Superpositions of quantum states 
The description of the microscopic world of atoms made by quantum physics is the greatest conceptual revolution of humanity. The laws of quantum mechanics, never challenged, have had enormous repercussions on the progress made over the past 80 years. It is thanks to these laws that we basically explain "everything that exists" in the universe, from the Big Bang to biological life (properties of particles, behavior of atoms and molecules, stability of matter, superconductivity, superfluidity, physicochemical processes, nuclear energy, radioactivity, electrical conductivity, magnetism, tunnel effect, laser, photosynthesis, etc.). The concepts of quantum physics (waveparticle duality, superposition of states, nonlocality, entanglement, measurement uncertainty, quantum teleportation, etc.) have been accepted by all physicists since the 1930s. But among these concepts, the principle of superposition of states is the fundamental principle of quantum physics. Behind this concept hides a disturbing property that allows a quantum system to be in several states at the same time. For example, the same atom can be simultaneously at two places x1 and x2 separated by a macroscopic distance, until a measurement is made to determine its position. The principle of superposition of states preexists in all exchanges of matter with light and this notion characterizes all particles. It is with it that the quantum revolution begins. In quantum physics, all microscopic objects (electron, photon, proton, etc.) can sometimes present properties of waves and sometimes properties of corpuscles. We are used to saying (wrongly) that quantum particles are both waves and corpuscles.   Although quantum particles are neither waves nor corpuscles, they end up causing a phenomenon of interference characteristic of waves as can be seen in the Young's slit experiment. Indeed, when the particles are viewed one by one on a detection screen, their impacts interfere like a wave. This interference phenomenon is the trace of superpositions of waves of the same frequency. It is for this reason that particles are better described by their undulatory rather than corpuscular character. This description provides the full state of the particle. The state of a particle describes all the aspects of this particle, i.e. all the properties, common or not, (mass, charge, speed, angular momentum, direction of the spin, position, energy, etc.), which we can obtain on the particle if we make experimental measurements on it. This state varies from one electron to another. However, the principle of superposition of quantum states obliges us to describe all these states in a probabilistic way. For example, before the measurement operation, the position of an electron free of any environment is given by a wave function Ψ(r,t) calculated using the Schrödinger equation. If its position, probabilistic before the measurement, becomes deterministic after the measurement, it is because the environment (screen, wall, observer or even air molecules) forces it to interact. In other words, the electron or rather its energy is suddenly reduced to a point which can then be located because it has taken on the appearance of a corpuscle. We then measure on the screen points of impact and not interference.   Image: Interference concerns all quantum particles. Interferences have been observed with electrons, photons, neutrons, atoms and even molecules. These interference phenomena, typical of waves, are all observed experimentally in the microscopic world but disappear in the classical macroscopic world. Two waves of the same type of propagation (mechanical or electromagnetic) are capable of being added together. It is for this reason that we observe interference. As soon as the particle arrives in the classical world, its energy is suddenly reduced to a point on the screen which can then be located because it has taken on the appearance of a corpuscle. It is after the reduction of the wave packet that one sees on the screen the points of impact of the particles sent one by one on the Young slits. 
Summary:  The particles are better described by their undulatory character than corpuscular.  The states of a particle can be added to each other. If a and b are two possible states of the system then (a+b) is also a possible state of the system. If a is a possible state of the system then λa is also a possible state of the system (λ=number). This is the principle of state superposition.  The principle of superposition of states preexists in all exchanges of matter with light and this notion characterizes all particles.  If we want to determine the state of a quantum system, we must observe it, but this observation has the effect of reducing the state in question. Before the measurement operation, the quantum properties of a particle have no physical meaning.  The particle is a wave of probability of presence as Max Born (18821970) said in 1927. We cannot know if the particle is at a precise place in space, but we can know the probability that it is there after the reduction of the wave packet.    Everything that can be known about particles can be extracted from their wave function. Erwin Schrödinger proposed an equation to find the wave function and the energy of any quantum system. This equation makes it possible, for example, to precisely calculate the spectrum of the hydrogen atom. That is to say, all the wavelengths of the absorption and emission lines its spectrum.  Entities capable of being added together are represented by vectors. The physical states of the particles are represented by state vectors which generalize the wave function. The physical state of a system is represented in an abstract mathematical vector space (Hilbert space) very different from the space of classical physics.  In quantum physics the observer and the observed object are linked (Copenhagen interpretation).  
Energy levels of the electron in the atom H 



n=1 
Fundamental level, lowest energy 
E1=13.6/1^{2} =13.6 eV 
n=2 
First excited level 
E2=13.6/2^{2} =3.4 eV 
n=3 
Second excited level 
E3=13.6/3^{2} =1.51 eV 
n=4 
Third excited level 
E4=13.6/4^{2} =0.85 eV 
n=5 
Fourth excited level 
E5=13.6/5^{2} =0.54 eV 
n=6 
Fifth excited level 
E6=13.6/5^{2} =0.38 eV 
Image: Schrödinger's equation devised in 1925 is a fundamental equation in nonrelativistic quantum mechanics. If we solve the equation, we know everything about the system. H(t)Ψ(t)> = iℏ d/dt Ψ(t)> allows for example to calculate precisely all the values of the energy that a hydrogen atom can take. 