# What is the meaning of probabilities in quantum mechanics?

In quantum mechanics, probabilities are associated with the detection of a physical event by a macroscopic device, or are events at the microscopic level also probabilistic? For example, the probability of scattering at a certain angle in a collision of particles is the probability of detecting a scattered particle in a certain place, or is it the probability that the particle will be scattered at this particular angle, regardless of observation?

• I don't think there is an agreement within the physics community about the meaning of probabilities in quantum mechanics. The different takes about this are usually referred to as interpretations of quantum mechanics, although some might say that they are not just interpretations, but just different theories with different underlying physical realities (if they posit a reality at all). Dec 31 '20 at 9:30
• I am pretty sure this is a recurrent question, here. Dec 31 '20 at 13:08

## 2 Answers

Your question is quite subtle and I believe the answer depends on the interpretation of quantum mechanics you want to go with. The probabilities of quantum mechanics - described by complex probability amplitudes - are different from traditional mathematical probabilities - which are non-negative real-valued mesaures whose integral (or sum in the discrete case) has to add up to 1. Any measurement will marry the two in some way. Think about Fermi's golden rule, (for explanations, see https://en.wikipedia.org/wiki/Fermi%27s_golden_rule) $$\Gamma_{i \to f} = \frac{2\pi}{\hbar} |\langle f | H' | i \rangle|^2 \rho(E_f).$$ Here, $$\Gamma_{i \to f}$$ is a classical probability (you can see that the right hand side contains only non-negative contributions). But the quantity $$\langle f | H' | i \rangle$$ is a "quantum probability", i.e. a probability amplitude. In the golden rule formula, you can even see why they coined it amplitude: only its modulus sqaured $$|\langle f | H' | i \rangle|^2$$ appears in the end result, much like the absolute amplitude squared of a wave gives its intensity.

So when you ask about the nature of probability at the microscopic level, you will run into the distinction between those two kinds of probability: real-valued probability and complex-valued probability, often described by the buzzword of "superposition". And this distinction is hard, because it depends on what you believe a world according to quantum mechanics is like, while all measurements can only tell us what a world according to quantum mechanics looks like.

Before diving into interpretations, let's distinguish the two types of time evolution a quantum system can undergo. There is unitary evolution: the state of the system is rotated in some way within the Hilbert space. That's what Schrödinger's equation describes: the wave function never changes length, which is why it's totally fine to describe it by normalized functions - the normalization must stay intact during evolution. (A more accurate idea is to actually consider pure wave functions to be rays in Hilbert space, but let's not go down that rabbit hole) This is what quantum systems seem to do between measurements. However, when we measure, that is when we pull information out of the quantum realm to make it available to our minds, a different time evolution happens, coined projective evolution by some. And projection is what happens: apparently, a quantum state $$|\psi\rangle$$ is decomposed into eigenstates $$\{|\phi_j\rangle\}$$ of an operator $$\hat A$$ corresponding to the measurement we make (called an observable). The measurement result is an eigenvalue $$a_i$$ of $$\hat A$$, and after the measurement the unitary evolution continues as if it started from one of the eigenstates $$\phi_i \in \{\phi_j\}$$ corresponding to the eigenvalue $$a_i$$. (let's go with a non-degenerate case to keep it simple. That is, there is exactly one eigenstate $$|\phi_i\rangle$$ corresponding to $$a_i$$). One can describe this by projecting $$|\psi\rangle$$ onto the eigenstate $$\phi_i$$, which gives a probability amplitude $$\langle \phi_i | \psi\rangle$$, the modulus squared of this amplitude is considered to be the probability of measuring the result $$a_i$$. And immediately after the measurement, the wave function is in the state $$|\psi\rangle_{\textrm{after}} = |\phi_i\rangle$$.

Now this is a set of mathematical prescriptions that works. We have rules for how the system behaves in between measurements and rules for how to predict measurement results and what the state immediately after a measurement is. But there is a large void to be filled: what is truly happening?

Now, there are different interpretations of this. None of which change the mathematical framework, just the way this mathematics is to thought about. Copenhagen takes everything quite literal: there is unitary evolution and then a measurement is like a sledgehammer, smashing the quantum egg the system is in and giving us a classical result. There is many worlds theory that says that the superposition that is encoded in the unitary evolution isn't actually destroyed but that the world is constantly in superposition, it's just our minds that cannot perceive it. And that, unfortunately, is just the distinction you want to clarify in your question. Is the probability a feature that is introduced by measurement or is everything probabilistic? For many worlds, superposition permeates reality and measurement doesn't change anything about it. It just branches reality further and further. For Copenhagen, superposition exists at the microscopic level, but is destroyed once we do a measurement to get a macroscopically readable results, and complex probability is replaced by real probability.

So, I'm sorry that there isn't a more definite answer to your question. I rather took the effort to show why it's hard to answer.

• I think that speaking about negative or complex-valued probability is misleading. The probability densities used in QM are always positive probability distributions. Its origin from a complex-valued function has nothing to do with the possibility of using standard probability formalism. An interesting question could be about the interpretation of the underlying probability theory. But apparently, this is not the aim of this question that looks more related to the interplay between probability and measurements. Dec 31 '20 at 10:26
• It is true that by Born's rule, you get rid of probability amplitudes when talking about measurement predictions. But to my mind, this is an axiom that describes measurement. It is true that, once applied, you are back at non-negative probability measures. But I understood "are events at the microscopic level also probabilistic?" to refer to what actually happens in a collision and how this relates to the measurement probabilities one encounters in QM. For that, you need to talk about where these probabilities come from and if there is a measurement-independent way to think about them Dec 31 '20 at 11:34
• I agree that there is a pre-probabilistic level made by complex-valued amplitudes. Still, that is not affecting the usual rules of probability theory. This is the point I wanted to stress. Dec 31 '20 at 11:48
• Ah ok, thank you. But a follow-up: what do you mean by the usual rules? Bell's theorem tells us that probability in quantum mechanics is quite different from what we expect from the classical probability theory that is grounded in experiences from classical mechanics. I think people are still discussing the relation between quantum and classical probabilities, I just found this nice essay: arxiv.org/pdf/1310.1484.pdf Dec 31 '20 at 12:11
• I think that Bell's theorem can be interpreted equally well as showing that there is the need for a different probability theory if one chose to keep the same events of the Classical Mechanics, or that one can continue using the usual Kolmogoroff probability theory (this is what I call the usual rules), suitably restricting the set of the events. As far as I know, these remain two alternatives on the same foot. Dec 31 '20 at 15:08

The only predictions a quantum mechanical theory can do , observatble in data, are probability distributions. These are built in in the postulates of quantum mechancs. . The quantum mechanical solution of any given system with its boundary conditions comes out with a wavefunction, the complex conjugate square of this function gives the probability for a particle to be at (x,y,z,t). So if one could measure , the probability is calculable even when experimentally one could not do the measurement.

see my answer here Understanding superposition principle

• However, when no measurement takes place, knowing the probability gives you incomplete information about the system's state. When you have particles going through a double slit, you can give the probabilities of (say) 50% going through the upper slit. If you do the measurement, you will find exactly that and particles that go through the upper slit will move on to a detector screen further away from the source in a corresponding way. If you don't do the measurement, the pattern detected on the screen will be different, because there is more to the state than the probability. Dec 31 '20 at 11:48
• the complex conjugate square of this function gives the probability for a particle to be at (x,y,z,t). to be at (x, y, z, t), or to be found at (x, y, z, t) as a result of measurement? Big difference. Dec 31 '20 at 11:49
• @АрманГаспарян as far as I know measurement is putative i.e. may or may not take place. The mathematics is there to tell the probability if it takes place. The wavefunction of the hydrogen atom exists whether i measure it or not.See the orbtals here, the possible positions an electron can occupy that could be measured, hyperphysics.phy-astr.gsu.edu/hbase/Chemical/eleorb.html Dec 31 '20 at 12:58
• The wave function exists, but the particle at a certain coordinate does not. Dec 31 '20 at 13:32
• @АрманГаспарян By the mathematics, it has a probaility to exist there. That is all QM can say. If the probaility is 30%, 30% of the time it is there. Dec 31 '20 at 13:36