Several concepts are tangled up in your question.
The electrons in their orbitals are a stable solution to a potential problem, the potential supplied by the charges in the problem. The orbitals describe the probability of finding the electron in a position (x,y,z) around the nucleus.
How can one see the electron orbital? In this link you can see that it is not a simple matter, but it has been done, by disturbing the electron with photons and getting statistically the original location.
Once one has seen/measured the electron's location the solution of the potential no longer holds for this free electron. The probability of finding it can be described by a traveling plane wave, until it is caught by another potential and radiates down to a ground state or an energy level that was empty in an ionized atom.
Actually atoms get incident by electromagnetic radiation at every instant, if we assume electrons to get superposed by electromagnetic wave, electron can't exist as stationary wave, but according to schrodinger model, electron is said to exist as stationary wave.
A photon hitting a bound ( probability standing wave) electron can expel it from the atom . A photon of appropriate energy can kick it up to a higher energy level , the photon disappears and the electron falls back to ground state by emitting that special spectral line. It is true that atoms, and this means the electron cloud about the nucleus, are continuously hit and interacting with photons, but the grand majority of photons do not have enough energy even to ionise the atoms, let alone free the electron.
The photons in our environment are bounded in energy to first order by the black body radiation of the earth and by the spectrum from the sun which supplies the energy to the surface of the earth. The high tail from the sun which can ionize atoms and thus is dangerous is mostly cut off by the atmosphere, our shelters and clothes.
So the only two outcomes on atoms from the "superposition" you imagine are either change of energy level and disappearance of photon, or ionization of atom and change in energy of photon. This last does destroy the standing wave probability function of the electron to a plane wave one.
A free electron hit by a photon can either scatter elastically or inelastically, but a free electron is no longer described by a probability standing wave, just by a plane wave propagating ( again , a probability wave), which is also a solution of Schrodinger's equation in the absence of a potential.
Electron as a standing wave
Yes, the electron is a standing wave. See atomic orbitals on Wikipedia: "The electrons do not orbit the nucleus in the sense of a planet orbiting the sun, but instead exist as standing waves".
I couldn't understand how come Bohr who interpreted electron as a particle, formulated an equation for electron's angular momentum which shows its mathematical proof to be a wave.
Maybe you need to check out De Broglie and matter waves: "All matter can exhibit wave-like behaviour. For example a beam of electrons can be diffracted just like a beam of light or a water wave". See this picture by artist Kenneth Snelson:
It isn't a totally accurate depiction. Electrons aren't actually thin coloured strips, but you should get the idea of these standing waves.
Simply when one compares first harmonic of wave on a string with electron moving around nucleus, from where the nodes shown in the figure arise in case of orbiting electron?
It's bit like a wave in a closed string. But it isn't a wave on a string, it's an electromagnetic wave that's configured as a standing wave. A field variation that's configured as a standing field. It has a Compton wavelength of 2.426 x 10⁻¹² m.
The electrons can only orbit stably, without radiating
Don't think of the electron as some little billiard-ball thing. Think of it as something more like a hula hoop.
So as we move towards some nth harmonic, the trajectory becomes complicated. Is it that case?
Yes. Check out spherical harmonics.
Also that how s,p,d, and f orbitals (May be, the way in which electron, a wave, moves around nucleus as a function of time taken?) do exist without interfering each other? I mean, an atom is so small and intact.. So, don't they mix up i.e., superpose?
Superposition is a wave thing. Two ocean waves can ride right over one another and then keep going. But for electrons in orbitals, like Acid Jazz said, the Pauli Exclusion principle applies. The simplest analogy I can think of for that is two whirlpools can't overlap.
Best Answer
An animation is worth a million words: