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I have thought Max Born rule as one of the axiom of quantum mechanics that says norm square of wavefunction gives the probability density. But I also found written somewhere that the rule says that probability of eigenstate is norm square of the amplitude of the eigenvalue corresponding to the eigenstate. First I thought maybe they are same thing but I am unable to find connection between them (former one takes the square norm of wavefunction whereas latter one do the same for amplitude of eigenvalue). Which one is actually Max Born rule? And also I want to know how we know the other one is true.

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    $\begingroup$ "But I also found written somewhere" - where? It's difficult to answer this question without context. $\endgroup$ Apr 17, 2020 at 17:36
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    $\begingroup$ isn't $\psi(x)$ the amplitude for the eigenstate of the $\hat x$ operator?, which means each statement is equivalent. $\endgroup$
    – JEB
    Apr 17, 2020 at 18:29
  • $\begingroup$ The second sentence in the wikipedia article that you link to tells it like it is: In its simplest form, it the Born rule] states that the probability density of finding a particle at a given point is proportional to the square of the magnitude of the particle's wavefunction at that point. $\endgroup$
    – my2cts
    Apr 17, 2020 at 20:47
  • $\begingroup$ @probably_someone here is the link preposterousuniverse.com/blog/2014/07/24/… $\endgroup$
    – Pirateking
    Apr 18, 2020 at 6:35

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They are the same thing. Be careful, I think the other book you’re reading by ‘amplitude of eigenvalue’ means probability amplitude of measuring that eigenvalue, not the amplitude of the eigenvalue itself (otherwise really large values of an observable would have really high probabilities which makes no sense).

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It is the same thing. The way I approach it is to start with a Hilbert space spanned by position states (actually I start a bit further back, by justifying Hilbert space, but the present argument starts with Hilbert space).

For any ket $|f\rangle \in \mathbb H$ we can define the magnitudes of the coefficients $\langle|f\rangle$ in position space according to the Born rule, $$ {|\langle x|f\rangle|^2 \over \langle f|f\rangle}=P(x|f)$$ If $|f\rangle$ is normalised this reduces to $$ |\langle x|f\rangle|^2 =P(x|f)$$ and $ \langle x|f\rangle$ is the probability amplitude.

When we do a measurement, $K$, we get a definite result, a terminating decimal or $n$-tuple of terminating decimals read off the measurement apparatus. Let the possible results be $k_i$ in $\mathbb Q^n$ for $i=1, \dots, m$. The $k_i$ are taken to be distinct; if $i \neq j $ then $k_i \neq k_j $. We assume that the dimension $\mathbb H$ of is greater than $m$.

Each physical state is associated with a ket, labelled by the measurement result, so that if the measured result is $k_i$ then the state is $|k_i\rangle$. The empirical determination of $|k_i\rangle$ requires that we draw from experimental data the value of the inner product $\langle k_i|f\rangle$ for arbitrary $|f\rangle$.

Without loss of generality, $|k_i\rangle$ and $|f\rangle$ are normalised. By assumption, measurement of $K$ is reducible to a set of measurements of position, so that each $k_i$ is in one to one correspondence with the positions $y_i$ of one or more particles used for the measurement (e.g. $y_i$ may be the positions of one or more pointers). Then $$|\langle k_i|f\rangle |^2= |\langle y_i|f\rangle |^2=P(y_i|f) = P(k_i|f) $$ is the probability that a measurement of $K$ has result $k_i$, given the initial ket $|f\rangle$. It follows from $\langle x|y\rangle = \delta_{xy}$ that $$\langle k_i|k_j\rangle = \langle y_i|y_j\rangle = \delta_{ij}$$ meaning that if the result is $k_i$ it is definitely $k_i$ and cannot at the same time be $k_j$with $i \neq j$.

To relate this to the eigenstates of a Hermitian observable we note that measurement with result, $k_i$, implies a physical action on a system and is represented by the action of an operator, $K_i$, on Hilbert space. If a quantity is measurable we require that there is an element of physical reality associated with its measurement, meaning that the configuration of matter necessarily becomes such that the quantity has a well-defined value. In practice this means that, in the limit in which the time between two measurements goes to zero, a second measurement of the quantity necessarily gives the same result as the first. It follows that $K_i$ is a projection operator $$ K_i=|k_i\rangle\langle k_i| $$ This is the projection postulate. The expectation of the result from a measurement of $K$, given the initial normalised ket, $|f\rangle$, is $$ \langle K \rangle = \sum\limits_i k_i P(k_i|f)=\sum\limits_i \langle f|k_i\rangle k_i \langle k_i|f\rangle = \langle f |K|f\rangle$$ where Hermitian operator, $$K= \sum\limits_i |k_i\rangle k_i \langle k_i|$$ is an observable with eigenvalues $k_i$.

(this has been extracted from my book The Mathematics of Gravity and Quanta and my published paper The Hilbert space of conditional clauses)

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