I am trying to understand the link between the wavefunction and probability density. I can understand the probability density $\rho$ would be some function of the wavefunction, but I am unable to understand why it would be specifically $\rho = \psi^{*}\psi=|\psi|^{2}$. Why not some other functional form, say $\rho=|\psi|$ or $|\psi|^{n}$, where $n>2$ is some real number? Is there any physical intuition behind this specific functional form?

  • $\begingroup$ See this re: Born rule. $\endgroup$
    – joseph h
    Apr 15, 2023 at 9:14
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    $\begingroup$ It's a postulate. It has its proofs, and also considered alternatives. See en.wikipedia.org/wiki/Gleason%27s_theorem, arxiv.org/quant-ph/0401062 and arxiv.org/abs/quant-ph/0401062v2 $\endgroup$
    – Avantgarde
    Apr 15, 2023 at 10:03
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    $\begingroup$ The ultimate answer is, as so often, that we postulate this because it agrees with the experimental findings. The ontological status of the wavefunction is not clear, so IMHO this is the only suitable explanation. $\endgroup$ Apr 15, 2023 at 10:13
  • $\begingroup$ The probability density is a misinterpretation of the Born rule. It is the Born rule for a measurement that is represented by the unity operator. There is, however, a difference between the free system and the system that is being measured: the measurement destroys the system by taking quanta of energy out of the quantum system, hence it seizes to exist. So there is no "probability density". There is only a probability distribution that we would measure if we could "stop the quantum system cold". In practice we can't. There is no "detector" the implements the unity measurement operator. $\endgroup$ Apr 15, 2023 at 16:58
  • $\begingroup$ Things make a bit more sense if you think of $\psi(x)=\langle x|\psi\rangle$ as the spatial components of the physical state being a Hilbert vector $|\psi\rangle$ - for an expectation value like $1 = \langle 1\rangle = \langle\psi|1|\psi\rangle$ you add $1=\int\,dx |x\rangle\langle x|$ and end up with $1=\int\,dx \langle\psi|x\rangle\langle x|\psi\rangle = \int\,dx \psi^*(x)\psi(x) = \int\,dx |\psi(x)|^2$ $\endgroup$ Apr 15, 2023 at 21:57

3 Answers 3


The standard view of quantum mechanics is that the Born rule is just a postulate of quantum mechanics. That postulate happens to be consistent with some other constraints on states, such as consistency with obtaining the reduced density matrix $\rho_1$ on Hilbert space $\mathscr{H}_1$ you get by doing a partial trace over $\mathscr{H}_2$ on a density matrix $\rho$ on the Hilbert space $\mathscr{H}_1\otimes\mathscr{H}_2$, see Box 2.6 on p.107 of "Quantum computation and quantum information" by Nielsen and Chuang.

If you want an actual explanation of the Born rule: an account of what features of reality make that rule appropriate, then you need an explanation of what is happening in reality in quantum mechanical experiments. People often claim there is more than one candidate explanation. These different explanations are called interpretations of quantum mechanics, but since most of them contradict quantum theory that doesn't make any sense.

Some of these interpretations, such as the Copenhagen or statistical interpretations, assume that the Born rule is true. Some other interpretations, such as the pilot wave theory which adds particles on top of the wave function, claim they can explain the Born rule and predict deviations from it:


Under the many worlds interpretation, which takes quantum theory literally as a description of reality and applies it consistently, there are a couple of candidate explanations of the Born rule.

One explanation uses decision theory. The probability of an outcome is a function of the state that satisfies particular properties required by decision theory, like if a person uses those probabilities you can't make him take a series of bets that will make him lower the expectation value of his winnings, the probability of an outcome can't be changed by a later measurement and some other assumptions. From this you can explain that the probability of the $𝑗$th outcome in the state $\sum a_j|j\rangle$ will be $|𝑎_𝑗|^2$:





There is another approach to deriving the probability rule called envariance, which uses properties of the state of a system that remain unchanged after interaction with the environment:


  • $\begingroup$ We know what is happening "in reality". During a measurement we are taking a quantum of energy out of the system. That's the process the Born rule describes. After that the systems seizes to exist the way it was before the measurement. There is no mystery here. There is, however, an educational problem: we are teaching the solution theory of QM without mentioning the kinds of physical systems that it describes. That only happens in courses about atomic, molecular, nuclear, solid state etc. physics. $\endgroup$ Apr 15, 2023 at 17:24

The probability density is defined as the square modulus of the wavefunction. This definition was guessed by Max Born. Experimental evidences have verified this definition, thus, it is defined the above-mentioned way.

In cases like this, it is good to say that the universe just works in this way and accept the definition.

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    $\begingroup$ I dont think your answer OP's question. $\endgroup$ Apr 15, 2023 at 10:28
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    $\begingroup$ They weren't "postulating". The problem was to find the correct physical theory for atomic spectra and a plethora of other experimentally well known phenomena. Quantum mechanics grew out of an entire century of unexplained physical data. $\endgroup$ Apr 15, 2023 at 17:30
  • $\begingroup$ Yes, you are correct. I have changed it to definition. $\endgroup$
    – user355398
    Apr 16, 2023 at 5:44
  • $\begingroup$ Can you please refer to some of the experimental evidences?@Agnibho Dutta $\endgroup$ Apr 17, 2023 at 5:10
  • $\begingroup$ All quantum systems showed the same definition of the probability density function. From those experiments, the rule was verified. $\endgroup$
    – user355398
    Apr 17, 2023 at 7:42

ψ is a complex number; a probability is always real and positive. Since the squared magnitude is the simplest possible way of constructing, Mother Nature decided to use it; later Born hit upon this rule for the same reason--Born again.

  • $\begingroup$ Yes, Born just guessed it in his reasearch paper. Then, experimental evidences further verified this definition. $\endgroup$
    – user355398
    Apr 16, 2023 at 5:41
  • $\begingroup$ Nature doesn't know anything about the wave function. The wave function describes a quantum mechanical ensemble, i.e. the infinite repetition of the same experiment. There is no such thing as an ensemble in nature. Just like in case of probabilities it's a human abstraction. $\endgroup$ Apr 16, 2023 at 5:42
  • $\begingroup$ @FlatterMann "Nature doesn't know anything about the wave function." You've asked Her? $\endgroup$ Apr 16, 2023 at 5:47
  • $\begingroup$ @SimonCrase That used to be my job. I am an experimental physicist. To me physics is only what nature tells me, not what's printed in the books. See "La Trahison des Images" by René Magritte for an artistic illustration of the fallacy of mistaking descriptions of reality for reality itself. $\endgroup$ Apr 16, 2023 at 5:56
  • $\begingroup$ @FlatterMann "...the infinite repetition of the same experiment...." Us engineers never have time, or the budget, to repeat something that many times... $\endgroup$ Apr 16, 2023 at 6:00

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