In non-relativistic quantum mechanics, the equation of evolution of the quantum state is given by Schrödinger's equation and measurement of a state of particle is itself a physical process. Thus, should be governed by the Schrödinger's equation.
But we predict probabilities using Born's rule.
Do we use Born's rule just because it becomes mathematically cumbersome to account for all the degree of freedoms using the Schrödinger equation, so instead we turn to approximations like Born's rule.
So, is it possible to derive Born's rule using Schrödinger's equation?
Answer
Indeed, in non-relativistic quantum mechanics, the equation of evolution of the quantum state is given by Schrödinger's equation and measurement of a state of particle is itself a physical process and thus, should and is indeed be governed by the Schrödinger's equation.
Indeed, people like to predict probabilities using Born's rule, and sometimes they do this correctly, and sometimes incorrectly.
Do we use Born's rule just because it becomes mathematically cumbersome to account for all the degree of freedoms using the Schrödinger equation?
Yes and no. Indeed sometimes you can just use the Born rule to get the same answer as the correct answer you get from using the Schrödinger rule. And when you can do that, it is often much easier both computationally and for subjective reasons. However, that is not the reason people use the Born rule, they use it because they have trouble knowing how to relate experimental results to wavefunctions. And the Born rule does exactly that. You give it a wavefunction and from it you compute something that you know how to compare to the lab. And that is why people use it. Not the computational convenience.
Is it possible to derive Born's rule using Schrödinger's equation?
Yes, but to do so you need to overcome the exact reason people use the Born rule. All the Schrödinger equation does is tell us how wavefunctions evolve. It doesn't tell you how to relate that to experimental results. When a person learns how to do that, then they can see that the job done by Born's rule is already done by the unitary Schrödinger evolution.
How are probabilistic observations implied by causal evolution of the wave function?
The answer is so simple it will seem obvious. Just think about how you verify it in the lab, and then write down the appropriate system that models the actual laboratory setup, then setup the Schrödinger for that system.
For the Born rule you use one wavefunction for one copy of a system, then you pick an operator, and then you get a number between zero and one (that you interpret as relative frequency if you did many experiments on many copies of that one system). And you get a number for each eigenvalue in a way that depends on the one wavefucntion for one copy of a system even though you verify this result by taking a whole collection of identically prepared particles.
So that's what the Born rule does for you. It tells you about the relative frequency of different eigenvalues for a whole bunch of identically prepared systems, and so you verify it by making a whole bunch of identically prepared systems and measuring the relative frequency of different eigenvalues.
So how do you do this with the Schrödinger equation? Given the state and operator in question you find the Hamiltonian that describes the evolution corresponding to a measurement of the operator (as an example my other answer to this question cites an example where they explicitly tell you the Hamiltonian to measure the spin of a particle). Then you also write down the Hamiltonian for the device that can count how many times a particle was produced, and the device that write down the Hamiltonian for the device that can count how many times a particle was detected with a particular outcome, and the device that takes the ratio. Then you write down the Schrödinger equation for a factored wavefunction system that has a huge number of factors that are identically wavefunctions, and also where there are sufficient numbers of devices to split different eigenfunctions of the operator in question and the device that counts the number of results. You then evolve the wavefunction of the entire system according to the Schrödinger equation. When 1) the number of identical factors is large and 2) the devices the send different eigenfunction to different paths make the evolved eigenfunctions mutually orthogonal, then something happens. The part of the wavefunction describing the state of the device that took the ratio of how many got a particular eigenvalue evolves to have almost all of its $L^2$ norm concentrated over a state corresponding to the ratio that the Born rule predicts and is almost orthogonal to the parts corresponding to states the Born rule did not predict.
Some people will then apply the Born rule to this state of the aggregator, but then you have failed. We are almost there. Except all we have is a wavefunction with most of its $L^2$ norm concentrated over a region with an easily described state. The Born rule tells us that we can subjectively expect to personally experience this aggregate outcome, the Born rule says this happens with near certainty since almost all the $L^2$ norm corresponds to this state of the aggregator. The Schrödinger equation by itself does not tell us this.
But we had to interpret the Born rule as saying that those numbers between 0 and 1 correspond to observed frequencies. How can we interpret "the wavefuntion being highly concentrated over a state with an aggregator reading that same number" as corresponding to an observation?
This is literally the issue of the question, interpreting a mathematical result about a mathematical wavefunction as being about observations.
The answer is that we and everything else are described by the dynamics of a wavefunction, and that a part of a wave with small $L^2$ norm that is almost entirely orthogonal doesn't really affect the dynamics of the rest of the wave. We are the dynamics. People are processes, dynamical processes of subsystems. We are like the aggregator in that we are only sensitive to some aspects of some parts of the rest of the wavefunction. And we are robust in that we are systems that can act and time evolve in ways that can be insensitive to small deviations in our inputs, so the part of the wavefunction that corresponds to the aggregator having most of the $L^2$ norm concentrated on having the value predicted by the Born rule (ant that state with that concentration on that value is what the Schrödinger equation predicts) is something that can interact with us, the robust information processing system that also evolves according to the Schrödinger equation interacts with us in the exact same way as a state where all the $L^2$ norm was on that state, not just most of it.
This dynamical correlation between the state of the system (the aggregator) and us, the interaction of the two, is exactly what observation is. You have to use the Schrödinger equation to describe what an observation is to use the Schrödinger equation to predict the outcome of an observation. But you only need to do that on states very very very close to get the Born rule since the Born rule only predicts the outcomes of an aggregator's response to large numbers of identical systems. And those states are exactly the ones we can give a purely operational definition in terms of the Schrödinger equation.
We simply say that the Schrödinger equation describes the dynamics, including the dynamics of us, the things being "measured" and the whole universe. The way a measurement works is that you have a Hamiltonian that acts on your subsystem $|\Psi_i\rangle$ and your entire universe $|\Psi_i\rangle\otimes |U\rangle$ and evolves it like:
$$|\Psi_i\rangle\otimes |U\rangle\rightarrow|\Psi_i'\rangle\otimes |U_i\rangle.$$
The essential aspects of it being a measurement is that when $|\Psi_i\rangle$ and $|\Psi_j\rangle$ are in different eigenspaces they are originally orthogonal, but that orthogonality transfers over to $|U_i\rangle$ and $|U_i\rangle$ in such as way as to ensure the Schrödinger time evolution evolutions of $|\Psi_i'\rangle\otimes |U_i\rangle$ remain orthogonal. (And also we need that $|\Psi_i'\rangle$ is still in the eigenspace.) That's our restriction on the Hamiltonians that are used in the actual Schrödinger
What is the problem?
The problem is that we had to say how to relate a mathematical object to us and where probability words entered. And there isn't any probability. We just have ratios that look like the ratios that probability would predict for us if there were probabilities. And we have to bring up how our observations and experiences relate to the mathematics.
Historically there were strong objections to this, that talking about how human people dynamically evolve should not be relevant to physics. Seems like Philosophy the old fashioned objections would go. But if you think of people as dynamical information processors, then we can characterize them as a certain kind of computer that interacts with the wavefucntion of the rest of the world in a particular way. And other kinds of computer are possible, things we call quantum computers. And now we can make this excuse no longer. We need to talk about the difference between a classical computer that is designed to be robust against small quantum effects, and one that can be sensitive to these effects so that it can vontinue to interact before it has gotten to the point in the evolution where the Born rule could be used.
We must now own up to the fact that the Schrödinger equation evolution is the only one we've seen, and that is what corresponds to what we actually observe in the laboratory experiments where the Born rule is used. And we must own it so that we can correctly describe what happens in experiments where the Born rule doesn't apply, where as always we must use the Schrödinger equation.
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