Did anyone has any idea how Born came up with the probabilistic interpretation of quantum mechanics. It is by all means very bizarre. And then it leads to the idea of copenhagen interpretation. Also no one emphasis on any kind of meaning attached to $\psi$ the wavefunction rather it's norm squared. One last thing why we work just with complex number why not quaternion if we are happy doing abstract thing without give any physical foundational meaning to complex number?
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$\begingroup$ I think it is easier to see what the QM equations means in $\mathbb{R}^2$ instead of $\mathbb{C}$. $\endgroup$– EmilCommented Apr 9, 2018 at 6:19
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1$\begingroup$ he explains a bit in his Nobel lecture nobelprize.org/nobel_prizes/physics/laureates/1954/… $\endgroup$– physicopathCommented Apr 9, 2018 at 8:06
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1$\begingroup$ If you’re interested in a possible interpretational justification of the Born rule, that would be a strength of the transactional interpretation of quantum mechanics. ayuba.fr/mach_effect/cramer1986.pdf $\endgroup$– GilbertCommented Apr 9, 2018 at 15:22
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2 Answers
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There are several questions in this post, some of which belong in https://hsm.stackexchange.com, others are more suggestions. However, the title is a question that belongs here. The short answer is that $\psi^*\psi$ is, up to the factor $q=\pm e$, the charge density. With the operational definition that the particle is where the charge is, the charge density is $e$ times the probability distribution for finding the particle.
To justify the expression of the charge distribution, consider the Schrödinger Lagrangian $${\cal L} = i\hbar \psi^*\psi − \frac{\hbar^2 } {2m} {\bf \nabla} \psi^* \cdot {\bf \nabla} \psi − V (x,t) \psi^*\psi $$ This lagrangian leads to the Schrödinger equation. The associated Noether charge current conservation law is $$\frac{\partial \rho}{\partial t} = - {\bf \nabla} \cdot {\bf j}, \quad \rho \equiv q\psi^*\psi, \quad {\bf j} \equiv \frac{q}{2im} \left( \, \psi^*{\bf \nabla} \psi - {\bf \nabla} \psi^*\psi \, \right)\,, $$ where $q=\pm e$.
Hence we see that the Born conjecture is very well founded.
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$\begingroup$ @thomasFritsch Thanks. I corrected the formulas. $\endgroup$– my2ctsCommented Feb 3, 2020 at 6:13
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$\begingroup$ I meant to say it should be $\frac{\partial \rho}{\partial t} = - {\bf \nabla} \cdot {\bf j}$. $\endgroup$ Commented Feb 3, 2020 at 6:56
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$\begingroup$ Everything seems to work the same way if you remove $e$ and simply speak of (particle) density. Moreover in this latter case the interpretation works also for neutral particles. What do we learn more for attaching the $e$ factor? $\endgroup$– lcvCommented Feb 3, 2020 at 13:26
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$\begingroup$ @lcv Charge is observable, particle density is derived from it. It can also be derived from other observables,such as energy-momentum. By itself particle density is not observable. $\endgroup$– my2ctsCommented Feb 3, 2020 at 17:05
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$\begingroup$ What about when photons impinge on a a photographic plate? $\endgroup$– lcvCommented Feb 3, 2020 at 17:58
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The quantum mechanics didn't start with Schroedinger equation, it had a long preceding history. The atomic experiments of the time could only be described by the discrete stationary levels of atoms with probabilistic quantum jumps. Thus the probabilities were already ubiquitous in the atomic physics before Born though they were associated with lack of our knowledge about fundamental dynamics.
In 1916 Einstein described the absorption and emission of light by atoms as a probabilistic process of absorption and emission of photons. The absorption probability is determined by the spectral energy density of light. In his paper Born explicitly points out Einstein ideas that the waves should be treated as a "ghost field" that determines the probability of the photon to take a certain path.
It is interesting to point out that in the original rapid communication Born intepreted $\Psi$ itself as a probability density. It's only in a footnote, probably added after the paper was finished, where he points out that the correct formula involves square. This was a rather obvious guess as the wavefunction is not only nonpositive, it's also complex in the collision processes that Born's paper was devoted to. This guess was supported then by calculations of scattering crossections compared to the experiments.
The physical meaning of the wavefunction is that it is a probability amplitude. To properly understand this you should realize that $P(x)=|\psi(x)|^2$ is not the only probability distribution that appears in the experiments! As a matter of fact quite often it is not the particle position that is measured. E.g. you can measure the particle momentum and then you will have to use, \begin{equation} P(p)=|\phi(p)|^2,\quad \phi=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty}dx\,e^{-ipx}\psi(x) \end{equation} You can't predict $P(p)$ knowing only $P(x)$! To get it you have to also know the phases of $\psi(x)$. The constant phase doesn't matter but relative phases in different points do. So measuring sufficient number of observables you can actually reconstruct those phases up to the constant. These phases also govern the dynamics of the quantum state - different relative phases mean different dynamics.
So the complex phases are not some abstract things - they can be measured.
As to the quaternion idea the simple answer is that no one needed it. The simple complex structure was enough. In principle you may consider the quaternion wavefunction as a 2-component wavefunction with all the basic principles of the quantum mechanics applied. The real question is whether your dynamics respects this quaternionic structure so that it's useful. I don't know of any example though may naively expect something like that working in the Dirac-Weyl equation.
