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Cross sections: visualizations are possible but reality is their generalization

In particle physics and similar disciplines, cross sections are quantities that determine the probability that a collision of two objects with a "particular desired outcome" is successful. The Symmetry Magazine wrote an article promoting the concept and saying how wonderful it is that it's used in several disciplines of science.



I want to use the example of the "cross section" concept to show in what sense quantum mechanics "builds upon" the pictures we could draw in classical physics; but it isn't quite one of the classical pictures. So yes, this is a blog post in the "foundations of quantum mechanics" category.

"Cross sections" play the very same role as general "probabilities" of some evolution, transition, or process except that they're optimized for a special class of initial states – initial states that look like two particles or objects heading for a collision.




Let's start with the simple classical picture. Imagine that you're shooting bullets at something or somebody. To have a particular example in mind, shoot bullets from a CZ 75 at a Slovak communist agent. Note that I am not mentioning any specific target by name – we're just fighting pure abstract evil. A bullet that at least touches the agent is considered a "success".




OK, you stand at some distance from him, aim at him, and shoot many times. How many times will you be successful?

It depends on the "size" of the communist agent in some way because the "larger" he is, the easier it is to hit him; and on the precision of your aiming because the more precisely you aim, the higher chances you have to neutralize him. What are the equations? If you shoot \(W\) bullets per second, we can call it a rate. What's more directly relevant is the "flux" \(\Phi\), i.e. the number of bullets per squared meter per second that are flying "somewhere near" the agent. Once again, the units of \(\Phi\) are "one [projectile] per squared meter and per second". A projectile is dimensionless – we just count them. In some rough counting, \(\Phi = W / A\) where \(W=N/t\), the number of bullets per second, and \(A\) is the effective area at the same distance where the flux of the projectiles gets dissolved.

The more precisely you're able to aim, i.e. the "smaller the area" \(A\) in the denominator is, and the higher is the flux \(\Phi = W / A\). Now, to count the number of "successful" bullets, you multiply the rate by a characteristic constant, the cross section \(\sigma\),\[

W_{\rm success} = \Phi \cdot \sigma,

\] which has the units of area. This \(W_{\rm success}\) is the number of projectiles that hit the agent each second. You see that it's proportional to all the natural factors – to the flux \(\Phi\) and to some "size" \(\sigma\) of the agent. This way to express the success rate is natural because it separates all the quantities that matter to two groups: the flux \(\Phi\) takes care of all the parameters describing your gun, its speed, your aiming, the distance etc. while \(\sigma\) includes all the information about the agent himself, as seen from the direction of the projectiles, and his local interactions with the bullet.

It's not hard to see that in the purely classical, geometric case, \(\sigma\) is just the area of the projection of the agent to a two-dimensional plane orthogonal to the direction of the projectiles, by the rays whose direction coincides with the direction of the projectiles. The cross section is literally the cross-sectional area of the agent or his projection.



Note that one can also "refine" the concept of the cross section according to the final state. For example, you may want to know the rate of the successful bullets that hit a squared centimeter of his brain at some location; or events in which the bullets hit him and recoil to hit another region in space afterwards. One may therefore divide the final state to a whole spectrum and describe the "success rate" as an integral over some variables such as angles or a coordinate describing a location in his brain. The integrand – usually in an integral over angles describing the direction of motion of some final objects – is called the "differential cross section".

Great. What I presented above is a pedagogic model of the cross section. You can learn how to think about the cross section, flux, and various rates if you have a particular projectile and a particular target in mind. However, the point I want to make is that the same mathematics applies more generally. The analogous quantities and equations may be helpful even if you shoot at a different target, e.g. a Japanese immigrant who fights against immigrants too assertively. Or a pirate. Or another de facto winner of the elections in a Central European country. ;-) The equations also work when you replace CZ 75 with a Vz. 58, the Czechoslovak superior answer to the Kalashnikov.

Some generalizations are obvious. The shape of the gun, projectile, and target may be anything. But when you're talking about the proton-proton collisions or other collisions in particle physics, you need to generalize all the ideas above a little bit more dramatically. First, when two protons collide, they don't have a sharp shape. One of the "slightly harder" but still easy generalizations allows you to make the boundary of the target (or projectile) fuzzy.

Imagine that you're colliding protons according to the laws of classical physics. They repel by the Coulomb force. What is the probability that the proton-projectile's direction of motion will change at least a tiny little bit? Well, it's actually 100%. You would need an infinitely unlikely conspiracy for all the effects of the other proton to cancel exactly. For example, you would need to shoot one proton precisely at the center of the other, and it's impossible to aim this precisely.

In fact, in classical physics, you change the direction of the projection at least a little bit, regardless of the impact factor – the cross section is therefore infinite – for any potential in classical physics that is nonzero at any distance. Even if the repulsive potential decreases as quickly as \(\exp(-Kr)/r\), the Yukawa potential, the probability is zero that the direction of the particle is precisely unchanged. The force is simply nonzero and an arbitrarily small nonzero force changes the direction of the motion of the projectiles.

As you may have guessed, my aim – figuratively, in this case – is to discuss the quantum mechanical generalization. What is the actual change or generalization that quantum mechanics requires? And what are the new consequences?

In quantum mechanics, you may still talk about "similar" initial states and "similar" final states. Also, you may calculate\[

W_{\rm success} = \Phi \cdot \sigma,

\] according to the same equation that we understood classically. What quantum mechanics changes is the character of the intermediate states. In classical physics, you could mentally trace the projectile at every moment of its trajectory and objectively say what was happening with the bullet, how the size of a hole in the communist agent's skull was increasing in a particular millisecond, and so on. Maybe you didn't rent a high-speed camera to record what was happening with the agent. But classical physics allows you to imagine that someone could have monitored it and this monitoring wouldn't change anything substantial about the assassination. Some pictures of the intermediate state "objectively exist" even if no one knows what they are, classical physics allows you to assume.

That's not the case in quantum mechanics. The intermediate states – when they're not observed – don't have any objective properties. And if you wanted to observe what some quantities or pictures are during the collision itself, you would generically change the experiment and its outcome. In quantum mechanics, something happens in the intermediate state – some black box is operational in the middle – but you may talk about the transformation of initial states (analogous to those in classical physics) to final states (also analogous to classical physics) and quantum mechanics gives you directly a novel prescription to calculate the intrinsic probability of the process, e.g. the cross section of a collision. The quantum mechanical prescription is built on the Born's rule (squared absolute value of some complex probability amplitude) involving complex amplitudes extracted as matrix elements and products of operators acting on a complex Hilbert space.

Let's ask: Does the need to abandon the intermediate pictures violate some well-known facts? Is it consistent? Should it have been expected?

Well, quantum mechanics is at least as consistent as classical physics. It talks about "analogous observable" initial and final states. It just replaces the hard work in between with a "black box". For consistency, the quantum mechanical probabilities need to obey laws such as\[

P(A\text{ or }B) = P(A)+P(B) - P(A\text{ and } B)

\] and so on. And one may prove that all these rules are obeyed by the Born-rule-based quantum mechanical formulae for the probabilities and the cross sections. You need the orthogonality of the post-measurement states; or the consistency condition for the "consistent histories" in the interpretation of quantum mechanics based on consistent histories. These conditions are morally equivalent and when you do things right, they just work. So all the rules for probabilities that guarantee the consistency of mathematical logic within classical physics are still obeyed although you need to play with operators and vectors in a complex Hilbert space to prove them in quantum mechanics. But once again, what matters is that these formulae work – the theory is consistent.

Is there some "need" for the intermediate pictures? Not really. By definition, the intermediate states of a transition e.g. collision are not observed. So if they're not observed, you can't have any empirical evidence about them – and about their existence. The empirical evidence is one that matters in science. Because you don't have the evidence for the existence of any particular properties of these hypothetical intermediate states (analogous to the growing hole in the skull of the agent etc.), you shouldn't claim that they exist or that they "should" exist. Quantum mechanics doesn't really allow them to exist, they don't exist, and there is absolutely nothing "wrong" about this novel statement. Instead, it's a deep discovery of the 20th century science.

I started with the cross sections but the same comments about the intermediate state – and the absence of particular pictures – holds for any process in quantum mechanics.

What are the differences in the predicted cross sections?

Because the character of the "intermediate history" is conceptually different in quantum mechanics, quantum mechanics tells us to calculate the cross sections and probabilities using very different formulae. They're not just a version of the classical "shooting at agents". You need operators on complex linear spaces. Are the resulting cross sections the same as in a classical theory?

Not quite. Some of them may be very similar and there exist classical limits of most quantum mechanical theories – extreme parts of some parameter spaces – where the quantum mechanical answers mimic the classical ones. Sometimes, e.g. exactly in the Coulomb scattering, the cross sections calculated quantum mechanically have exactly the same form in classical physics – these situations are rare and special, however.

In general, the quantum mechanical answers may differ so deeply that one may prove that no classical scattering – with particular pictures in between – could produce exactly the same answers. But there are some easy-to-understand and rather general differences in the quantum mechanical cross sections. One of them is that finite cross sections are omnipresent in quantum mechanics. What do I mean?

Do you remember that I argued that the "total cross section" is infinite for any potential in classical physics because the force is nonzero and always causes "at least a slight" change of the direction of the projectile? Well, so this conclusion is basically wrong for "one-half" of the potentials in quantum mechanics. The potentials with a finite cross section are known as resulting from the "short-range forces" while the potentials with the infinite cross section, like in classical physics, follow from "long-range forces".

First, examples. The Coulomb \(1/r\) potential is a long-range force. Classically, the force always changes the direction of the charged particle at least a little bit. That's why you don't need to aim at all if your only goal is to change the direction at least a little bit: you always succeed. So it's like hitting an infinitely fat agent. You can't miss him. I also mentioned that the formula for the Coulomb cross section happens to be the same in quantum mechanics. So the total cross section is infinite in quantum mechanics, too.

What about the Yukawa potential, \(\exp(-Kr)/r\)? In classical physics, it's still true that the force is nonzero at any distance, so it's virtually guaranteed that the direction of the projectile changes even if you don't aim precisely at all. It's different in quantum mechanics. Why? In quantum mechanics,

arbitrarily weak but guaranteed effects are replaced with finite (not so tiny) changes that have a small probability.
And for this reason, the cross section of the scattering governed by the Yukawa potential is finite! The point is that you may calculate the probability that the direction changes at all, given some "impact factor" (distance between the axis of the initial motion from the center of the target-object). And the probability is strictly in between 0 and 100 percent. It's because the tiny force acts differently than it did in classical physics. In classical physics, a tiny force created a tiny change of the velocity of the projectile.

In quantum mechanics, the tiny force changes the probabilities that the direction of the motion is some vector or another vector. But the probability that the velocity remains precisely the same as it was in the beginning remains nonzero. One suppresses the probability amplitude for that default velocity a little bit but it doesn't instantly drop to zero. It drops by a finite amount – and the probabilities for different directions rise from zero to a finite amount.

This is a characteristic quantum mechanical behavior. You may see it in related situations and descriptions. You know, when a particle changes its velocity, you may understand the change as the emission of some electromagnetic waves. But in quantum mechanics, one emits photons and at a given frequency – and the frequency may be "approximately given" in the case of scattering – one can't have arbitrarily weak electromagnetic waves. One photon is the minimum amount of energy that electromagnetic waves of that frequency can carry!

So instead of emitting \(N\) photons where \(N\) is an arbitrary positive real number, you may only emit \(N\in\ZZ\) photons, an integer, and the small number resulting from the tiny Yukawa potential manifests itself as the tiny probability that the number of emitted photons changes from \(N=0\) to \(N=1\) at all! When the force is really weak, the probability of \(N=2,3,\dots\) is negligible.

The most general lesson I want to convey is that the particular classical theories and "pictures" may be good to visualize the basic concepts such as probabilities and cross sections – they may be useful for pedagogic reasons. But physics – and quantum mechanics – may recycle many of the formulae and rules while completely denying the pictures that were used to "visualize" the cross section and other things. These pictures aren't really needed and, if you study things carefully, you find out that they don't exist according to the laws of Nature as we've known them for almost a century.

So quantum mechanics generalizes classical physics with its picture in a similar way as one axiomatic system in mathematics generalizes another – when some of the axioms are simply dropped. The main axiom of classical physics that is dropped is that "there exists an objective, particular picture of reality at every moment of the intermediate evolution". This axiom is not only dropped; one can indeed prove that this classical assumption is false within the framework of quantum mechanics. It's an example of a revolution in physics but there's nothing wrong about the new, quantum framework because the new framework is as internally consistent as the classical framework was; and it doesn't contradict any empirically proven facts, either. It's just new and may be philosophically challenging for many people – but that's not a rational reason to refuse a physical theory.

And that's the memo.

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