The nquantum numbers are used to describe the quantum state of the electrons in the atom and originate from the solution of the Schrödinger equation for the simplest of all: hydrogen.
The Schrödinger equation is a differential equation, the solutions of which are wave functions and are denoted by the Greek letter ψ. Infinite solutions can be proposed, and their square is equal to the probability of finding the electron in a small region of space, called orbital.
Each orbital has definite characteristics that distinguish it from the others, such as energy, angular momentum and spin, an entirely quantum property of the electron and which is responsible, among other things, for magnetic effects..
The way to identify each orbital is to distinguish it by a set of numbers that describe it, and these are precisely the quantum numbers:
-n: is the principal quantum number.
-ℓ: the azimuthal quantum number.
-mℓ, is the magnetic number.
-ms, the spin number.
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Quantum numbers are used to describe the state of the electrons inside the atom. That atomic model in which the electron goes around the nucleus is inaccurate, because it is not consistent with atomic stability or with a large number of observed physical phenomena..
That is why the Danish Niels Bohr (1885-1962) made an audacious proposal in 1913: the electron can only be found in certain stable orbits, the size of which depends on an integer called n.
Later, in 1925, the also Austrian physicist Erwin Schrödinger (1887-1961) proposed a differential equation in partial derivatives, whose solutions describe the hydrogen atom. They are the wave functions ψ mentioned at the beginning.
This differential equation includes the three spatial coordinates plus time, but when this is not included, the solution of the Schrödinger equation is analogous to that of a standing wave (a wave that propagates between certain limits).
The time-independent Schrödinger equation is solved in spherical coordinates and the solution is written as the product of three functions, one for each spatial variable. In this coordinate system, instead of using the coordinates of the Cartesian axes x, Y Y z the coordinates are used r, θ Y φ. In this way:
ψ (r, θ, φ) = R (r) ⋅f (θ) ⋅g (φ)
The wave function is intangible, however quantum mechanics tells us that the squared amplitude:
| ψ (r, θ, φ) |two
That is, the module or absolute value of the wave function, squared, is a real number that represents the probability of finding the electron, in a certain region around the point whose coordinates are r, θ Y φ.
And this fact is something more concrete and tangible.
To find the wave function, you have to solve three ordinary differential equations, one for each variable r, θ Y φ.
The solutions of each equation, which will be the functions R (r), f (θ) and g (φ), contain the first three quantum numbers mentioned.
We briefly describe the nature of each quantum number below. The first three, as previously stated, arise from the solutions of the Schrödinger equation.
The fourth issue was added by Paul Dirac (1902 - 1984) in 1928.
It is denoted by n and indicates the size of the allowed orbital, as well as the energy of the electron. The higher its value, the further the electron is from the nucleus and its energy will be higher as well, but in return it reduces its stability.
This number arises from the function R (r), which is the probability of finding the electron at a certain distance r of the nucleus, which is determined by:
-Planck's constant: h = 6.626 × 10 -3. 4 J.s
-Electron mass mand = 9.1 × 10-31 kg
-Electron charge: e = 1.6 × 10-19 C.
-Electrostatic constant: k = 9 × 10 9 N.mtwo/ Ctwo
When n = 1 corresponds to the Bohr radius which is approximately 5.3 × 10−11 m.
Except for the first layer, the others are subdivided into sub-layers or sublevels. Each shell has an energy in electron volt given by:
In theory there is no upper limit for n, but in practice it is observed that it only reaches n = 8. The lowest possible energy corresponds to n = 1 and is that of the fundamental state.
Denoted by the italic letter ℓ, this number determines the shape of the orbitals, by quantifying the magnitude of the orbital angular momentum of the electron.
It can take integer and positive values between 0 and n-1, for example:
-When n = 1, then ℓ = 0 and there is only one sublevel.
-If n = 2, then ℓ can be equal to 0 or 1, so we have two sublevels.
-And if n = 3, then ℓ assumes the values 0, 1 and 2 and there are 3 sublevels.
It can be continued in this way indefinitely, although as said before, in practice n goes up to 8. The sublevels are denoted by the letters: s, p, d, F Y g and they are increasing in energy.
This number decides the orientation of the orbital in space and its value depends on that of ℓ.
For a given ℓ, there are (2ℓ + 1) integer values of m ℓ, that correspond to the respective orbitals. These are:
-ℓ, (- ℓ + 1),… 0,… (+ ℓ -1), + ℓ.
If n = 2, we know that ℓ = 0 and ℓ = 1, then m ℓ takes the following values:
-For ℓ = 0: m ℓ = 0.
-For ℓ = 1: m ℓ = -1, m ℓ = 0, m ℓ = +1
The n = 2 orbital has two sublevels, the first with n = 2, ℓ = 0 and m ℓ = 0. Then we have the second sublevel: n = 2, ℓ = 1, with 3 orbitals:
The three orbitals have the same energy but different spatial orientation.
When solving the Schrödinger equation in three dimensions, the numbers already described emerge. However, in hydrogen an even finer structure is observed that these numbers are not enough to explain.
For this reason, in 1921 another physicist, Wolfgang Pauli, proposed the existence of a fourth number: the spin number ms, which takes values of + ½ or -½.
This number describes a very important property of the electron, which is the spin, word that comes from english spin (to turn on itself). And the spin in turn is related to the magnetic properties of the atom.
One way to understand spin is by imagining that the electron behaves like a tiny magnetic dipole (a magnet with north and south poles), thanks to a rotation around its own axis. The rotation can be in the same direction as the needles of the clock, or in the opposite direction.
Although Pauli suggested the existence of this number, the results of an experiment carried out by Otto Stern and Walter Gerlach in 1922 had already anticipated it..
These scientists managed to divide a beam of silver atoms in two by applying a non-uniform magnetic field.
The value of ms does not depend on n, ℓ and m ℓ. In graphical form, it is represented by an arrow: an up arrow indicates a clockwise turn and a downward arrow indicates a counterclockwise turn..
The behavior of the electrons in the atom is summarized in the Pauli exclusion principle, which states that two electrons in an atom cannot exist in the same quantum state.
Therefore each electron must have a different set of quantum numbers n, ℓ, m ℓ and ms.
The importance of quantum numbers and this principle lies in the understanding of the properties of the elements in the periodic table: electrons are organized in layers according to n, and then in sub-layers according to ℓ and the rest of the numbers.
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