Classical physics is well known because of works of Newton in “Mechanics” and famous Maxwell equations of “electromagnetism” and also Kepler’s laws of “Planetary Motion”. These works all had done before 20th century, starting by Kepler’s famous laws in 17th century. There is a stage between classical physics and modern physics which is not completely belong neither to classical physics and nor to modern physics. This stage is “Statistical Mechanics” and represents new definition for temperature and rearranges the “Thermodynamics Laws” by statistical point of view. These works mainly had done by “Boltzmann” and Maxwell and Others.
The modern physics is started after discovering of atoms and studying its behaviors. If we want to subdivide the modern physics in just two section, we should name “Quantum Mechanics” and “Relativity”.
Quantum mechanics started by works of Bohr and Max Plank and continued by works of De Broglie, Schrodinger, Pauli, Dirac and others. At the time of max plank, it was discovered by Maxwell that visible light is also an electromagnetic wave. Max plank hypothesis is that every electromagnetic wave has an energy which is related to its frequency by a constant (E=hf), which “f” is frequency of wave and “h” is plank constant.
Bohr considered that electrons which circulate around nucleus, only can have some certain radius (Bohr famous formula for radius of electron) and each radius has an energy level. This hypothesis was somehow start point of thinking quantum. This says that radius of electron is not continuous function and is quantized to some energy level, but Bohr didn’t use any quantum mechanics concepts and formulations. The official start of Quantum Mechanics is by considering electrons and protons and other particles as waves and its formulation had done by “Erwin Schrodinger”. Schrodinger equation is almost most important equation in quantum mechanics and should be solved for different situation. By studying the Hydrogen Atom model and solving Schrodinger equation for hydrogen, we get a lot of ideas of behaviors of atoms, but we should note that solving Schrodinger for atoms with more than one electron, is not easy at all. Considering momentum (linear and angular) conservation laws and considering spins of electrons, we will find 4 quantum number for each particle (this time electrons). Pauli rule say that no two particles (fermion and this time electron), can have same quantum number. We should mention that all fundamental particles are “fermions” or “bosons”. Fermions are that kind of particles that can have fractional numbers for their spins. For example spin of electron is ½. Bosons can have only integer numbers (0, 1, 2, …) for their spins. For example spin of photon is 1.
The next revolutionary idea of modern physics is relativity. It discovered by Albert Einstein that the speed of light is constant in all condition and in non-inertial frames (moving frame with zero acceleration). This is special relativity and in macroscopic variables of real life, like Newtonian Mechanics, it cannot be seen easily. But when we approach to speed of light, like movement of electrons and etc, we should revise all of equations of Newtonian Mechanics by relativistic ones. Soon after discovery of special relativity, Einstein discovered general rule of relativity which is a revised equation for gravitational forces between masses.
There was a challenge among some experiments in early 20th century. As quantum mechanics supposes, every particle moves like a wave and has a wave equation of motion. But some phenomenon like “Photoelectric effect”, showed different properties of light as an electromagnetic wave. Photoelectric effect showed that when we bombard a metal by light wave, we cannot scatter electron from metal by increasing the intensity of light. If light is a wave, it is expected by increasing its intensity, we must can emit electrons from atom. But experiment showed that electrons emitted only by electromagnetic waves (or light) with a frequency more than a threshold frequency. Einstein supposed that light is formed from a particle called “photon” in motions of a wave. Einstein received Nobel Prize in 1921 for discovering Dual behavior (particle-wave behavior) of light (and in general electromagnetic wave), which is explaining the photoelectric effect.
If you are an engineering student, you are only familiar with classical physics, but modern physics is taught especially for physics major (and physics-related majors like Nano-Engineering). What I mentioned in this article is classical courses of Bachelor of Physics. There are still some unsolved problems in physics as you move forward to graduate studies, you will learn more about them.