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Interpretation of Wave Function

  • This post is first of its kind in a series of posts called "Dialogues".
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I must caution that the analogy of intensity of light used in context of interpretaion of the \(|ψ|^2|) as position  probability density is not quite accurate. The intensity of light, or of a beam of particles, is closer to the probabilty current density, and not the probability density itself.  

First of all there are some grammar mistakes which may be overlooked. Second the concept of wave function was interpreted by assuming an elctric fluid rather than pointlike electrons. Then the mass density of the fluid \(\rho(x)m\psi*\psi\) were related to current by continuity equation. Max Born had a completely different idea of the wave function. and elevated it from a fluid desity to a probaility density for a particle to be at various points of space. A good reference is the book by OMnes.

Please provide a complete referenece of the book by Omnes.

The interpretation of  \(|\psi(x,t)|\,dx\) as probability at time \(t\) of finding particle at position between \(x\) and  \(x+ dx\) is inconsistent with conservation of probability as derived from the Schr\"{o}dinger equation.

If \(|\psi(x,t)|\) is the probability density for position at time \(t\), then the probability that the position will have any value will be given by 
\(\int_{-\infty}^\infty |\psi(x,t)|dx \). For such an interpretation, we must use the  normalization

   \[\int_{-\infty}^\infty |\psi(x,t)|\, dx =1\]

for the wave function. But such a normalization is, in general, inconsistent with the time dependent Schr\"{o}dinger equation, because the left hand side of the above equation is not independent of time and the total probability will not be conserved. This would mean if the total probability is one at some instance, it will not remain one at all times for general solutions of the time dependent Schr\"{o}dinger equation. Note that only for stationary states the total probability will  remain constant in time.