The wave function is perhaps more accurately also called a probability amplitude. It is related to probability by the Born rule, and is equivalent to the probability of a particular result in an measurement given known initial conditions (this approach to quantum mechanics was taken by Dirac and von Neumann). This means that there is nothing more mysterious to wave function collapse than there is in the change to a probability once a result is known.

The more mysterious part is why should the wave function obey the Schrödinger equation, from which follows the mathematics of wave mechanics. The answer is buried in the mathematical foundations of quantum mechanics. This is not generally covered in physics courses which are usually more concerned with practical application than with mathematical structure, but the general form of Schrödinger's equation can be derived from the Dirac–von Neumann axioms. An outline of the derivation is given at Derivation of the Schrödinger equation. I have given a detailed derivation in The Hilbert Space of Conditional Clauses.

The key postulate is that probabilities are given by the Born rule (or expectations given by the inner product). One also requires that the fundamental physical behaviour of matter does not change. This enables one to show that the probability interpretation requires unitary time evolution satisfying the conditions of Stone's theorem, and the general form of the Schrödinger equation follows as a simple corollary.

The key reason for this difference in mathematics from classical probability is that classically probabilities are determined by unknown variables, whereas in quantum theory probabilities result from the absence of determinism in the fundamental interactions of matter.

The wave function is perhaps more accurately also called a probability amplitude. It is related to probability by the Born rule, and is equivalent to the probability of a particular result in an measurement given known initial conditions (this approach to quantum mechanics was taken by Dirac and von Neumann). This means that there is nothing more mysterious to wave function collapse than there is in the change to a probability once a result is known.

Probabilities always "collapse" with a change of information. If you throw a dice then the probability of getting a $6$ changes from $1/6$ to either $1$ or $0$ when you see the result. The change in probability depends on information. If your friend throws a dice behind a screen where you cannot see it, then the probability (for you) remains $1/6$ until the screen is removed. But your friend can see the dice when it lands, so for him the change in probability takes place earlier. (We may compare this to Wigner's friend, who has a different knowledge of a quantum result from Wigner, who is outside the room, and to Alice and Bob in Bell tests. When Alice observes her result, she obtains a different wave function for Bob's particle, but Bob observes no change as he cannot know Alice's result).

The real question is "Why should the wave function obey the Schrödinger equation?" from which follows the mathematics of wave mechanics. The answer is buried in the mathematical foundations of quantum mechanics. This is not generally covered in physics courses which are usually more concerned with practical application than with mathematical structure, but the general form of Schrödinger's equation can be derived from the Dirac–von Neumann axioms. An outline of the derivation is given at Derivation of the Schrödinger equation. I have given a detailed derivation in The Hilbert Space of Conditional Clauses.

The key postulate is that probabilities are given by the Born rule (or expectations given by the inner product). One also requires that the fundamental physical behaviour of matter does not change. This enables one to show that the probability interpretation requires unitary time evolution satisfying the conditions of Stone's theorem, and the general form of the Schrödinger equation follows as a simple corollary.

The key reason for this difference in mathematics from classical probability is that classically probabilities are determined by unknown variables, whereas in quantum theory probabilities result from the absence of determinism in the fundamental interactions of matter.

The wave function is perhaps more accurately also called a probability amplitude. It is related to probability by the Born rule, and is equivalent to the probability of a particular result in an measurement given known initial conditions (this approach to quantum mechanics was taken by Dirac and von Neumann). This means that there is nothing more mysterious to wave function collapse than there is in the change to a probability once a result is known.

The more mysterious part is why should the wave function obey the Schrödinger equation, from which follows the mathematics of wave mechanics. The answer is buried in the mathematical foundations of quantum mechanics. This is not generally covered in physics courses which are usually more concerned with practical application than with mathematical structure, but the general form of Schrödinger's equation can be derived from the Dirac–von Neumann axioms. An outline of the derivation is given at Derivation of the Schrödinger equation. I have given a detailed derivation in The Hilbert Space of Conditional Clauses.

The key postulate is that probabilities are given by the Born rule (or expectations given by the inner product). One also requires that the fundamental physical behaviour of matter does not change. This enables one to show that the probability interpretation requires unitary time evolution satisfying the conditions of Stone's theorem, and the general form of the Schrödinger equation follows as a simple corollary.

The key reason for this difference in mathematics from classical probability is that classically probabilities are determined by unknown variables, whereas in quantum theory probabilities result from a fundamental lack of determinism in the fundamental interactions of matter.

The wave function is perhaps more accurately also called a probability amplitude. It is related to probability by the Born rule, and is equivalent to the probability of a particular result in an measurement given known initial conditions (this approach to quantum mechanics was taken by Dirac and von Neumann). This means that there is nothing more mysterious to wave function collapse than there is in the change to a probability once a result is known.

The more mysterious part is why should the wave function obey the Schrödinger equation, from which follows the mathematics of wave mechanics. The answer is buried in the mathematical foundations of quantum mechanics. This is not generally covered in physics courses which are usually more concerned with practical application than with mathematical structure, but the general form of Schrödinger's equation can be derived from the Dirac–von Neumann axioms. An outline of the derivation is given at Derivation of the Schrödinger equation. I have given a detailed derivation in The Hilbert Space of Conditional Clauses.

The key postulate is that probabilities are given by the Born rule (or expectations given by the inner product). One also requires that the fundamental physical behaviour of matter does not change. This enables one to show that the probability interpretation requires unitary time evolution satisfying the conditions of Stone's theorem, and the general form of the Schrödinger equation follows as a simple corollary.

The key reason for this difference in mathematics from classical probability is that classically probabilities are determined by unknown variables, whereas in quantum theory probabilities result from the absence of determinism in the fundamental interactions of matter.