First I want so say that I'm new to quantum physics and have basic knowledge about waves .
The question : When the lecturer wanted to describe a wave of photons he always described and drew it as a single wave but when we we began to learn about wave nature of electrons he described the wave of electrons as a packet of waves . So my question is why describing the wave of photons and electrons is different and why aren't electrons wave described by a single wave?
Thank you.
In the initial stage of learning a new subject, it is normal to be puzzled by many different concepts. This situation is even worse in quantum physics, particularly when we deal with photons.
In many sources, from popular science to introductory quantum electrodynamics (QED) textbooks, a photon is introduced as a "particle of electromagnetic radiation," "a quantum of radiation," characterized by a well definite frequency and momentum. This way of introducing it may be the origin of the idea that when we associate a wave to the photon, this wave should be monochromatic and with a unique wavevector, i.e., a plane wave.
Things are quite different for at least four independent reasons. They may be summarized with the following four statements:
In the following, I'll try to provide some more details based on contemporary physics.
Plane waves cannot exist in nature
This is an observation valid for all waves, also classical waves like sound waves. A plane wave would have an infinite extent. Considering that a wave carries energy and the energy density is associated with the intensity of the wave (i.e., to the square of the amplitude), a plane wave would carry infinite energy. It would have no start and no end—definitely an unphysical object. Still, plane waves are handy conceptual tools, especially for linear waves. The Fourier analysis shows that every real, finitely extended perturbation may be described as an infinite sum or integral of plane waves.
A photon (and also an electron) is not a wave
Single-photon or single-electron diffraction experiments show that the famous (or infamous) duality between particle and waves has to be intended in a way less trivial than saying that quantum particles are at the same time particles and waves. Nobody ever measured $1/3$ electron or $1/10$ of the photon spin. Some physical properties are always quantized as if particles carried them around. However, the dynamics of those particles has an evident wave-like behavior. Diffraction figures are built by a huge number of single-particle spots. They do not come by the superposition of single-particle diffraction figures.
Quantum field theory (QFT) provides a unified treatment for quantum particles. Within QFT, it is possible to show that a quantum field's behavior provides a faithful description of the experimental evidence.
A convenient (but not unique) way to describe a quantum field's state is by using a kind of Fourier analysis in terms of excitations having a well definite momentum and energy. In the classical limit, it is possible to interpret the energy as a frequency and the momentum as a wavenumber. Notice, however, that the QFT description does not explicitly require a wavefunction. Therefore, a statement like "there is one photon (or one electron) of frequency $\omega$ and wavevector $\bf k$ should not be intended as saying that there is a plane wave $e^{i({\bf k}\cdot{\bf r} - \omega t)}$. The two things are decoupled.
When can we say that there is a three-dimensional wave? When we have many bosons (like the photons) and combined in a so-called coherent state. In such a case, the behavior of this superposition of quanta behaves like a classical wave. If the momenta involved in the coherent state are clustered around $\bf p$ and energies are all equal to $E$, the resulting wavepacket will have a wavevector close to ${\bf p}/\hbar$ and a frequency equal to $E/\hbar$. In summary, the wave is the space-time description of the collective behavior of this coherent state.
Wavefunction for electrons and photons
In addition to the above-mentioned difficulty in identifying a particle with a wave, there is an additional difficulty with the idea of assigning a plane-wave to a photon. At the level of ordinary quantum mechanics, we can associate a quantum wavefunction to an electron in the sense that Schrödinger's description of the dynamics of a single electron is based on a three-dimensional wave-like probability amplitude. The same procedure requires some care when particles are massless, and the fully relativistic regime is the only possible description. Some important points related to such an issue are in this post, although there is more on this site. It is enough to search with the keyword "Photon wavefunction."
Does a photon have a unique frequency?
A final objection against the idea of associating a monochromatic plane wave to a photon comes from the fact that from the technical point of view, a photon is an eigenstate of the so-called Number operator with eigenvalue equal to one.
It turns out that we can obtain such a state as a superposition of many states with the same $\omega$ and different wavenumbers. Still, we can equally build one-particle states as a superposition of state at different energies/frequencies. Although often ignored in the introductory textbooks of QED, such polychromatic photons exist and are used in quantum Optics.
Where to find an expanded description of these comments in a book on *Electrons and Photons from A to Z"?
Unfortunately, I am not aware of a book at the undergraduate level. In advanced textbooks on Quantum Optics, you may find most of what I have written, although at a more technical level.
