Nature of expectation values and Born's rule and the measurement problem Suppose we take a normalised quantum mechanical wave function of $\Psi (\mathbf{r} ,t)$.
If we expand it in a certain form of spatial functions $\psi_{n} (r)$ which is complete orthonormal. Then we can assume the expansion coefficients vary in time. So we can get
$$\Psi (\mathbf{r} ,t)=\sum_{n} c_{n}\psi_{n} (\mathbf{r})$$
From the fact that $\Psi (\mathbf{r} ,t)$ is normalised, we can deduce the following expression: $$\int^{\infty }_{-\infty } \left\vert \Psi (\mathbf{r} ,t)\right\vert^{2}  d^{3}r=\int^{\infty }_{-\infty } \left[ \sum_{n} c^{\ast }_{n}\psi^{\ast }_{n} (r)\right]  \times \left[ \sum_{m} c_{m}\psi_{m} (r)\right]  d^{3}r=1$$
Because of the orthogonality of the basis functions, only terms where $n=m$ would be able to get out from the integrating operator. As they are orthogonal, the result from any such term will be $\left| c_{n}(t)\right|^{2}$. Hence we deduce the following expression; $$\sum_{n} \left| c_{n}\right|^{2}  =1$$
On measurement of a state, the system collapses into $n$th eigenstate of the quantity being measured with the probability of $P_n=\left| c_{n}\right|^{2}$. However, how and why this is a solid tool which works? It is a problem that we do not certainly know as it is partially a philosophical problem, I am aware. Any philosophical interpretative answer and a mathematical one is appreciated. I am also looking forward to some quantum mechanical explanation with mathematics included. I am aware of its existence although it might not be suited for my level of understanding the subject. Any explanation of why Born's rule is efficient with certain complex quantum mechanical concepts in a simpler way would be appreciated. Thank you, have a lovely day.
 A: 
How and why this is a solid tool which works?

It always holds, since it's considered to be a postulate of the theory itself (for example in the so-called orthodox interpretation). The idea at the basis of quantum mechanics is describing nature and its (apparently casual and accidental) behavior using wavefunctions. Differently from what happens in Newtonian mechanics, systems can also behave as a superposition of states, so that the wavefunction describing them can be mathematically written as a sum of possible states which the system can have access to, weighted with opportune coefficients (complex in general) whose modulus squared is interpreted as the probability of that specific state to be the result of some measurement. Nothing new up to here.
The reason why the Born rule is so widespread relies on the fact that it's able to provide immediately a link between the mathematical formalism of QM, Hilbert space and so on, and experiments, which are linked to predictions of quantum physics itself. By introducing Born rule, the system isn't deterministic anymore and thus the fundamental theory produced is intrinsically probabilistic. Historically, it was I think the first time in which indeterminism entered fundamental physics.
You might think that since the outcome of a certain measurement is in principle unpredictable, the whole theory turns out to be incomplete; you would be quite in good company, since Einstein himself at the beginning thought this was the case.
The "problem" - if you want to call it like that - is that we are looking at the quantum world through our knowledge of classical mechanics: one cannot say that the Born probabilities are either subjective or objective, because the situation is more subtle and has no alternative counterpart in classical physics or probability theory.
Finally, if you want a mathematical justification of Born rule, there's been a huge work by Hartle and many others to derive it from basic postulates. They started studying infinite sequences of measurements and at the end they discovered that the Born rule was automatically satisfied.
However, a plausible critique is that the whole reasoning might be some kind of circular argument, so that no definite answer to this derivation has been given.
A: Most introductory text books on quantum mechanics have some variant of the following postulates for quantum mechanics:

*

*The state of a system $S$ can be represented by a vector $|\psi\rangle$ in a Hilbert Space $H$. If $S$ can be decomposed into two smaller systems $S_A$ and $S_B$, which can be represented by Hilbert spaces $H_A$ and $H_B$ respectively, then $H = H_A \otimes H_B$

*$|\psi\rangle$ evolves according to a unitary operator determined by the Schrodinger equation except when measurements occurs. $|\psi(t)\rangle = U(t)|\psi(0)\rangle$.

*When a measurement is performed, the state of the system collapses into a new state which corresponds to a definite result of the measurement. (Projection Postulate)

*The probability of collapse to a given state is proportional to the magnitude of the overlap squared between the initial and final states (Born Rule).

Postulates 3 & 4, collectively referred to as the measurement postulates, are the source of most discomfort and confusion in quantum mechanics, and much controversy.
In the past two decades people have found several ways to derive the measurement postulates from axioms 1&2, often slightly modifying postulate #1, in an interpretation independent way. I personally love Zurek's approach using envariance (see paper here, which is reasonably accessible). Here is another, more recent, derivation of the measurement postulates.
Many physicists (see Bell's paper "Against Measurement" for example) consider the traditional postules to be incomplete, because it doesn't include a prescription for what configurations of wavefunctions result in something that "counts" as a measurement. For this reason, different modifications to the above postulates have been made over the years, and each of which allows different derivations of Borns rule. These modifications are often called "interpretations" even though they are physically different theories which make theoretically testable different predictions (and sometimes actually testable, for example we've all but ruled out local hidden variables by testing Bell's Inequality). These include:

*

*Objective collapse theories Of these, I believe GRW is the most well developed, and there has been some interesting experimental work testing the theory in the last decade (see for example this)

*"No Collapse" theories. In this approach, only postulates 1 & 2 are assumed, and postulates 3&4 are derived from them under the additional assumption that the entire universe can be thought of as a quantum system. This is also called "many worlds" because, when deriving 3&4, you find that multiple classical realities are present in the wave function of the universe. Early efforts to derive Borne's rule within the no collapse approach were circular, but I don't believe this is the case for modern derivations. Derivation of Born's rule in many worlds typically use Gleason's theorem to "count" the number of different worlds where each outcome occurs.

*So many others (Consistent Histories, etc)

A: The system doesn't "collapse". During a measurement it is connected to an external system that adds quanta of energy to it (absorption spectroscopy) or removes quanta of energy from it (emission spectroscopy). What you are looking at here is merely the solution theory of this class of linear differential equation. It is NOT the physics that is actually happening when we are doing a measurement. The Born rule only gives you a set of hypothetical measurement scenarios. It does not describe real-world experiments that can actually be performed in the lab. We can, for instance, not measure  the electron density directly even though the Born rule operator for that is the trivial identity operator. Instead we have to build quite complicated diffraction experiments for that purpose and even those don't give us an unambiguous density function.
There is no philosophical problem here. You are simply not given the totality of the knowledge needed to use this in an actual physics lab. When we are measuring the spectrum of the hydrogen atom for instance, we aren't actually measuring the eigenvalues of the SE. Instead we are only getting energy differences between states with an orbital angular momentum difference of one Planck unit. We then have to reconstruct the absolute energy values of the SE states from that experimental data. How this is done is usually described in textbooks about atomic physics.
One can get "closer" to actual measurement with different mathematical formulations of quantum mechanics that are commonly used in quantum field theory. The most "natural" form produces scattering cross sections, which is what the spectroscopist in the atomic/molecular physics lab and the high energy physicist at CERN actually observes. The mathematics of that is not nearly as neat and simple as the solution theory of linear wave equations, though, so it is not suitable as teaching material in an undergrad course on QM. At least I have never seen an introductory undergrad QM class that tried to teach scattering calculations first. I doubt that it would be enjoyable to most students if we were teaching QM that way.
So, yes, you are missing something here, but it is entirely technical. No philosophy and no new interpretations are needed.
