This will likely get slightly math-y, however please bear with me. I’ll attempt to preserve it (as Einstein would say) “so simple as potential, however not less complicated”.
Begin with sound waves that come from an orchestra. The strain in your eardrum goes up and down many occasions per second in a sample that appears chaotic to the attention.
That is the uncooked sign that the ear is receiving, however the ear hears musical notes! Tiny hairs within the ear are tuned every to a unique frequency, and choose up simply that a part of the sign that’s (for instance) an E flat above center C on the piano.
What the ear does mechanically may also be completed with arithmetic. The process was found by Jean-Baptiste (“Joe”) Fourier and first revealed in 1822. The process includes an excessive amount of arithmetic, although it may be written in a really compact notation utilizing integral calculus. You don’t have to know something concerning the process besides that it’s referred to as a Fourier Rework, and it turns that strain operate — how robust was the strain at every second of time — into one other operate that tells you ways a lot of every musical notice was current within the sound that the ear heard.
A pure query to ask is about when every notice was performed by the orchestra. The Fourier rework incorporates this info as nicely. It’s handled as a really low frequency. For instance, the orchestra could play 5 notes C D E F G in a single second. The ear hears this as 5 completely different notes in a rhythmic sequence. To the ear, the time scale of hundredths of a second defines the notice, however the time scale of tenths of a second is heard as separate notes. However to the Fourier rework, these two time scales are a part of a continuum. The Fourier rework treats tenths of a second and hundredths of a second with the identical arithmetic. The result’s that the Fourier rework incorporates details about which notes are performed and likewise when they’re performed, all coded in the identical mathematical operate.
These two graphs, the 2 features include precisely the identical info. They’re completely different mathematical descriptions, however they correspond to precisely the identical bodily scenario. Each are methods of describing the sound waves that impinge in your ear.
We all know that they’ve the identical info as a result of you may carry out this mathematical trick, the Fourier rework, and convert the strain illustration into the notes-of-the-scale illustration. (A physicist would name the previous the “time area” and the latter the “frequency area”.)
The Fourier rework converts the graph within the time area to a graph within the frequency area. In the event that they actually include the identical info, it must be potential to carry out one other mathematical process, name it the “inverse Fourier rework”, and convert the frequency area again to the time area. The inverse rework means that you can get better the primary graph from the second graph.
“Joe” had this all labored out. The Fourier rework converts the time graph to the frequency graph. The Inverse Fourier rework converts the frequency graph again to the time graph.
Now comes the mathemagic that will need to have shocked and delighted Joe, even because it surprises and delights physics college students 200 years later.
The Fourier Rework and the Inverse Fourier Rework are precisely the identical. No matter you do to the primary graph to get the second graph may also be completed to the second graph and, lo and behold, you get again the primary graph.
Keep in mind, the Fourier Rework is an intricate recipe involving oodles of arithmetic that you simply carry out on the numbers that make up the primary graph. Once you comply with this recipe, you find yourself with the second graph. Now when you take the second graph and also you apply the identical recipe, you get again the primary graph.
The identical Fourier Rework, utilized twice, takes you proper again to the unique graph that you simply began with.
The lesson we’ve discovered to this point is that the identical info is contained in these two graphs. The 2 graphs are equal methods of describing a single bodily scenario. Both description will do.
Right here’s an instance that we’ll come again to later. Suppose the orchestra is on strike, and one man comes out on stage with a tuning fork tuned to A 440. He performs a single notice. The graph #1, strain on the ear, is a sine wave. It seems to be like this.
What does graph #2 appear to be? For each notice that’s not A 440 the contribution is zero. So graph #2 is zero in every single place aside from a pointy spike at A 440.
I’ve drawn a wedge, however I ask you to think about that the spike at A440 is infinitely skinny.
Can we Fourier rework this spike again to get the sine wave? Right here’s trick that ought to work: We all know this spike is in frequency house, however we mentioned that the identical Fourier math works ahead or backward. It shouldn’t matter if it’s in frequency house or it’s pressure-vs-time. So let’s think about the spike is in time. It seems to be like a giant strain pulse at one time, with nothing earlier than or after. There’s a pulse of sound at only one second, like a bang or a clap or a click on. The Fourier rework of that ought to appear to be a graph of what frequencies are current in a clap. The ear doesn’t hear any explicit frequency, and the reply is that all frequencies are contributing equally. So once we Fourier rework the clap we get a continuing operate, a line that’s the identical in every single place.
Woops — we had been anticipating to get again the sine wave. What occurred? The 2 are kind of alike, in that they unfold evenly, however one is wavy and the opposite is straight.
The reply is that I cheated you after I mentioned that the Fourier rework is only a recipe for lots of arithmetic operations on the numbers in graph #1. The graph is simply an extraordinary plot of actual numbers. What I didn’t inform you is that, for all of it to work out as I mentioned, the numbers need to be complicated numbers, two-part numbers, actual and imaginary, a + bi.
Now that I’ve mentioned this, I ask you to depart it at the back of your thoughts. It doesn’t matter for a lot of follows. I’ll let you recognize if it is advisable take into consideration imaginary numbers, and principally you gained’t.
Apart: Fourier and Holograms
The place is the knowledge? The strain in your eardrum at each cut-off date relies on all of the completely different notes which might be being performed, as a result of they mix collectively.
This will likely remind you of the best way a hologram works. You possibly can flip {a photograph} right into a hologram, and each level on the hologram incorporates details about the entire picture. A laser beam can create a hologram from the picture, and the identical laser beam can re-create the picture from the hologram.
The similarity will not be a coincidence. A hologram is a Fourier rework in two dimensions.
I hope I’ve satisfied you that the sound wave is one factor, and you may signify it both within the frequency area or within the time area. Both one incorporates the identical info. Maintain that thought as we transfer on to quantum physics.
In acoustics, Joe Fourier’s math works for us, giving us two methods we are able to take a look at any scenario. We are able to take our choose, and work with whichever whichever illustration is extra handy. However in quantum mechanics, Joe’s math works in opposition to us. It’s the identical math, nevertheless it’s not about two completely different representations of the identical factor. It’s about two completely different issues, and we want each of them however Joe tells us we are able to solely have separately. That is the story arising.
Consider a billiard ball bouncing round a desk. If I need to know all the things about how that ball is transferring, all I want is 2 pairs of numbers.
Decide a time, any time. The primary pair of numbers tells me the place the ball is. We are able to consider the x and y coordinates that outline the ball’s place at your chosen time. The opposite factor we have to know is how briskly the ball is transferring, and in what path. That’s one other pair of numbers, that we are able to consider as a pace and an angle, or else, simply nearly as good, we might get the identical info from its velocity within the x path vx and velocity within the y path vy.
As soon as we all know (x, y) and (vx , vy), we all know all there’s to know concerning the scenario. We are able to comply with the billiard ball across the desk. We are able to know the place it will be and how briskly it’s transferring at any time sooner or later.
It’s not sufficient to know (x, y) or (vx , vy) individually. We have to know each the place and the rate at anyone time. Then we’ve all the knowledge we have to calculate the ball’s trajectory eternally after.
With this basis, we’re prepared for one of many nice paradoxes of quantum physics. In QM, we’re not working with the precise place of the billiard ball. As an alternative we’ve a “wave operate” that tells us the likelihood of the the billiard ball being at a given place on the inexperienced desk. The wave operate associates a likelihood with each level on the desk. If the wave operate is small across the edges and enormous within the heart, that implies that there’s a excessive likelihood that the ball is close to the middle of the desk.
Now we’re prepared to usher in the Fourier math. Right here’s a profound fact that Neils Bohr bequeathed to us: There’s a wave operate for the place of the ball, as we simply described. The place wave operate tells you the likelihood the ball shall be discovered at any given location. There’s additionally a wave operate for the rate of the identical ball. The speed wave operate tells you the likelihood of the ball having any given velocity.
The speed wave operate is the Fourier rework of the place wave operate.
If this doesn’t make any sense to you, you then’re understanding one thing essential. We need to know the place the ball is, and QM tells us we are able to’t comprehend it precisely. We are able to solely know a likelihood distribution. OK we are able to stay with that. Now, we additionally need to know the rate of the ball. And Bohr is telling us that this isn’t a separate likelihood distribution. It’s already decided once we specified the likelihood distribution for the ball’s location.
Bohr referred to as it “complementarity”, however having a reputation for it doesn’t make it any extra smart.
Make this sensible. Say we pin down the ball within the heart of the desk. We all know precisely the place it’s. So its wave operate is a skinny spike, as we described above for the sound of a clap.
The unconventional dictate of the Complementarity Precept tells us that we don’t need to measure the ball’s velocity. We already know all that may be recognized about it, and that’s the velocity wave operate. The speed wave operate is the Fourier rework of the place wave operate. Above, we did that train — we mentioned {that a} clap has no explicit sound frequency, so it incorporates equal quantities of all frequencies. The Fourier rework of the spike is the fixed operate, the identical in every single place.
Translate the maths to physics, and we get this very inconvenient, very unwelcome limitation. Once we know the precise place of the ball, its velocity is as unknown as it may be. It has the identical likelihood of going quick or sluggish, of going left or proper or backwards or forwards. The speed is totally random.
However wait a second, you say. Let’s simply measure the rate. Let’s connect a speedometer to this ball and learn how quick it’s transferring.
There. Now we all know precisely how briskly the ball is transferring.
The foundations of QM let you do that, however there’s a catch. This second measurement makes the primary out of date. We now know the ball’s velocity precisely, however we all know nothing concerning the place. We now have to replace the place wave operate. Now it’s the Fourier rework of the rate wave operate that we simply measured. We measured the rate precisely, so the rate wave operate is a spike. Meaning the place wave operate is identical in every single place. The ball could be anyplace on the desk, with equal likelihood.
We are able to know the place precisely, after which we all know nothing about velocity. Or we are able to know velocity precisely, after which we all know nothing about place.
Are you pondering of a compromise? How a few bell-shaped curve? Nice minds assume alike. Werner Heisenberg was pondering the identical factor. We’re used to pinning down a variety of possibilities with a bell-shaped curve. Let’s measure the ball’s place roughly, so we get a bell-shaped curve for the likelihood of its place.
Right here’s one other lovely outcome from arithmetic: The Fourier rework of a bell-shaped curve is one other bell-shaped curve. The Fourier rework of a slim bell-shaped curve is a large bell-shaped curve. (And, after all, the Fourier rework of a large bell-shaped curve is a slim bell-shaped curve.) It is a mathematical theorem, unknown to Joe Fourier, however proved by Werner Heisenberg a century later, as he was growing the maths he wanted for quantum physics. Once you translate the maths into physics, what do you assume it tells us?
Sure, that is the well-known Heisenberg Uncertainty Precept. A slim bell curve for the place means a large bell curve for the rate, and vice versa. The higher you recognize the rate, the more serious you recognize the place. The higher you recognize the place, the more serious you recognize the rate.
So our dream of with the ability to predict the movement of the billiard ball on the inexperienced desk is eternally thwarted. We are able to solely know possibilities, and as time goes on, the ball would possibly unfold itself all around the desk.
Muss es sein? Should it’s? Can’t we’ve a compromise for the place wave operate and the rate wave operate that stays put and doesn’t change? The reply is sure, and one instance is Schrödinger’s resolution for the wave operate of a lone electron in a hydrogen atom. However that’s a narrative for one more day.
The unusual truth of quantum physics that I want to depart you with is that when you take a snapshot of an electron at a single second in time, and the snapshot has some decision, in order that you recognize with some fuzzy decision the place the electron is situated, then that very same snapshot tells you all that it’s potential to learn about how briskly the electron is transferring and in what path. Even stranger, when you simply connected a speedometer to the electron and also you measured how briskly it’s transferring, once more with some approximate precision, they you’ll even have details about the place the electron is situated, once more with a likelihood distribution.
