It is not clear to me that the 'collapse of the wave function' even makes much sense in the 'Copenhagen Interpretation'.
This perspective is presented in the canonical reference  (which is sometimes referred to as the closest elucidation of Bohr's views, for example in ), which I will summarize below for convenience. Please see  for more details. It will hopefully become clear from my summary that, except in a very qualified (and in fact, trivial) sense discussed below, the 'collapse' is really referring to an idealized collapse that never really happens.
A famous article by Bell  specifically 'criticizes' some of the passages I will describe below, in the sense of saying they adequately represent the incompleteness of standard quantum mechanics he is not satisfied with (and he talks about "jumping" which simply does not seem to be correct, one can judge for themselves), so one can see how authoritative the perspective in  is, even to it's famous critics. One can see some further commentary in the literature on this approach for example in  page 12.
To see what the 'collapse' really means we should examine the theoretical description of what it means to actually do a measurement in quantum mechanics as described in , specifically section 7. (The justifications for why this is the approach to measurement that they take are given in the earlier sections 1, 2 and 6 - I would get side-tracked if I start justifying all this too, e.g. why it's unavoidable that we must assume the existence of classical objects, so please consult  for that, below we will just take it as a given that we must describe measurement in the following way).
A measuring device is a classical system with a quasi-classical wave function. Assuming the measurement process can 'completely describe' the quantum system (i.e. within the limits of quantum mechanics) it means the quasi-classical wave function is part of a complete basis of eigenfunctions characterizing a measurement process, i.e. eigenfunctions associated to the possible eigenvalues of the measuring apparatus. Assuming it's spectrum is discrete for simplicity it means a collection of
represent the possible states of the measuring apparatus, and we can say with certainty that the classical apparatus is described by one, and only one, of these wave functions, if we know the value of a measurement.
In other words, the classical nature of the measuring apparatus means we can be absolutely certain that the measuring apparatus is in a given state $n$ and so has as it's wave function the stationary state
Before a measurement of a system, the apparatus and system are independent subsystems of a total system, and so the total wave function is a product of their wave functions,
$$\Psi(x,y) = \Phi_n(x) \psi(y)$$
for $\psi(y)$ the wave function of the system we want to measure (I will always refer to this as the 'system', and refer to the 'total system' when I want to include the measuring apparatus with it, referring to the measuring apparatus separately when I talk about it).
After the measurement, which involves an interaction between the apparatus and system we're measuring, the total wave function $\Psi'(x,y)$ is a complete mess in general, however because the systems are no longer interacting the apparatus is again independent and so we can Fourier expand the total wave function in terms of the $\Phi_n$ basis via
$$\Psi'(x,y) = \sum_n A_n(y) \Phi_n(x).$$
Now again we invoke the classical nature of the classical measuring apparatus to say the following. If we measured a single eigenvalue from the discrete spectrum of the classical measuring apparatus with certainty, then the classical apparatus after the measurement in fact has a definite wave function, it is again a single eigenfunction from the spectrum of possible eigenfunctions, so in fact this sum 'collapses' down to a single term
$$\Psi'(x,y) = A_m(y) \Phi_m(x).$$
It is important to note that, because of the classical nature of the measuring apparatus, and the fact we know the value of a measurement from the discrete spectrum with certainty, the initial sum that 'collapsed' was never really there, it was just a convenient tool to understand what's going on. Obviously if the measurement process was such that we can only be sure it was one from a set of possible measurements, the sum collapses to that set.
What we really care about is the case where we know the precise eigenvalue of the classical measuring apparatus after the measurement, but it's important to keep in mind the case where we don't know the precise value in what follows below (I will mention it explicitly when this case arises).
This immediately implies, since the systems are again independent after the measurement, that $A_m(y)$ is proportional to the wave function of the system we measured after the measurement. It is only proportional because $A_m(y)$ not only has to account for the state of the system after the measurement, it also has to account for the probability that we would find the $m$'th reading of the classical apparatus. We will see this explicitly below.
Therefore we can set it equal to a multiple of the true normalized wave function of the system, $\phi_m(y)$, after measurement
$$A_m(y) = a_m \phi_m(y).$$
An obvious implicit assumption here is that $A_m(y)$ does not depend on the initial wave function $\phi(y)$. In other words, $A_m(y)$ (and as a consequence, $\phi_m(y)$,) is completely determined by the measuring process alone, assuming the measuring process can completely describe the state of the system as I said at the beginning of this post (otherwise the initial conditions could clearly affect things).
However, the linear nature of the equations of of quantum mechanics implies that there still should be a linear relation between the wave function before measurement, $\psi(y)$, and the wave function after measurement, $A_m(y)$. In other words, $\psi(y)$ should evolve into $A_m(y)$ under some evolution operator which we can write as
$$A_m(y) = \int K_m(y,y') \psi(y') dy'.$$
Since $A_m(y)$ is completely determined by the measurement process, it means $K_m(y,y')$ is completely determined by the measurement process.
We now have two different interpretations of $A_m(y)$, the 'collapse' interpretation and the 'evolution' interpretation, i.e.
$$A_m(y) = a_m \phi_m(y)= \int K_m(y,y') \psi(y') dy'.$$
They must clearly be the same thing so that we can set
$$K_m(y,y') = \phi_m(y) \psi_m^*(y')$$
$$a_m = \int \psi_m^*(y')\psi(y') dy'.$$
At this stage (ignoring the obvious implication that the notation suggests for now) all we can say about these $\psi_m^*(y')$'s is that they depend on the measurement process.
But this is all just saying that the abstract wave function of the total system after a measurement, allowing for situations where we could not measure the eigenvalue of the classical measuring apparatus even with any certainty (i.e. an extreme version of the special case I warned about above), is
$$\Psi'(x,y) = \sum_n a_n \phi_n(y) \Phi_n(x)$$
where the $\phi_n(y)$ are normalized functions (which represent the total system we measured after a measurement, note counter-intuitively they are not actually 'eigenfunctions' of anything), the $\Phi_n(x)$ are normalized eigenfunctions of the measuring apparatus, so that the $a_n$'s are just the usual coefficients representing probabilities and satisfying
$$\sum_n |a_n|^2 = 1.$$
The fact that this last relation should hold, coupled with the fact that $a_n$ is defined by $a_n = \int \psi_n^*(y')\psi(y') dy'$, means that the $\psi(y')$ should expand in a complete basis of $\psi_n(y')$'s, but the $\psi_n(y')$'s were determined by the measuring process. In other words, the wave function after measurement should expand in a basis of eigenfunctions of an operator characterizing the measurement process.
But again, invoking the classical nature of the measuring apparatus for a (at least theoretically) precisely known measurement from the discrete spectrum, the sum $\Psi'(x,y) = \sum_n a_n \phi_n(y) \Phi_n(x)$ thus 'collapses' to $\Psi'(x,y) = a_m \phi_m(y) \Phi_m(x)$ (i.e. it was always of this form for the specific interaction between the classical apparatus and system where we know the precise eigenvalue of the measuring device after measurement), meaning the measuring apparatus gave the $m$'th eigenvalue associated to $\Phi_m(x)$, but since $a_m = \int \psi_m^*(y')\psi(y') dy'$ this tells us that the wave function $\psi(y')$ was actually 'measured' to be in the state $\psi_m(y')$ when we did the measurement. In other words, this is the best we can say about the state of a quantum system when we do a measurement, meaning we cause a measuring apparatus to interact with the system. All we can infer about the quantum system is that in the process of interacting, the $\psi_m(y')$ wave function 'rubs off' onto the measuring apparatus under the time evolution of the total system, in the sense that the quasi-classical wave function evolves (through interacting with the system) from one 'stationary state' to another in the process of interacting. It's not "jumping", one is completely ignoring the fact the measuring apparatus is interacting during the measurement and so obviously can linearly evolve (not discontinuously jumping) to a new state.
We are completely shielded, in principle, from saying anything more about what the system was 'really' doing, all we can do is infer from the final measurement of the apparatus what the system was doing from how it caused the measuring apparatus wave function to register the measured eigenvalue. Further the wave function after this measurement process is also given by this discussion, it is this new wave function $\phi_n(y)$, which in general is completely different from the initial wave function $\psi(y)$. Everything is encoded in the above discussion.
Thus you see the idea of 'collapse of the wave function' is nonsensical if one means anything other than the trivial collapse of the Fourier expansion discussed above. It would completely contradict the linearity of the equations of quantum mechanics if there was some jarring 'collapse' of the wave function going on. The above process completely accounts for this properly. The initial wave function of the system, $\psi(y)$, just evolved into $A_m(y) = a_m \phi_m(y)$ via a linear evolution operator $A_m(y) = \int K(y,y') \psi(y') dy'$ where $\phi_m(y)$ is the normalized wave function of the system after measurement, and $a_m$ encodes the (experimental) fact that we measured a certain value due to the system somehow 'rubbing off' onto the measuring apparatus during the interaction. That's intrinsically the best we can say about the state of a quantum system within quantum mechanics. It would thus completely contradict the linearity of quantum mechanics to think the wave function of the system actually 'jumps', and it's always stated in a completely hand-waving unjustified manner, unlike the discussion above where everything fits into place.
The really non-trivial thing going on here, that the concerns about 'quantum jumping' are really expressing, is that the fact we can measure anything at all. This is intrinsically caused by the fact that quantum mechanics can only be defined in the first place by assuming the existence of classical mechanics, to which it must reduce in the 'classical limit'. That is the real 'jarring' thing about this. This assumption of the existence of classical mechanics means we must have the measuring process is such that the total wave function (expanded in a basis of eigenstates of the measuring apparatus) always (when we measure the precise eigenvalue of the measuring apparatus) 'collapses' to a single term, but the 'collapse' doesn't really happen what really happens is 'classical mechanics' says only one term was there the whole time.
The wave function just evolves from one wave function into another wave function via linearity and due to the interaction between the (classical) apparatus and the system (we're measuring), that something 'rubs off' onto the measuring apparatus during this is simply an experimental fact the theory is trying to capture.
So, through all this, there is no 'collapse' of the total wave function. It is just a mathematical tool allowing us to say that the total wave function 'jumps' from the complete Fourier sum down to a single individual term in the sum if we keep things general in the beginning. If that sum doesn't 'collapse' then we can never measure anything, or actually even say the system even had a wave function, indeed how can we even talk about a measurement. In other words, nothing makes sense without the 'classical limit'.
If you assume classical mechanics exists however, then there is never any 'jump', the abstract 'Fourier expansion' we proposed actually just contained one term the whole time. Again, the point is: without classical mechanics, that Fourier sum can never be said to really just be a 'single term', so we get completely stuck and just have no theory.
Not only is it the case that the initial wave function $\Psi(y)$ transitioning into a new wave function $\phi_m(y)$ is just a consequence of the fact that a classical measuring apparatus interacts with a quantum system, i.e. an interaction makes it evolve into a new wave function, it is absolutely vital to notice that it is precisely only because of the classical nature of the measuring apparatus that means we can even be aware of the fact that the system evolved from one wave function to a new wave function.
In other words, without the existence of classical mechanics, there is absolutely no theory of quantum mechanics at all. Without the quantum system "rubbing off" onto a classical apparatus through the interaction, we just have nothing. Any discussion of "collapse" that mentions it in some hand-waving fashion is either just a misunderstanding of the above description of the measurement process, or an 'alternative interpretation' of quantum mechanics which you can likely bet isn't even internally logically consistent (compared to Copenhagen).
There is a(n unbelievable) claim that 'decoherence' allows us to understand how the 'classical world' arises from quantum mechanics, e.g. via diagonal entries on a density matrix. At least from the above discussion, it is very likely completely circular and nonsensical from the above perspective. One has to trust that an 'alternative' perspective (e.g. those mentioned in ) to the Copenhagen interpretation given above (from reference  below) is as logically internally consistent as this while also somehow defining quantum mechanics without classical mechanics and properly accounting for the measurement process without a contradiction. It is hopefully clear from this discussion why people sometimes say there is no alternative to the 'Copenhagen interpretation'.
A side comment is to notice how absolutely vital it is that the continuous spectrum eigenfunctions in quantum mechanics are actually "wave functions" (a view actually commonly denied even on this site, see my answer here for how serious some of the the flaws are with this perspective, on top of the critical flaw in the whole theory that would occur due to the measurement process according to this post). Indeed re-reading the above discussion for this case, if they aren't, we can never even know what the wave function of a system after a measurement is, the measuring apparatus could never even have a definite value thus we can never even fix the wave function of the system after measurement.
- Landau and Lifshitz, "Quantum Mechanics", 3rd Ed., Sections 1, 2, 6, 7.
- Zinkernagel, "Niels Bohr on the wave function and the classical/quantum divide".
- Weinberg, "Quantum Mechanics", 1st Ed., Sec. 3.7.
- Bell, "Against Measurement", 1990 Phys. World 3 (8) 33.