Is an electron in an hydrogen atom being measured by the nucleus? In an hydrogen atom, the electron interacts with the nucleus by multiple forces, for example the Coulomb force. Does that mean that the nucleus makes quantum measurements of the electron?
EDIT: I became aware that the word ‘measurement’ is not present in all theories of quantum mechanics. Therefore this question is ill-defined. Instead I would like to know whether the wave-function of the electron is influenced by the interactions with the proton, in a way that is not part of the Schroedinger equation.
 A: No. Effects of interaction are described by the Hamiltonian. Quantum measurements are not, they require some "projection", or "collapse" or "choice by hand".
A: The answer depends on what interpretation of quantum mechanics you subscribe to. Different interpretations have different definitions of 'measurement'.
In most of the 'wavefunction collapse' interpretations (Copenhagen interpretation), the answer is 'no'. An 'observation' requires some extra criterion to be met separating the quantum and classical domains - that the observer is 'big enough' in some way. Mass, spatial extent, number of internal states, the cumulative effects of some slight non-linearity in the equations exceeds some threshold, or (not scientifically favoured, but still popular) the consciousness or intelligence of the observer system. Proposals are speculative, and experimentally unobservable.
In the Everett (Many Worlds) interpretation, then yes, the interaction could potentially be considered to be a measurement. For example, (and oversimplifying the physics heavily,) the uniform motion of the centre of mass of the electron-proton system in a Hydrogen atom means their positions are correlated. Joint states in which the electron is offset one way have the proton offset the other way. The proton's offset from the centre of mass can be considered 'knowledge' of the electron's position. The atom remains in a superposition of states with the electron offset in every direction - from the outside point of view, the situation remains uncertain. But the proton's point of view may be broken down into a superposition of orthogonal (and hence mutually invisible and non-interacting) states, in each of which the proton 'observes' a different position of the electron.
If a wavefunction for a single particle is spread out over a region of space, then the different bits of the wavefunction do not interact with one another. They cannot 'see' one another. When an electron passes through both slits of the 'double slit' experiment, the electron passing through one slit does not 'see' itself passing through the other. There is, for example, no electromagnetic repulsion between the charges, affecting the interference pattern. This is a simple consequence of the linearity of the equations.
It is as if each possibility happened in its own separate world. The worlds can only interact with each other by interference, which manifests itself only in the apparent probability of events happening. Other interactions between different orthogonal parts of a wavefunction, like electrostatic repulsion, are forbidden. So it appears to each component as if all the other options have disappeared, and one outcome has been selected at random. However, all the options are still there, and to any outside system that is not correlated with the internal state of the atom (like us humans), the wavefunction still appears to be uncollapsed, as if no measurement had taken place.
The different interpretations of quantum mechanics all make exactly the same predictions about what we will observe in any given situation, and so are experimentally indistinguishable. It is therefore not a scientific question. The Everett interpretation has simpler rules (no mysterious collapse triggered by who knows what, or faster-than-light backwards-in-time paradoxes), but a far more complicated state (billions of unobservable alternate realities). Which picture you prefer is up to you. The idea of interactions and measurements being the same comes from the Everett interpretation and its close relatives. Collapse interpretations treat them as entirely different sorts of events, although they don't agree among themselves on what the distinction is.
A: 
I would like to know whether the wave-function of the electron is
influenced by the interactions with the proton, in a way that is not
part of the Schroedinger equation.

In all interpretations without wavefunction collapse the wavefunction can be influenced only by Schroedinger's equation. Only two things affect the wavefunction: Schroedinger's equation and collapse (in those interpretations where it exists). Note though that a prolonged in time process similar to wavefunction collapse is often called "decoherence". It can be understood as "smooth collapse", as the system interacts with multiple low-energy quanta which measure only part of the information about the system's wave function and interact not with the detector directly but with environment. Under decoherence, the wavefunction reduces gradually.
In Copenhagen interpretation you definitely can use a proton to perform measurement of the position of electron (would be difficult as proton is more massive, but nevertheless). This will make the wavefunction of electron to collapse.
In atom the proton's fields, electromagnetic, and theoretically, gravitational (but to a much lesser degree) affect the form of electron's wave function, but this is described by Schroedinger's equation.
