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Let's consider a discrete quantum system such as the harmonic oscillator or the hydrogen atom. According to quantum mechanics, when a measurement of the energy of the system is made the wave function collapses to one of the eigenfunctions of the hamiltonian operator. But if we consider instead a non-discrete system, such as a free particle, the eigenfunctions of the hamiltonian operator are non-normalizable functions, so the wave function can't collapse to one of those. What does quantum mechanics says in this case? Can it be determined, knowing the result of the measurement, what is the new wave function (like we can in a discrete system using accurate-enough instruments)?

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  • $\begingroup$ You might want to read this regarding normalisation. Also I don't see how it follows that the inability to normalise leads to the inability to find the system in a particular state. $\endgroup$ – Charlie Aug 14 '20 at 20:06
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    $\begingroup$ The hydrogen atom has a continuum of unbound states. $\endgroup$ – G. Smith Aug 14 '20 at 20:16
  • $\begingroup$ Non-normalizable function are not acceptable solutions so they can't represent the wave function of a particle unless you consider a supeposition of them. My question is: after i make a measurement and obtain a certain energy, which superposition of the eigenfunctions did the system collapse to? $\endgroup$ – Rhino Aug 14 '20 at 20:18
  • $\begingroup$ @G.Smith right, my question also applies to the hydrogen atom when I measure an energy greater that zero $\endgroup$ – Rhino Aug 14 '20 at 20:21
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It depends on the type of measurement you perform. The idealized case is described by the PVM (projector valued measure) uniquely associated to the observable viewed as a selfadjoint operator through the spectral theorem. The PVM is a collection of orthogonal projectors $P_E$, where $E$ is a Borel set of the real axis, typically a finite interval, defined in the practice by the precision of the instrument. This collection of projectors satisfy some mathematical properties similar to those of a positive measure.

If the initial state is represented by $\psi$ and the outcome is $E$, the post-measurement state is always described by the vector $P_E\psi \neq 0$ up to normalisation.

Here $||P_E\psi||^2$ is the probability to obtain the outcome $E$ when the initial state is represented by the normalized vector $\psi$.

All that is nothing but the Luders-von Neumann postulate.

If the spectrum is continuous, single points $E=\{\lambda \}$ have automatically zero projector $P_E=0$, so that "non-normalizable vectors" cannot be produced this way.

For instance, for the position operator, if the position measurement produced the outcome $E=[a,b]$, the corresponding projector is $$(P_E \psi)(x) := \chi_E(x) \psi(x)$$ where the function $\chi_E$ is zero outside $E$ and $1$ in $E$.(Notice that if $E$ is a single point, the associated projector is zero since single points have zero Lebesgue measure.)

It is worth stressing that this description is valid also when the spectrum is a point spectrum. In that case single points (eigenvalues) have non-zero projectors: the ones onto eigenspaces.

A more realistic description is provided by a POVM (positive operator valued measure) and its decomposition (it is not unique) in terms of Kraus operators, but also this description does not give rise to non-normalized state vectors.

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  • $\begingroup$ So it depends on the presition of the instrument and it's just a slice of the initial state renormalized? And what do "PVM and POVM" mean? $\endgroup$ – Rhino Aug 14 '20 at 20:42
  • $\begingroup$ To every observable viewed as a selfadjoint operator there is an associated a PVM. i.e. a map sending every Borel set $E$ to an orthogonal projector $P_E$ satisfying some properties. The measurement defines $E$ in relation to the concrete instrument. Notice that this description is valid also when the spectrum is a point spectrum. In that case single points (eigenvalues) have finite projectors: the ones onto eigenspsces. $\endgroup$ – Valter Moretti Aug 14 '20 at 20:46
  • $\begingroup$ A POVM is a map associating Borel sets (outcomes) to positive operators (PO) instead of projectors (P). PVM = projector valued measure...But this description is a bit more complicated and I do not want to enter into the details. $\endgroup$ – Valter Moretti Aug 14 '20 at 20:50

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