Quantum box

In Physical, in Mechanical quantum, in the Model of Bohr, the quantum boxes is the places in the atomic orbital which can be occupied by an electron, or a pair of complementary electrons of spin.

Quantum numbers

The electronic Nuage of each Atome can be defined by four numbers known as “quantum”, since they take only certain quantified values:

N : quantum number “principal” which can take all the whole values equal or higher than 1 is 1,2,3,4… It mainly defines the energy of the electron as well as the size of the orbital which increases as N increases.

L : quantum number “secondary” which can take all the whole values ranging between 0 and N -1. If N = 2, L can be equal to 0 or 1. This number defines mainly the form of the orbital one. With L = the 0 form is spherical, with 1 a bilobate form, L = 2 form having 4 lobes, L = 3 eight lobes and so on. It also helps to determine the energy of the electron and its average distance to the core.

m_l (or m ): quantum number “magnetic” which can take all the whole values ranging between - L and + L , it lays down the orientation of orbital described by the electron. The higher L is, the more there are possible different orientations for this orbital.

m_s (or S ): quantum number of “spin” which represents the Spin electron, namely +1/2 or -1/2. It is independent of the other quantum numbers. One can classically interpret it like a “intrinsic kinetic moment. ” It actually represents rather the symmetry of the Fermion.

According to the quantum numbers to which one has access, one can more or less precisely define part of this cloud:

If the 4 quantum numbers are known, the state allowed of the electron is defined;
If the first 3 quantum numbers are known, then the orbital one of the electron is defined.

To more easily indicate orbital or a state of the electron, a notation symbolic system is used. The principal number N is indicated by a figure, L by a letter and the value of m is in index. The spin is noted by an indent for the value -1/2, and nothing for value +1/2.

“S” for L = 0 (sharp)
“p” for L = 1 (principal)
“D” for L = 2 (diffuses)
“F” for L = 3 (fundamental)
…

For example:

$4f_ {- 2}$ means N = 4, L = 3, m_l = -2, m_s = +1/2;
$2p_ {+1}$ means N = 2, L = 1, m_l = +1, m_s = +1/2.

Influence atomic number on energy

More the electron occupies orbital low, like orbital the 1s, fundamental, more the energy which it contains is weak. The size of orbital depends much on its energy.

For the Hydrogen, energy corresponds only to the quantum number N . For this atom, and only him, the formula of the energy provided by quantum mechanics is exactly the same one as that of the model of Bohr, is (in eV):

$E_n = \ frac {- 13.6} {n^2}$

The value -13,6 eV, which corresponds at the fundamental state of hydrogen, is the Constante of Rydberg. One can extend this model to the ions known as “hydrogénoïdes”:

$E_n = \ frac {- 13.6 Z^2} {n^2}$

For the other atoms, the energy of the electrons depends on N and L (not of m_l nor of m_s ). The rule of Klechkowski is used to determine it - but there exist many exceptions. Actually, it makes it possible to know which electron has more energy than the precedents. For hydrogen, the energy level 2s = 2p but helium 2s < 2p because of L .

The more one atom has protons, the more the electrons are difficult “to remove” i.e. less they have energy. Thus the energy level of hydrogen 1s is much higher than the energy level 1s of helium, while with the same atoms, the level 2s is about on the same level (actually, a little low). The more there are protons, the more the sizes of orbital decrease.

Electronic configuration in a fundamental state

The Principe of exclusion of Pauli states that in the same atom, two electrons cannot have their four identical quantum numbers; N , L and m_l defines a “quantum box” (the two possible values of m_s lay down an orientation of the electron). Each quantum box can contain 0,1 electron (spin up ) or 2 electrons (of opposite spins).

It is the first rule of the “ Aufbau Prinzip ”: nuclear engineering.

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