e-iEt.
Let me take a second to explain the significance of this.
In quantum mechanics, physical quantities are invariably found by taking the absolute square of complex quantities. This is why you can have a complex wave function and an equation of motion with an i in it, and still end up with a universe quite free of imaginary numbers.
But when you take the absolute square of e-iEt, you end up with e-iEt · eiEt = 1. What’s important here is that the time dependence seems to fall away.
A way to see this is to notice that y = e-ix, when graphed, looks like a point on a unit circle in the complex plane.
So e-iEt, when graphed, is just a point repeatedly spinning around the unit circle. The larger E is, the faster it spins.
Taking the absolute square of a complex number is the same as finding its distance from the origin on the complex plane. And since e-iEt always stays on the unit circle, its absolute square is always 1.
So what this all means is that quantum mechanics tells us that there’s a sense in which our universe is remarkably static. The universe starts off as a superposition of a bunch of possible energy states, each with a particular weight. And it ends up as a sum over the same energy states, with weights of the exact same magnitude, just pointing different directions in the complex plane.
Imagine drawing the universe by drawing out all possible energy states in boxes, and shading these boxes according to how much amplitude is distributed in them. Now we advance time forward by one millisecond. What happens?
Absolutely nothing, according to quantum mechanics. The distribution of shading across the boxes stays the exact same, because the phase factor multiplication does not change the magnitude of the amplitude in each box.
Given this, we are faced with a bizarre question: if quantum mechanics tells us that the universe is static in this particular way, then why do we see so much change and motion and excitement all around us?
I’ll stop here for you to puzzle over, but I’ve posted an answer here.
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