Determining What Eigenfunction Occurs With Wavefunction Collapse

Question

Suppose an operator $O$ has eigenfunction normalized $f$ corresponding to eigenvalue $n.$ Of course, any function $cf$, with $c$ on the unit circle, is also a normalized eigenfunction. Thus, if a measurement of $O$ on some system returns a value of $n$, we have the state collapse to some $cf$. How is the constant $c$ determined? Does $c$ end up being physically irrelevant for the later time evolution of the system? (I can see it being physically irrelevant for future observables, but what about interference with other waves?)

Similarly, suppose an operator $O$ has degenerate spectrum at eigenvalue $n$, with orthonormal eigenfunctions $f_1$ and $f_2$. When a measurement of $O$ returns a value of $n$, can we in general determine what linear combination $c_1f_1+c_2f_2$ the collapsed state is in? Is the ratio $\frac{c_2}{c_1}$ perhaps given by the ratio $\frac{\langle f_2|S\rangle}{\langle f_1 | S\rangle}$, where $S$ is the state at the time of collapse?

Answer 1

As you guessed in your comment, the usual formalism is to just discard all terms inconsistent with the measurement outcome and then re-normalize. Nothing is changed about the phases of existing terms consistent with the measurement.

A concrete example: If an operator $O$ has two eigenfunctions with eigenvalue 1, say $|1a\rangle$ and $|1b\rangle$, and the state before measurement is

$|\psi\rangle=\alpha_1|1a\rangle+ \alpha_2 |1b\rangle + \alpha_3|2\rangle$

then if the eigenvalue 1 is found, the state after measurement is

$|\psi'\rangle=N(\alpha_1|1a\rangle+ \alpha_2 |1b\rangle)$

where $N=1/||\alpha_1|1a\rangle+ \alpha_2 |1b\rangle||$.

I feel it is my duty to warn you that this works nicely for discrete states as above, but I consider it highly untrustworthy for continuous variables like position. Yet there is no well-accepted alternative. This is just one manifestation of what is known as the measurement problem.

Answer 2

Consider the wavefunction of the form $\psi=e^{i\theta_1}f_1+ e^{i\theta_2}f_2$. Since we can represent any unimodular complex number $c$ in this way as $e^{i\theta}$. Let us see what the probability density of this state looks like: $$|\psi|^2=|f_1|^2+|f_2|^2+ e^{i(\theta_2-\theta_1)}f_1^*f_2+ e^{i(\theta_1-\theta_2)}f_1f_2^*$$

Notice that the interference effect only depends on the difference between the phases. Or as you rightly say, the ratio between the coefficients. This is because the phases cancel out in the direct terms and only the cross terms give the interference.