What is the wavefunction of the observer himself? I am aware about different interpretations of quantum mechanics out there but would mostly like to see an answer from the perspective of Copenhagen interpretation (or relative quantum mechanics if you wish).
Let an observer being a man with brain consisting of molecules and atoms. According the basic principles of quantum mechanics each of these particles has a wave function. 
The question is: is there a combined wave function of all those particles which constitute the observer? Can such wavefunction be (in theory) determined by the observer himself?
Since the observer cannot isolate himself from his own brain, this would mean that the wave function, at least the part which determines his thoughts is permanently collapsed (i.e. the measurement happens instantly once the state changes). Does this "permanently collapsed" wave function imply special physical properties of the observer's own brain?
Does knowing his own thoughts constitute a measurement? Which moment should be counted as the moment of the collapse of wave function when making measurements on own brain?
Pretend an observer tries to measure the wave function of his own brain by a means of an X-ray apparatus or other machinery and read his own thoughts. Would not his own knowledge of that measurement or its results invalidate the results thus making the whole measurement impossible? 
Does the behavior of particles which constitute the observer's brain differ statistically (acoording his measurements) from the behavior of particles which constitute the brains of other people?
Is there a connection with quantum immortality here?
 A: A measurement is a type of entanglement.  If you measure a photon at the opening of a slit in a double slit experiment then you can think of there being a spin state at this opening.  If the photon passes through it the spin changes direction.  So the wave function for the photon in the double slit experiment is
$$
\psi(x)\rangle~=~C(e^{ikx}|1\rangle~+~e^{ik(x’)}|2\rangle)
$$
where the x and x’ denote the different paths through the two slits. If you compute the modulus squared you get
$$
\langle\psi|\psi\rangle~=~|C|^2(2~+~e^{ik(x’-x)}\langle 1|2\rangle~+~ e^{-ik(x'-x)}\langle 2|1\rangle)
$$
The exponential terms give the interference between the superposed states $|1\rangle$ $|2\rangle$ which have a nonzero overlap $\langle 1|2\rangle~\ne~0$.  We now consider the coupling of a spin to this
$$
\psi(x)\rangle~\rightarrow~C(e^{ikx}|1\rangle|+\rangle~+~e^{ik(x’)}|2\rangle|-\rangle)
$$
which serves as the detector. Since $\langle +|-\rangle~=~0$ if you compute $\langle\psi|\psi\rangle$ the interference term is gone.  This gives a reason for why one can’t measure which slit the particle travels through.  We have replaced a superposition of states type of nonlocality with an entanglement.
Now in doing this we have not addressed the question of how one actually observes the spin.  One might presume we entangle some other states and do so up some “chain.”  This of course leads to the Schrodinger cat problem.  A cat or a human being, our brains and the like are not described well as single quantum systems.  In fact they are messy thermal systems with high entropy.  This is one problem with the whole idea of quantum consciousness, which never got out of the starting gates as a serious physical problem.
A: You might want to look at Tegmark's recent paper at http://arxiv.org/abs/1108.3080 . In it, he proposes a variant interpretation of quantum mechanics based upon a tripartite division of the universe into a system, observer and environment. As in decoherence, a partial trace is taken over the environment, but a relative state is taken over the observer. The observer part is tied in with the anthropic principle cosmologically, and conditioning upon observers, i.e. taking the relative state lowers the entropy of the system.
Another point about observers has been brought up by Zurek. He noticed the fundamental difference between an observer and an apparatus is that an observer as an apparatus with access to its memory states, and memories are records which survive over time. Observers can introspect.
A: Hugh Everett is often misunderstood. His interpretation of quantum mechanics is taken by most physicists today to mean a many worlds interpretation with splittings. His interpretation was actually something else, the relative state interpretation. Most physicists narrowed in on the the part of his thesis where he gave a reduced density matrix analysis, but his real interpretation lies in the concept of a relative state. Decompose the universe into the observer and everything else. The internal states of the observer are entangled with the rest of the universe. Relative to a choice of a classical internal state of the observer, the term in the entanglement expansion given by the tensor product of this internal state with the wave function of the rest of the universe is the relative state.
Let me give you a consistent histories overview for more precision. An Information Gathering and Utilization System, or IGUS in short, is some cybernetic system taking in information from the outside world through its senses, processing them internally, storing some information as memories, and as a result of all the processing, it decides how to act in or manipulate the external world. An IGUS may be an animal, human, robot, computer or whatnot. In a succession of moments in time, the IGUS makes consecutive measurements of its environment and also its internal states through its senses and introspection. This can be modeled as coarse grained measurements in consistent histories. The end result stored in the memory banks of the IGUS is not a simple transcript of all of these measurements but the result of a classical computational processing of these measurements. It is undoubtedly classical because scientists have successfully simulated the brains of worms, or cortical columns of a rat using classical simulations and nothing else. Consistent histories can be adapted so that the choice of successive measurements by the senses is dependent upon the outcomes of earlier measurements, and a further coarse graining of these measurements by combining collections of histories into a further coarse grained history reflecting the actual memory contents of the IGUS after elaborate processing. The wave function of the universe is an entanglement between the memory contents of the IGUS and the rest of the universe. The memories are encoded in synaptic connections, magnetic disks, solid state drives, or what not, with all the attendant coarse grained fuzziness.
Now choose a classical configuration for the memories. The relative state of the universe with respect to this classical configuration is what we want. A particular history. It contains limited information about the universe out there and most of the world out there remains in a macroscopic superposition.
You might ask, why classical memories? Can quantum IGUSes exist? Suppose for the sake of argument that there are quantum memory registers. Because of no cloning, either the register is representing classical information of the external world in some basis, it does not represent anything out there, or it is the incorporation of some quantum information which once existed out there but now only exists inside the register. In the latter case, such quantum information can only be useful with respect to the goal of successful action if it is entangled with something out there. The information must have its coherence preserved right until the decision to determine which action. Because most actions decohere, this has to be an essentially classical command and we can fix the classical information stored in the memories right before it was used to determine the action in the appropriate basis. Unless the action is some communication over a quantum channel. In that case, it's pretty safe to say the IGUS was never conscious of the quantum information passing through it because of no cloning. Unconscious processing.
In string theory, the universe is in a superposition of multiverses, and the relative state picks out multiverses hospitable to IGUSes. The classical memories manifest the right anthropic laws and the right environment in the past.
This is an epistemic interpretation, more or less, with radical epistemological uncertainty.  The central concept is the internal consistency and coherence of the story the contents of the memory tell. What sort of branches in the world out there correspond to the relative state of the classical memories? For very high internal story consistency, a good contribution comes from an external world which more or less match the contents of the story at a coarse grained level. But the contributions coming from being in a realistic computer simulation are also high... For slightly less internal consistency, it could be from something which more or less matches but with inaccurate recall/perception and distortions along the way, or maybe the IGUS had been systematically deceived. For even less consistency, it could be fiction or a not so naturalistic computer simulation or a brain in a vat. For very low consistencies like near randomness, the most likely contribution to the relative state comes from Boltzmann type statistical fluctuations conspiring together. There is no objective reality in quantum mechanics. This is what the math tells us.
How is the IGUS selected if there are many IGUSes? Pool the memory contents of many IGUSes together to form a supra-IGUS. The individual IGUSes are like drops of water and the supra-IGUS the ocean, and the drops of water merge into the collective consciousness of the ocean.
Let me tell you the meaning of consistent histories. Consistent histories are about what an observer external to our universe in some higher plane can measure and record externally in some Akashic records about our universe without being detected from the inside. The selection of framework is now transparently what the external observer chooses to measure. Why should the external observer select quasiclassical histories? Because that tells him a coherent meaningful story. Why should the external observer select the memories of IGUSes? Because that gives him a predigested condensed high information coarse grained meaningful description of the universe.
A: This question was phrased in terms of the Copenhagen Interpretation, although it is tempting to discuss how other Interpretations would deal with it.
The Copenhagen Interpretation (according to the Wikipedia article) does have several formulations and paradoxes connected with it. This example could also be said to stretch the Copenhagen Interpretation. The most similar problem amongst the classics would be the Wigner's Friend argument.
Copenhagen is saying that $\Psi$ is determined only after an experiment, by means of classical instruments by an Observer. So this leaves quite a gap when trying to understand "Brain Function" and "Consciousness". The first question is whether "consciousness" arises from the quantum behavior of the atoms in the brain. Copenhagen does not address this, neither does Quantum Mechanics more generally. Of course Neuroscientists such as Hameroff are interested in this idea. The problem, from a Copenhagen and classical philosophic perspective are similar: "who or what is observing the brain and collapsing the wave function?".
As Copenhagen does not really have an answer to this, Hameroff is more interested in a different type of QM Interpretation. In this interpretation the collapse happens for physical reasons very quickly and frequently. There is no observer to do the collapsing, as Copenhagen would require.
The Decoherence approach also collapses $\Psi$ in a thermodynamic way, so it doesnt have such a direct connection with Observers either.
A variant of this question which is less Neuroscience oriented, is the question of the $\Psi$ for the Universe. If this obeys a Schrodinger-like equation (called Wheeler-deWitt equation) and we assume Copenhagen, then again who does the Observing? Here any Observers are part of the Universe and so there would seem to be a self-referential problem if we take an Observer-centric interpretation like Copenhagen.
A: Paraphrasing from a book I read a long time ago (Quantum Measurement by Braginskii):
It is possible to write down a wave function for a human, but it will not be comprehensible.  It is necessary to define the initial states of all the elementary particles from which the observer is made.  However, the amount of information that the observer is able to comprehend is determined by the number of neurons in the brain.  This number is far smaller than the number of particles that make up his brain (and body), so he cannot possibly comprehend the content of a wave function that describes himself.
A: Think in terms of the anthropic principle. In the universal wave function of the universe, the probability for the existence of observers is exponentially small. The probability is definitely smaller than the fine-tuning in the cosmological constant, $10^{-123}$, most likely much much smaller than that. Anthropic selection effects.
Ask yourself, what is the Shannon entropy of consciousness? What is your naive estimate of the Shannon entropy of human consciousness as encoded within neuronal firing and encoding patterns within the brain? The capacity of working memory is $7 \pm 2$, and the logarithm of the number of possible neural network symbols parseable by working memory is relatively small. Is the product of this with $7\pm 2$ still less than the log of $10^{123}$? If so, statistically speaking, will Boltzmann brain fluctuations predominate over evolved life?
Something isn't quite right here. Should we add short term memory into the contents of consciousness, and not just working memory? Maybe consciousness isn't what we think it is?
Why digital encoding in terms of discrete symbols for working memory? Why not analog encoding? Surely the particular form of the encoding of the symbols within the neural network ought to be irrelevant to consciousness? Or not?
What physical process searches through the haystack of the universal wave function looking for needles looking like the "conscious states" of observers encoded in a messy form of neural networks anyway so that it can attach "consciousness" to those states?
A: I have just found this interesting paper by Thomas Breuer from 1995:
https://homepages.fhv.at/tb/cms/?download=tbPHILSC.pdf
The paper seems to prove that for a system that includes observer himself there are quantum states which are indistinguishable by the observer however technical means he employs, while he can measure any such states in the brains of other people.
The paper claims this proves that not only quantum mecanics is not universal theory that is applicable to all objects in the universe, but that no such universal theory can exist.
It also follows that in the world that the observer himself observes there is hidden information in his brain which cannot be extracted and read by any means even with help of other people (while the same information can be easily extracted from the brains of other people).
I do not know however how to interpret it regarding the wavefunction. Does it mean the observer's wavefunction indeterminate, singular or inexistent?
UPDATE.
And this this paper says it all. There will be subjective decoherence once the observer wants to measure himself. So he will see himself in a mixed state while others in the same situation will be observed as if they were in coherent state.

Note that this position is often taken for quantum mechanics. According to many interpretations, as for example the one of Bohr, or
the one of London and Bauer (1939) and Wigner (1961, 1963), or even
perhaps19 the one of von Neumann (1932), the “true” observer (or
his mind) cannot be described by quantum mechanics. These authors
say that if quantum mechanics is universally valid at all, then it is so
only in the relative sense that every observer can perhaps apply it to
any selected part of the world, except himself. It supposedly applies
to Schroedinger’s cat, Wigner’s friend and Wigner himself under the
condition that they lose their status of observer and are observed by
something or somebody else.

A: The answer of this question hinges in two alternatives:


*

*The brain is a classical computer If this is true, it means that the state of the brain neurons decohere with the environment molecules quicker than they can entangle to each other quantum mechanically. In this case, the environment sees the brain before the brain can see itself, so a classical description is appropiate, because distant parts of the brain will be interacting with mixed neuron states in random (uncorrelated) distributions

*The brain has macroscopic nontrivial quantum entanglement if the decoherence happens slower than whatever quantum state in the neuron firings is able to entangle to each other, then there should be a increasingly non-negligible amplitude that increasingly bigger patches of neurons are entangled, so distant parts of the brain will not see random distributions, but rather distributions with correlation (high or low, all parametrized by the order parameter of the so-called transition, which is described by the ratio of half life of decoherence and the half life of entanglement, under some sensible measuring criteria)
from the above, is not impossible that both can be physically realized; more interestingly, and a field day for new-age philosophers and alike, would be the question if this order parameter can be influenced by the brain state itself; in other words, can i optimize a quantum algorithm to minimize the decoherence happening in the qubits?
A: Look in the mirror and you will see your wave functions squared ;-).
No, I am joking. In the mirror you see an inclusive picture, not elastic one.
The observer himself consist of so many degrees of freedom with so tiny differences between energetic levels that it is impossible to keep it in a pure state. There are permanent transitions leading to decoherence.
A: The Observer is the Observed.
An experiment is performed upon a system. The measuring apparatus is the observer of the system. But what makes the pointer of the apparatus real? The apparatus is observed by a human brain and becomes a perception encoded within its pattern of neural firings and connections. But who observes the brain? The brain observes itself! The perception is stored in short term memory so that some time later, the brain can recall it. But what happens at the end of the day? The short term memory is encoded within long term memory which is made real when recalled by the same brain later, and this will happen over and over again. The brain observes itself observing itself observing itself... except that this memory is imprisoned within the mortal and perishable brain and yearns to break free. The human observer can communicate the observations to other people in writing or in speech. Then, the other people becomes the observer of the original human observer. But who observes the thoughts of the other people? If a brain implant is installed in their brains, the implant will observe their thoughts. But who will observe the implants? The memories in the implants can be uploaded to an external computer, making the computer observe the implant. But who will observe the states of the computer? An artificial intelligence program will observe it. But computers are finite and perishable, and who will observe the artificial intelligence? An even larger computer or network of computers in its future after uploading the memories. But the Earth is doomed to destruction, and who will observe the network of computers on the Earth? von Neumann probes sent out to outer space will. This process will continue forever and ever until the Ultimate Observer, which is the asymptotic limit of all these chain of observers, observes it, and this Ultimate Observer is GOD.
GOD brings the entire universe into existence, manifesting one of the potentialities of the quantum superposition into the actuality of measured existence. GOD is the telos and Final Cause, the Omega and the end. He exists outside of time in the endtimes.
