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Where does the idea come from, that possible results of quantum measurement are eigenvalues of the operator corresponding to the observable?

Mathieu Krisztian
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  • For those who state that the question is duplicate : it is not the case. The other questions so-called 'duplicate' are discussing about the reason that probability is the square of the eigenvectors. I deal here mostly with reason for choosing eigenvalues as solutions : I don't discuss on the associated probability. – Mathieu Krisztian Sep 04 '19 at 12:24

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Postulates come in a sense from nowhere, but I will make a case. De Broglie proposed that since light can behave as consisting of particles in the photoelectric effect, it would make sense that all particles could behave like waves. This inspired Schrödinger to replace energy and momentum by the well known operators in the classical energy expression $E=p^2 /2m +V$. It turned out right. The resulting Schrödinger equation has quantized solutions, so called eigenfunctions, and describes (nonrelativistic) physics correctly. And there you are: nature is quantized.

There are two phenomena to distinguish. Quantization is the observed fact that the energy of a system is a sum of quanta of its eigenfrequencies. The eigenfrequencies can be discrete or continuous. This is governed by boundary conditions or, related, discrete symmetries.

If you solve the SE on a discrete basis of functions then the eigenvalue equation becomes a matrix equation.

my2cts
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  • My question may not be clear. The fact that nature is quantified was coming from Planck law. ok, good, but then, where is coming the idea that the solutions are those coming from the diagonalization of a matrix and from eigenvectors ? There is an infinite number of possibilites of ideas : what was the motivation to explore the diagonalization of the matrix ? Where is the idea coming from ? Do you mean that quantized solutions mean necessary that there are eigenvectors, so eigenvalues and matrix ? – Mathieu Krisztian Sep 03 '19 at 21:45
  • It seems that quantified solution has nothing to do with eigenvector. It seems that it comes only from limit conditions. If potential is null the solutions are continuous. Do you agree/disagree ? Do you have comments ? – Mathieu Krisztian Sep 04 '19 at 06:52
  • Thanks, I updated my answer to address this. – my2cts Sep 04 '19 at 08:09
  • I'm not sure about your sentence : "Quantization is the obsserved fact that the energy of a system is a sum of quanta of its eigenfrequencies." : energy is not the sum : according to quantum mechanic, there are an infinite number of possible energy values, but these values are discrete. About "The eigenfrequencies can be discrete or continuous.". OK. But then, "This is governed by boundary conditions or, related, discrete symmetries." But the problem is that if there is no boundary condition, there is no discrete solution. – Mathieu Krisztian Sep 04 '19 at 12:30
  • So we come back to the beginning : the quantified solution do not come from the genuine Schrodinger equation, but from fixing additional condition limits. So Schrodinger equation is not intrisically a motivation to search for eigenvector of a matrix. Or maybe I am lost in that. Could you reformulate your answer taking into account my comment ? Thank you – Mathieu Krisztian Sep 04 '19 at 12:30
  • I don't see how you arrive at "this inspired Schrödinger to replace energy and momentum by the well known operators..." from the wave/particle realization of De Broglie. – interoception Sep 04 '19 at 13:16
  • @Danny1 This assertion is based on the first sentence of E. Schrödinger PRL28 (1926) p. 1049. See https://people.isy.liu.se/jalar/kurser/QF/references/Schrodinger1926c.pdf – my2cts Sep 04 '19 at 14:44
  • The discrete energy levels come from boundary conditions or discrete symmetry. The quantization comes from observation. – my2cts Sep 04 '19 at 17:35