Could Wittgenstein Speak about the Wave Function $\Psi$ of Quantum Mechanics?

I cannot speak about $\Psi$ because it is not something which is the case.

Physicists generally like to speak about the time-dependent Schrödinger wave function $\Psi (x,t)$ as a solution to the initial value problem with $t$a time variable and $x$ a multidimensional spatial variable: 

• $i\dot\Psi = H\Psi$  for $t>0$ and all $x$
• $\Psi (x,t) =\Psi_0(x)$ for $t=0$ and all $x$,

where $\dot\Psi =\frac{\partial\Psi}{\partial t}$ is the time derivative of $\Psi$,  $H$ is a Hamiltonian differential operator acting in space and $\Psi_0(x)$ is a given initial value. The existence of $\Psi (x,t)$ is said to be solidly guaranteed by mathematical analysis because the Hamiltonian is a linear. 

But determining a specific $\Psi (x,T)$ at some specific time $T$ requires (i) specification of $\Psi_0(x)$ for all $x$, and (ii) time stepping of the equation $i\dot\Psi =H\Psi$. 

Since $x$ has $3N$ dimensions for a system with $N$ electrons, both (i) and (ii) represent daunting tasks. In fact both appear to be impossible as soon as $N$ is not small, certainly impossible for a system with more than $10$ electrons. 

This means that speaking about a wave function $\Psi (x,t)$ for an atomic system with many electrons such as a big molecule, is like speaking about something with features which are forever hidden to inspection. 

What would Wittgenstein have said about such a situation? Let us recall the first and last sentence of his Tractatus:

• The world is everything that is the case. (1)
• Whereof one cannot speak, thereof one must be silent. (6.57)

We are led to ask: 

• Is $\Psi$ something that is the case even if $\Psi$ cannot be computed/determined? 
• Can we speak of $\Psi$ if $\Psi$ cannot ever be computed/determined?

Maybe a theoretical physicist would say yes without hesitation, but maybe a chemist would like to speak about something more concrete like chemical bonding between atoms into molecules. It is here RealQM comes in as something which definitely is the case, and so something to speak of?

To clarify the meaning of "what is the case", Wittgenstein continues from (1):

• The world is the totality of facts, not of things. (1.1)

This opens to a world that is not only materialistic consisting of things like atoms, but also contains facts about atoms in play when atoms form molecules.

The famous quantum physicist Anton Zeilinger claims in Einstein's Schleier. Die Neue Welt der Quantenphysik (2003) that Wittgenstein had little understanding of modern physics in particular quantum mechanics in its StandardQM form, and that (1) should better read:

• The world is everything that is the case, and all that can be the case. (2)

But is (2) really better than (1)? Of course the purpose of adding "all that can be the case" is to cover up the main weakness of quantum mechanics as a science about physics, namely the probabilistic interpretation of the wave function as a description of "what can be the case" rather the "what is the case".

Which world view is most profound, that of Wittgenstein or Zeilinger?  Certainly, speculations about "what can be the case" is what all planning is about, which takes up a major part of the day as more or less constructive day-dreaming, but if you do not make a distinction between dream/fiction and reality/case then you will end up in deep trouble. 

RealQM gives a new form to quantum mechanics in the sense of classical physics, which has a meaning within Wittgenstein's perspective, and does not need Zeilinger's extension. 

PS The idea of proposition (6.57) above is to avoid confusion, resulting from too much speaking. Both mathematicians and physicists have a habit of doing so, which has led to a lot of confusion. For example, mathematicians speak about the infinitely large and infinitely small often without clear qualification which has caused lots of confusion for students. And quantum mechanics is filled with words like duality, complementarity, identity, entanglement, probability, exchange, correlation all creating lots of confusion. The idea of RealQM is to reduce the confusion by showing that it is not necessary to speak about all theses things when speaking about atoms and molecules.