I am studying QM-I these days. Now, I just think of the wave function as just a mathematical function that defines the state of the particle at an instant and from it you can extract various observables. I have been told that it is just a misnomer calling it an actual wave, and there is no wave there, its just that the wave function satisfies the general wave differential equation in 3d space. Now is this true? Is there really no wave? What about QFT, when we say electron is an excitation in a Quantum Field, do we mean to say, it is sort of a pulse in that field medium? Like a photon is a pulse in an EM wave and essentially electron is just a pulse, which we see as a particle? Or is there no wave there ? Is this related to $E=\gamma mc^2$ by relating energy in a wave to a particle.
Except in very elementary examples (single particles), the QM wave function has nothing to do with a wave (apart from the historical origin).
For a system consisting of $N>1$ particles, the wave function is a function in configuration space (with 3N variables), not one in 3-space (whose coordinates are positions $x$ with 3 components). This can be read in any textbook on quantum mechanics. Whatever oscillates in configuration space has (therefore) little to do with oscillations of waves in space and time.
In quantum field theory, one has true waves, which are oscillations of expectation values of field observables or their products. But these have nothing to do with wave functions either. Indeed, the analogue of a QM wave function in QFT is a wave functional, which are functions $\psi(\Phi)$ depending not on space position $x$ and time $t$ (as the wave function of a single particle) but on all fields $\Phi$ (which themselves depend on $x$ and $t$). These wave functionals are not easy to work with, so you don't find them even mentioned in introductions to QFT. A recent (but not elementary) reference is Phys. Rev. D 77.085007 (2008). A much older, but more readable reference is Phys. Rev. D 37 (1988), 3557-3581.
When we say that an electron is an elementary excitation in a quantum field, it has a similar meaning as when we say that a sine wave is an elementary excitation of a string. The spectrum of an ideal string consists of a ground frequency and its overtones - integral multiples of the frequencies. The spectrum of a real string (or plate, etc.) is more complicated but the ground frequency is still a well-defined mode.
The spectrum of the quantum field in its rest frame consists of a discrete spectrum and a continuous spectrum. The reason for a continuous spectrum is that multiple excitations may move in different directions (though the rest frame stays fixed), and their kinetic energy adds a continuous amount to the frequency $\nu$, according to the formulas $E=h\nu$ and $E=E_0+mv^2/2$ (for a nonrelativistic motion with rest energy E_0 and speed v). The discrete part of the spectrum is made up of modes which are called particles - by tradition, although their properties don't resemble much those of little balls.
Single modes of quantum fields fields are described by classical wave equations with space-time arguments, and hence have a natural (and classically describable) interpretation in terms of waves. These classical wave equations are quantized to get the quantum fields. But a mode of a quantum field is something very different from a wave functional: A wave functional is a superposition of tensor products of an arbirary number of modes; the modes are just their building blocks.
I have been told that it is just a misnomer calling it an actual wave, and there is no wave there, its just that the wave function satisfies the general wave differential equation in 3d space. Now is this true? Is there really no wave?
The confusion comes from extending our intuitive view of classical waves, as in water, and sound and light, to the microcosm of quantum mechanics.
We have found mathematical equations that we call "wave" equations, and their solutions end up with sines and cosines describing the motion of energy either in a medium, as water and sound waves, or through space by propagation of energy carried by electromagnetic fields in electromagnetic radiation. The key word is energy propagation.
In the microcosm, we have found equations that are wave equations, i.e. their solutions have sine and cosine behaviors as with classical waves, but there is no energy transferred with these waves. We use the solutions of these equations with the postulate that the square of the solution ( amplitude) gives the measure of probability of finding a specific value when one measures the observable. Nothing to do with energy propagation per se. The individual particle can be described with a wavefunction, and it will depend on the boundary conditions of the problem ( one slit, two slits for example) whether we will see an interference pattern . In a sequential experiment with single electrons through a double slit
single electrons at a time
the interference that gives a wave pattern appears after many electrons have been shot at the slits. So it is a wave in space, but not in energy, and not in the particle itself, which appears whole when measured by the hit at the screen. It is a probability wave.
What about QFT, when we say electron is an excitation in a Quantum Field, do we mean to say, it is sort of a pulse in that field medium?
The probability postulate holds through all the quantum mechanical formulations
Like a photon is a pulse in an EM wave
No , a photon is not a pulse, it is an elementary particle with zero mass, and similar double slit experiments hold for it too. It has energy given by E=h*nu. How the classical electromagnetic field emerges from the ensemble of photons constituting it is not a simple correspondence and needs extra maths.
and essentially electron is just a pulse, which we see as a particle?
The electron is an elementary particle too. Elementary particles display billiard ball particle properties at times, and probability-wave properties at other times, depending on the boundary conditions for the observation.
Is this related to E=γmc2 by relating energy in a wave to a particle.
Well, everything in physics is correlated or emerges from a lower framework, but the relationship is not quantum mechanical. The Schrodinger equation works well for non relativistic energies and special relativity had to be incorporated in different equations for high energies where it holds. As I said the wave nature of a particle is not in its energy field but in its correlation with the boundary conditions imposed by the observation, and it is a probability wave seen after many repetitions of the observation.
You are right, there is no actual wave that the, say, electron turns into whenever it chooses. The only property that forms wave-like images is the probability density function which is given by $$|\Psi ( x , t) |^2$$
This behaviour of the p.d.f. can be plotted as it resembles what we call a wave. Here are some examples from 1D, 2D and 3D potential wells which contain a particle:
In the case of a single particle the wave function can be interpreted as a real wave. This is the De Broglie pilot wave interpretation of QM. This interpretation can provide interesting views on the double slit experiment, but it breaks down for more complex systems. When considering multi-particle systems it is clear that the wave is in configuration space rather than physical space. It is just a coincidence that the configuration space can be identified with physical space in the double slit experiment since both spaces are 3 dimensional in that case. However, the pilot wave is still central in the Bohm interpretation of QM, even though the wave is no longer in physical space. The Bohm interpretation has various extensions including applications to QFT. There is a thesis by Struyve on more recent developments in this area. There is also number of books on the Bohm interpretation and its extensions.
The Bohm interpretation has never really caught on. To the best of my knowledge it is really just an interpretation in that it is equivalent to standard QM in all experimental predictions at least for pure QM systems, but there seems to be some dispute on this.
It is a real wave associated with electron. And we adopt its mathematical description using wave function in configuration space