Born's Interpretation of Wavefunctions

[QuantumBunnii]

(The following is a purely qualitative consideration of Quantum Mechanics)
In a particular Quantum Mechanics text, I've come across the following quote which I'm having some difficulties interpreting.

"We describe the instantaeous state of the system by a quantity $\Psi$, which satisfies a differential equation, and therefore changes with time in a way which is completely determined by its form at a time t = 0, so that its behavior is rigorously causal. Since, however, physical significance is confined to the quantity $\Psi^{*} \Psi$, and to other similarly constructed quadratic expressions, which only partially define $\Psi$, it follows that, even when the physically determinable quantities are completely known at a time t = 0, the initial value of the $\Psi$ function is necessarily not completely definable. This view of the matter is equivalent to the assertion that events happen indeed in a strictly causal way, but that we do not know the initial state exactly."
-- Max Born

I understand that $\Psi$ contains some information that cannot be obtained from $\Psi^{*} \Psi$ (for instance, the imaginary part, or its sign), but this doesn't mean that $\Psi$ is not deducable from the Hamiltonian through the Schordinger equation.
Is the point that Born is trying to make that, although we can deduce the quantity $\Psi$ mathematically with some prior physical knowledge (namely, the Hamiltonian), we can never directly measure $\Psi$ itself? What does he mean by "not completely definable"? This seems a very important point.

Thanks.

 

[mfb]

There are different Ψ, leading to the same observations for all possible measurements. Therefore, you cannot know Ψ exactly. You can find some Ψ in agreement with your measurements, however.

 

[QuantumBunnii]

I see...
And I guess this is a direct result of the nature of the Schrödinger equation?
Also, is Born suggesting that "causality" breaks down (or, rather, that "causality is... empty") because the evolution of the probability function depends on this undefined quantity $\Psi$ which (although we can deduce certain representative possibilities) can never be found to exist in nature?

 

[Jano L.]

And I guess this is a direct result of the nature of the Schrödinger equation?

No, the ambiguity is not due to the equation. In principle one could use Schroedinger's equation for $\Psi$ playing role of a definite physical quantity.

The difficulty is that often there is no good general rule for choice of initial $\Psi$. Since it gives mainly averages and probabilities, its determination would typically require huge number of measurements, and even then there would remain the gauge freedom in the choice of its phase (now that I think of it this is partially due to the form of the Schroedinger equation).

Also, is Born suggesting that "causality" breaks down (or, rather, that "causality is... empty") because the evolution of the probability function depends on this undefined quantity Ψ which (although we can deduce certain representative possibilities) can never be found to exist in nature?

I can't find any support for this in the above quote. Is there more to it in your book?

 

[QuantumBunnii]

I can't find any support for this in the above quote. Is there more to it in your book?

Firstly, I had forgotten to include the final sentence in Born's quote:

"... In this sense the law of causation is therefore empty; physics is in the nature of the case indeterminate, and therefore the affair of statistics."

Secondly, after providing the reader with the above quotes, Resnick articulates as follows:

"His second point, about not being able to completely define the space dependence of the wave function at the initial time, [follows from the fact that] if we know a probability density from an initial set of measurements on a system, we still canot determine uniquely an initial wave function to associate with the system. All we can determine is the sum of the squares of the real and imaginary parts of the wave function."
(Quantum Physics: Of Atoms, Molecules, Solides, Nuclei, and Particles, Second Edition, Robert Eisberg and Robsert Resnick)

Admittedly, this doesn't shed much light on his statement regarding "causality" (indeed, it is tantamount to your assertion that we can always choose an arbitrary phase), and this is why I posted.
Undermining causality in virtue of an underterminate, immeasurable quantity ( $\Psi$ ) giving rise to measureable results ( $\Psi^{*} \Psi$ ) might make some sense (?), but it is a very profound expression.
Of course, I can also be completely misinterpreting it. :p