The answer by Craig Gidney is quite adequate for the question, but I want to address the word "collapse" in the title, since search engines will be homing in on it. From webster.com
1: to fall or shrink together abruptly and completely : fall into a jumbled or flattened mass through the force of external pressure <a blood vessel that collapsed>
2: to break down completely : disintegrate <his case had collapsed in a mass of legal wreckage — Erle Stanley Gardner>
3: to cave or fall in or give way <the bridge collapsed>
4: to suddenly lose force, significance, effectiveness, or worth <fears that the currency may collapse>
5: to break down in vital energy, stamina, or self-control through exhaustion or disease; especially : to fall helpless or unconscious
6: to fold down into a more compact shape <a chair that collapses> Definition of the word "collapse", Webster's dictionary
Note how the word describes a physics situation.
A wave function is a mathematical formula with complex numbers, posited for all particles in the quantum mechanical framework, from which the classical dimensions we live in emerge. It is a mathematical expression of a very successful model which, when squared with its complex conjugate gives a probability density distribution for observing with real numbers the problem at hand. In the case of the double slit experiment, the probability of finding the photon at the specific (x,y) of the screen.
The wavefunction exists in our copy books and our computers as a mathematical formula valid continuously. It is an unfortunate label that the word "collapse" has been attached to any property of the wavefunction. The wave function does not break down in any of the senses of the definition of the world collapse. It is always there, in our copy books and computers. A single measurement picks up an instance, and accumulation of measurements gives the probability distribution that the wavefunction so successfully models.
Example:
Take this paper which gives Probability of delivery within x days of a given date, the date given by the doctor from the data the mother gave.
The birth of a baby will be an instance of this plot, which accumulated with more instances should verify the distribution shown. Is anything in any logical way collapsing, according to the definitions of Webster?
I hope this makes clear that collapse is a wrong word to use for a mathematical distribution, attributing reality values. It is at worst an anthropomorphic word, giving human attributes to a mathematical formula, at best a misguided identification of the complex mathematical formula to the real formula of a collapsing balloon. The wave function is not a balloon either.
I think this question arises from a simple misunderstanding of what a wave function is. The wave function of a particle doesn't need to be "wavy". The description of a system in quantum mechanics is always given via its state-vector in the Hilbert space and that can always be translated to the wave function of the said system in a basis of your choice, e.g., the position basis or the momentum basis.
A wave function $\psi(x)$ of a particle in position basis simply gives you the probability amplitude of the particle at position $x$ which is a complex number, i.e., it gives you two bits of information:
- The magnitude gives you the probability (density) that you would find the particle in the vicinity of $x$ if you measure its position.
- The phase gives you the information that you'd need on top of the probability (density) to construct the wave function in some other basis, e.g., the momentum basis, so that you can calculate the probabilities (probability densities) associated with the measurement of its momentum.
So, the point is that there is always a wave function of a particle -- regardless of whether it is very localized and point-like or not.
As to why wave functions are nonetheless called wave functions, I think it's a historic relic. There are two tangible historic reasons that resulted in this naming, I think:
- The position-basis wave function of a particle that has a definite momentum is $\sim e^{ipx}$ and it is actually wavy. These are the famous de Broglie matter waves.
- The time-evolution equation that all wave functions satisfy is called the Schrodinger wave-equation (because it was the equation that was followed by the de Broglie waves, I suppose). One should note that the Schrodinger equation is not exactly a wave-equation although it admits wave solutions. It's more like a diffusion equation with an imaginary diffusion coefficient.
Best Answer
The effect you are describing in your question is known as wave-particle duality and is a form of complementarity, it has been observed in various experiments. Realisations of Wheelers delayed choice thought experiment are what I find most interesting.
In a delayed choice experiment the particles are not measured before they go through the slits but labeled so which slit they go through is known. The only time a quantum system is not disturbed by a measurement is when no new information is gained from the measurement, labeling ensures which slit the particle went through can be known without disturbing the quantum interference$^1$ of the wavefunction. In this context the purpose of any measurement would be to tell which slit a particle went through anyway.
If a particle has a label when it is detected (at the screen) there is no interference and particle-like behavior is observed. If there are no labels there is interference or wave-like behavior, even if the labels are erased after the particles pass through the slits and it cannot be known which slit they passed through.
It appears that it is not possible to see interference and know which way the particles have gone simultaneously. If there is some which way information that enables a better than blind guess at which slit the particles went through the visibility of the interference is reduced.
However, observation of wave-particle duality does not really require wavefunction collapse. Has wavefunction collapse been observed? In my opinion no, but the publicists for this Nature paper disagree. Collapse is connected to interpretations of quantum mechanics.
Collapse of the wave function would imply that the wave function is real (ontic) as opposed to only representing what we know about a quantum system (epistemic). This is an open question, some physicists think one some the other. There is no experimental evidence either way yet, until there is some physicists might say "Collapse? Ontic? Epistemic? that's all about interpretation, shut up and calculate"
If the wavefunction is purely epistemic then there is nothing real to collapse, only the state of knowledge. If it's ontic then wave function collapse would be a possibility, but even then wavefunction collapse is not required to explain quantum measurement.
$^1$Interference is like the pattern on the left in your question, the wave-like behavior.