I am studying the different probability interpretations (frequentist, bayesian etcetera) but for me it keeps bugging that since the probability density function in quantum mechanics is found by finding the modules of the wave function that satisfies schrödingers equation it seems to me that these wave functions are more fundamental than the concept probability itself.

In that sense probability emerges from a physical theory. What is the significance of the wave function being more fundamental than the probability in quantum mechanics?

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    The concept of probability is older than quantum mechanics and has been applied to many fields other than physics, also before the arrival of quantum mechanics. You may want to have a look at Bernoulli's Ars Conjectandi, en.wikipedia.org/wiki/Ars_Conjectandi Commented Nov 28, 2023 at 12:58
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    Probability does not emerge from a physical theory. You still need a pre-existent probability theory to interpret square modulus as a probability density of something. The Born rule, which is one of the QM postulates, presupposes a probability concept, it does not "emerge" it. But see SEP, Quantum Mechanics as a Probability Calculus on peculiar notion of probability that is specific to QM.
    – Conifold
    Commented Nov 28, 2023 at 13:21

3 Answers 3


In qm the position (until measurement) "is not decided". It's not that it "is not known" in the sense that we do not know where exactly it is. This situation cannot be described in a logical way : it's not that the particle is left or right for example; other states are valid too!

So this is described with a complex number. in our simple example, you can have a value for 1 if "the particle is in the left side" and a value of 0 if not; but since other states of this statement are valid too, you need a complex number to attribute the validity of the statement. The absolute square of the complex number is the probability for the statement to be true. The sum of the two probabilities (particle is left..., is right...) must be 1.

Ontologically speaking, the state is a potentiality (undecided position) and the probability is the possibility of specific position when an observation is to be made.


Yes, in quantum mechanics the value of the probability density derives from the wave-function. As you write, the squared modulus of the wave-function equals the probability density.

That the wave-function contains more information than the probability density is already indicated by the fact that the latter is a real number while the former is a complex number, hence determined by two real parameters, phase and modulus.

The deeper reason from the mathematical viewpoint: The wave-function develops according to a differential equation, the Schroedinger equation. The probability density can only be obtained after the differential equation has been solved.

Of course, the question whether the squared modulus of the wave function should be interpreted as a probability density - following Max Born - is another topic.

  • Thank you, does this suggest that the concept of probability is defined in terms of the quantum mechanical wave function? I have been reading plato.stanford.edu/entries/probability-interpret however this barely comes up in that text. Seems like I am misunderstanding something about the philosophy of probability. Commented Nov 28, 2023 at 11:42
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    @bananenheld Probability is a separate mathematical concept, independent from but with applications in physics. The interpretation of the probability "p" in quantum mechanics: If you prepare a sample of particles in the same way for a given experiment, then p is the fraction of observations with a fixed outcome in the given experiment. The statement generalizes to the case of a continuous probability distribution.
    – Jo Wehler
    Commented Nov 28, 2023 at 11:51

Probability is most definitely not a concept derived from the wave function. Quantum theory is a relatively recent development; the idea of probability pre-dates it by centuries. There are countless applications of the theory of probability that have no bearing whatsoever on quantum theory.

The wave function in quantum mechanics does provide probabilistic information about the state of a system, but that is quite different from saying that it is the source of the concept of probability.

  • Yes but since the wave function is more fundamental than the probability density function and you could argue that the concept probability in the world either reduces to something related to missing information of the observer or an outcome of a physical experiment my reasoning would be that probability gets reduced to the wave function in that limit. Commented Nov 29, 2023 at 6:46
  • @bananenheld I wonder whether it is not even more radical: QM formalizes the laws of microphysics as a world of possibilities. It computes for each possible preparation of a given system the probabilities for the different values, which a physical observable may realize as the result of a measurement.
    – Jo Wehler
    Commented Nov 29, 2023 at 7:34

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