What is Born's postulate
Born's rule and Schrödinger's equation
In non-relativistic quantum mechanics, the evolution equation of the quantum state is given by the Schrödinger equation, and the measurement of a particle state is itself a physical process and should and is therefore actually determined by the Schrödinger equation.
In fact, people like to predict probabilities using Born's rule, and sometimes they do it right and sometimes they do it wrong.
Are we using Born's rule just because it becomes mathematically cumbersome to take into account all degrees of freedom with the Schrödinger equation?
Yes and no. In fact, sometimes you can just use Born's Rule to get the same answer as the correct answer you get using Schrodinger's Rule. And when you can, it's often a lot easier, both computationally and for subjective reasons. However, this is not the case the Reason why people use Born's rule, but because they have difficulty relating experimental results to wave functions. And Born's rule does just that. You give it a wave function and use it to calculate something that you can compare to the laboratory. And that's why people use it. Not the computational friendliness.
Is it possible to derive Born's rule using the Schrödinger equation?
Yes, but in order to do this you have to overcome the exact reason people use Born's rule. The Schrödinger equation only tells us how wave functions develop. It doesn't tell you how to relate that to experimental results. When a person learns how to do this, they can see that the job according to Born's rule is already done by the uniform Schrödinger evolution.
How do probabilistic observations imply the causal development of the wave function?
The answer is so simple it seems obvious. Just think about how to check it out in the lab, write down the appropriate system that models the actual lab setup, and then set up the Schrödinger for that system.
For Born's rule, you use a wave function on one copy of a system, then choose an operator and get a number between zero and one (which you interpret as a relative frequency if you have done many experiments with many copies of that one system) . And you get a number for each eigenvalue in a way that depends on the one wave function for a copy of a system, although you check this result by taking a whole collection of identically prepared particles.
So this is what Born's rule does for you. It informs you about the relative frequency of different eigenvalues for a number of identically prepared systems. You check this by creating a number of identically prepared systems and measuring the relative frequency of different eigenvalues.
How do you do that with the Schrödinger equation? Given the state and operator in question, find the Hamilton operator that describes the evolution that corresponds to a measurement of the operator (as an example, my other answer to this question cites an example that specifically asks the Hamilton operator to use the To measure the spin of a particle). . Then also write down the Hamilton operator for the device that can count the number of times a particle has been produced and the device that records the Hamilton operator for the device that can count the number of times a particle has a given result was detected, and the device that picks up the ratio. Then write down the Schrödinger equation for a factorized wave function system that has a large number of factors that are identical wave functions and that have a sufficient number of devices around different eigenfunctions of the operator and device in question That counts the number of split results. You then develop the wave function of the entire system according to the Schrödinger equation. When 1) the number of identical factors is large and 2) the devices that send different eigenfunctions to different paths that make the developed eigenfunctions orthogonal to each other, something happens. The part of the wave function that describes the state of the device that has received the ratio of how many a certain eigenvalue evolves to have almost all L.2 The norm focuses on a state that corresponds to the ratio that the Born's rule predicts, and is almost orthogonal to the parts that correspond to the states that Born's rule did not predict.
Some people then apply the Born rule to this state of the aggregator, but then you've failed. We are nearly there. Except all we have is a wave function with most of it L.2 norm concentrating on a region with an easy to describe state. Born's rule tells us that we can subjectively expect to experience this overall result personally. The Born rule states that this will almost certainly happen, since almost every L.2 standard corresponds to this state of the aggregator. The Schrödinger equation itself does not tell us this.
However, we had to interpret Born's rule in such a way that these numbers between 0 and 1 correspond to the observed frequencies. How can we interpret that "the wave function is heavily focused on a state where an aggregator is reading the same number", which corresponds to an observation?
This is literally the problem of interpreting a mathematical result via a mathematical wave function as an observation.
The answer is that we and everything else are described by the dynamics of a wave function and that part of a wave is small L.2 An almost completely orthogonal norm doesn't really affect the dynamics of the rest of the wave. We are the dynamic. People are processes, dynamic processes of subsystems. We are like the aggregator in that we are sensitive to only some aspects of some parts of the rest of the wave function. And we are robust in that we are systems that can act and evolve over time that they are insensitive to small deviations in our inputs L.2 The norm that focuses on getting the value predicted by Born's rule have (and this state with this focus on this value is what the Schrödinger equation predicts) can interact with us, the robust information processing system, which also develops according to the Schrödinger equation, interacts with us just like in a state, in which all L.2 norm was in this state, not just most of it.
This dynamic 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. Use the Schrödinger equation to predict the outcome of an observation. However, you only need to do this for states that are very, very close together to get Born's rule, since Born's rule only predicts the results of an aggregator's response to a large number of identical systems. And it is precisely these states that we can define purely operationally using the Schrödinger equation.
We are simply saying that the Schrödinger equation describes dynamics, including the dynamics of us, things "measured", and the entire universe. A measurement works in such a way that you have a Hamilton operator acting on your subsystem | ⟩ich⟩ and your entire universe | Ψich⟩⊗ | U.⟩ and develop it as follows:
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