Let me give a vague analogy to illustrate the issues in talking about observation.
Imagine that I have just thrown a stone into a pond and I ask you, can you see the wave it makes? You say, yes of course, why? I say, no you didn't really see all of the wave. You only saw what you saw from your position and angle. And is what you saw the wave? Or is it just an image of the wave that is in your mind? How do you even know that that image is an accurate reflection of the actual wave? In fact we know our eye has a limit on its resolving power and its sensitivity.
So you didn't actually see any wave. You just saw something which both of us do call a "wave" in English. The word is merely a reference to the entity, not the entity itself. You might ask, is it possible to observe an entity directly and not through any intermediate instrument such as our eyeball? But what really is "directly"? If your mind can somehow touch the wave (with what, may I ask?), is it enough for you? Or is your mind itself merely an instrument which you use to interact with the world?
Anyway that dives straight into the deep end of philosophy, though you'll have to somehow answer that before you can specify precisely enough what you mean by "observe".
On the other hand, what if we both agree there was a wave, and I then ask you, what is the position of the wave? And you stare blankly at me. But that might well be the same question you would be tempted to ask about an electron. What if the electron indeed has an underlying reality that corresponds more closely to its wavefunction rather than a single point in space? Do you even have the slightest evidence that it is more like a point? No.
You could say, let's take the highest point of a water molecule (assuming it's sufficiently point-like) as the position. If so, then it's not going to have any nice properties at all, and would randomly jump around the pond. A better idea would be to take the average position of the water molecules that are above the average water level of the pond. 795
Then we can 'see' that it moves in the direction of the wave more or less along with the crest. We can even touch the wave crest as it goes past, which means that we can sort of estimate the position as defined this way. There is some uncertainty here, not unlike the uncertainty you see in measuring the position of a classical wave packet or in measuring the position of a particle (<PERSON>'s uncertainty principle), in the sense that despite the position of a pond wave being well-defined (assuming point-like water molecules or more generally some mass density function for the water), we cannot even classically measure it accurately because anything we do will disrupt the wave.
Similarly, we can define velocity of the wave as the flux, the average flow of the water (according to the evolution of the mass density function over time). As with position, we cannot even classically measure the velocity accurately without changing it.
Now, we could go another way around the problem. Instead of trying to observe all of the wave at once, we repeat the stone throw many times, and each time we observe just one small part of the wave. Of course, now that we are skeptical we'll question whether we really can repeat something in exactly the same way each time. It's of course impossible in general but we hope it's not too wildly different.
This is exactly what scientists have done in order to observe (in this sense) the wavefunction of an electron. It was done very long time ago, and I do not know the history, but supposedly IBM was one of the first to arrange impurity molecules on a metal surface and then using a Scanning Tunneling Microscope to image the electron density. They have some pictures here, including the well-known quantum corral:
(http://researcher.watson.ibm.com/researcher/files/us-flinte/stm16.jpg)
I do not know whether they edited the raw data (highly likely, when I did it before I had to edit to remove noise and artifacts from the imperfect STM tip). There are other images on the internet like:
(http://nisenet.org/catalog/media/scientific_image_-_quantum_corral_top_view)
But of course all colour or 3d effects in STM images are computer generated. Recently (2013), some have claimed to be able to image atomic orbitals, such as:
(http://physicsworld. 795