We have a relation, $E=h\nu$, where $\nu$ is the frequency and $E$ is the energy of any particle.
If we have a function, $\psi$, that is an eigenstate of energy, the particle has a definite energy, and frequency. This means that it is a regular wave with no superposition.
If we increase $\nu$ we get a higher energy, but is there any physical correspondence to this? If it is even possible, or am I thinking of this whole thing wrong?
Yes, you can increase the energy of a particle, which will increase its frequency. For example, you could accelerate an electron by passing it through the plates of a capacitor with an applied voltage.
In quantum mechanics, increasing the energy of a particle has pretty much the same effect you would expect in classical mechanics. The particle has more "oomph" if it has more energy. As a concrete example, a photon with less than $13.6\ {\rm eV}$ of energy will not be able to ionize a Hydrogen atom, while a photon with an energy above $13.6\ {\rm eV}$ does have enough energy to be able to do that.
Not quite sure what you're asking, but here's an idea.
If you have an energy eigenstate, that suggests your particle is confined. If you shrink the "box" it's in, that increases 𝜈 and raises the energy of the eigenstate.
I'm not sure this is the right way to go about it. A wavefunction of well-defined energy can still be written as a superposition, in any basis that isn't made of energy eigenstates.
For example, take the nitrogen atom in ammonia. It has two "position" eigenstates (on each side of the hydrogens), each one being a superposition of two energy eigenstates. You can reverse the description and say that both energy eigenstates are superpositions of "position" eigenstates.
Due to the Planck-Einstien relation that you mentioned, changing the energy will change the frequency and vice versa.
A lot of processes can do that. Due to temperature, atoms can routinely oscillate beween several energy levels. Or you can direct a well-chosen radiation toward a gas to ionize it (electrons will climb up the energy levels and get out), and so on.
1: Boost yourself into a frame that measures the desired $\nu$ by having a velocity parallel to or antiparallel to the particle's trajectory.
2: With the plane of the reflector at right angles to the trajectory, boost a reflector into the frame that measures the desired $\nu$. The reflection will be doppler shifted to the desired $\nu$ in the un-boosted observer frame. This is the mirror doing work on the particle (mirror trajectory and beam trajectory are antiparallel), or the particle doing work on the mirror (trajectories are parallel).
3: Do work with an externally applied force field to increase the energy, as accelerating an electron with an electric field. The energy changes equal to the work done by the force field.
4: Take a frame that measures a pseudoforce at points along the trajectory of the particle. The energy changes equal to the work done by the pseudoforce. (If the psuedoforce is gravity and the particle is a photon, this is gravitational redshift).
