So far, in the context of 1920s quantum physics, I’ve given you a sense for what an ultra-microscopic measurement consists of, and how one can make a permanent record of it. [Modern (post-1950s) quantum field theory has a somewhat different picture; please keep that in mind. We’ll get to it later.] Along the way I’ve kept the object being measured very simple: just an incoming projectile with a fairly definite motion and moderately definite position, moving steadily in one direction. But now it’s time to consider objects in more interesting quantum situations, and what it means to measure them.
The question for today is: what is a quantum superposition?
I will show you that a quantum superposition of two possibilities, in which the wave function of a system contains one possibility AND another at the same time, does not mean that both possibilities occur; it means that one OR the other may occur.
Instead of a projectile that has a near definite motion, as we’ve considered in recent posts, let’s consider a projectile that is in a quantum superposition of two possible near-definite motions:
- maybe it is moving to the left at a near-definite speed, or
- maybe it is moving to the right at a similar near-definite speed.
This motion is along the x-axis, the coordinate of a one-dimensional physical space. If the projectile is isolated from the rest of the world, we can write a wave function for it alone, which might initially look like
in which case its evolution over time will look like this:
Again I emphasize this is not the wave function of two particles, despite what you might intuitively guess. This is the wave function of a single particle in a superposition of two possible behaviors. For a similar example that we’ll return to in a few weeks, see this post.
Because the height and speed of the two peaks is the same, there is a left-to-right symmetry between them. We can therefore conclude, before we even start, that there’s a 50-50 chance of the particle going right versus going left. More generally, whatever we observe to the left (x<0) will happen with the same probability as what we observe to the right (x>0).
Today I will show you that even though the wave function has one peak moving to the left AND one peak moving to the right, nevertheless this wave function does not describe a projectile that is moving to the left AND moving to the right. Instead, it means that the projectile is moving to the left OR moving to the right. Superposition is an OR, not an AND. In other words, in pre-quantum language, we have either
We never have both.
But don’t take my word for it. Let’s see how quantum physics actually works.
First Measurement: A Ball to the Left
Our first goal: to detect the projectile if it is moving to the left.
Let’s start by doing almost the same thing we did in this post, which you may want to read first in order to understand the pictures and the strategy that I’ll present below. To do this, we’ll put a measurement ball on the left, which the projectile will strike if it is moving to the left.
Since we now have a system of two objects rather than one, the space of possibilities for the system now has to be two-dimensional, to include both the position x1 of the projectile and the position x2 of the ball. This now requires us to consider a wave function for not just the projectile alone, as we did in Figs. 1 and 2, but for the projectile and the ball together. This wave function will give us probabilities for each possible arrangement of the projectile and ball — for each choice of x1 and x2.
We’ll put the ball at x2 = -1 initially — to the left of the projectile initially — so that the initial wave function looks like Fig. 4, which shows its absolute value squared as a function of x1 and x2.
This wave function has the same shape in x1 as the wave function in Fig. 1, but now centered on the line x2=-1. A collision between projectile and ball will become likely when a peak of the wave function approaches the point x1=x2=-1.
As usual, let’s try to think about this in a pre-quantum language first. If I’m right about wave functions, we have two options:
- The projectile is heading to the left and the measurement ball will react OR
- The projectile is heading to the right and the measurement ball will not react.
Since our wave function is left-to-right symmetric, each option is equally likely, and so if we do this experiment repeatedly, we should see the ball react about half the time.
Here are the two pre-quantum options shown in the usual way, with
- a picture in physical space on the left, and
- a picture of the same event in the space of possibilities on the right.
In the first possibility (Fig. 5a), the projectile moves left, strikes the ball, and the ball recoils to the left. As the ball moves to the left in physical space, the system moves down (toward more negative x2) in the space of possibilities.
OR
In the second possibility (Fig. 5b), the projectile moves right and the ball remains unscathed; in this case, viewed in the space of possibilities, x2 remains at -1 during the entire process while x1 changes steadily toward more positive values.
What about in quantum physics? The wave function should include both options in Figs. 5a and 5b.
Here is an actual solution to the Schrödinger wave equation, showing that this is exactly what happens (and it has more details than the sketches I’ve been doing in my measurement posts, such as this one or this one.) The two peaks spread out more quickly than in my sketches (and I have consequently adjusted the vertical axis as time goes on so that the two bumps remain easily visible.) But the basic prediction is correct: there are indeed two peaks, one moving like the pre-quantum system in Fig 5a, changing direction and moving toward more negative x2, and the other moving like the pre-quantum system in Fig. 5b, moving steadily toward more positive x1.
Importantly, even though the system’s wave function displays both possibilities to us at the same time, there is no sense in which the system itself can be in both possibilities at the same time. The system has a near-50% probability of being observed to be within the first peak, near-50% probability of being observed to be within the second, and exactly 0% probability of being observed within both.
Second Measurement: A Ball to the Right
Now let’s put a ball to the right instead, at x=+1. This is a different ball from the previous (we’ll use both of them in a moment) so I’ll color it differently and call its position x3. The pre-quantum behaviors are the same as before, but with x2 replaced with x3 and with the collision happening at positive values of x1 and x3 instead of negative values of x1 and x2.
OR
The quantum version is just a 180-degree rotation of Fig. 6 with x2 replaced with x3.
Third Measurement: A Ball on Both Sides
But what happens if we put a ball on the left and a ball on the right? Initially the balls are at x2=-1 and x3=+1. What happens later?
Now there are four logical possibilities for what might happen:
- The ball on the left responds while the ball on the right does not
- The ball on the right responds while the ball on the left does not
- Neither ball responds
- Both balls respond
Where in the space of possibilities do these four options lie? The four logical possibilities listed above would put the ball’s positions in these four possible places:
- Option 1: x2 < -1 and x3 = +1 (and x1 negative, as in Fig. 5a)
- Option 2: x2 = -1 and x3 > +1 (and x1 positive, as in Fig. 7a)
- Option 3: x2 = -1 and x3 = +1 (and x1 is ???)
- Option 4: x2 < -1 and x3 > +1 (and x1 is ???)
The fact that it is not obvious where to put x1 in the last two options should already make you suscpicious; but just setting their x1 to zero for now, let’s draw where these four options occur in the space of possibilities. In Fig. 9 I’ve drawn the lines x2=-1 and x3=+1 across the box, with option 3 at their crossing point. Option 1 lies below down and to the left of option 3; option 2 is found to the rigt of option 3; and option 4 is found down and to the right.
What does the wave function actually do? Can the simple two-humped superposition at the start, analogous to Fig. 4, end up four-humped?
Not in this case, anyway. Fig. 10, which depicts the peaks of the absoulte-value-squared of the wave function only, shows the output of the Schrödinger equation. Compare the result to Fig. 9; there are peaks only for options 1 and 2, in which one ball moves and the other does not.
With balls on either side of it, the projectile cannot avoid hitting one of them, whether it goes right or left, which rules out option 3. And the wave function does not put a peak at option 4, showing there’s no way the projectile can cause both balls to move. The two peaks in the wave function move only in the x1 direction as the projectile goes left OR right; then the projectile collides with one ball OR the other; then the ball with which it collided moves, meaning that the system moves to more negative x2 (i.e. down in Fig. 10) OR to more positive x3 (i.e. to the right in Fig. 10), just as expected from Fig. 9.
Actually it’s not difficult to get the third option — but we don’t need quantum physics for that!
We simply change the original wave function to contain three possibilities: the projectile moves left, or it moves right, or it doesn’t move at all. If it doesn’t move at all, then neither ball will react, a third option even in pre-quantum physics:
If the projectile were isolated, we would encode this notion in a wave function which looks like this:
and when we include the two balls we would see the wave function with three peaks, one sitting still at the point marked “Neither Ball Moves” in Fig. 9. But this isn’t particularly exciting or surprising, since it’s intuitively obvious that a stationary projectile won’t hit either ball.
Every Which Way
There simply is no wave function you can choose — no initial superposition for the single projectile — which can cause the projectile to collide with both balls. The equations will never let this happen, no matter what initial wave function you feed into them. It’s impossible… because a superposition is an OR, not an AND. There is no way to make the projectile go left AND right — not if it’s a particle in 1920s quantum physics, anyway.
Yes, the wave function itself can have peaks that appear at to be in several places at the same time within the space of possibilities, as in Figs. 6, 8, and 10. But the wave function is not the physical system. The wave function tells us about the probabilities for the system’s possibilities; its peaks are just indicating what the most likely possibilities are.
The system itself can only realize one of the many possibilities — it can only be found (through a later measurement) in one place within the space of possibilities. This is always true, even though the wave function for the system highlights all the most probable possibilities simultaneously.
A particle, in the strict sense of the term, is an object with a position and a momentum, even though we cannot know both perfectly at any moment, thanks to Heisenberg’s uncertainty principle. It can only be measured to be in one place, or can only be measured to be traveling in one direction, at a time. In 1920s quantum physics, these statements apply to an electron, which is viewed as a strict particle, and so it cannot go in two directions at once, nor can it be in two places at once. The fact that we are always somewhat ignorant of where an electron is and/or where it is going, and the fact that quantum physics puts ultimate limitations on our ability to know both simultaneously, do not change these basic conceptual lessons… the lessons of (and for) the 1920s.
