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SSPD_Chapter1_Part9_Continued_RELATIVISTIC EFFECTS IN THE HYDROGEN ATOM.
Relativistic Hamiltonian can be fully solved for the Hydrogen atom. Here only the end results are enumerated.
As already seen azimuthial quantum number decides the shape of the subshell corresponding to different l. Now we know from Kepler’s second law [Appendix XXXIII] that a body in elliptical orbit sweeps equal area in equal time. Hence in spherical orbit electron moves with uniform linear velocity all the time but in ellipsoidal orbit apogee velocity is the slowest and perigee velocity is the fastest. If relativistic considerations are made then n fold degeneracy is removed among the different values of l but same n.
Relativistic considerations lead to spin angular momentum of electron. The spin quantum number is s = ±(1/2)ћ.
Table 1.6. Quantum Numbers of an Atomic Electron.
Name | Symbol | Possible Values | Quantity determined |
Principal | n | 1, 2, 3, …. | Electron Energy |
Azimuthial or Orbital | l | 0,1, 2, 3, 4,….(n-1); | |
Magnetic | m _{l} | -l, -( l -1),…0….( l -1), l | Spatial direction of L in presence of Magnetic field |
Spin Angular Momentum | s | -1/2, +1/2 | Electron Spin Angular Momentum |
From the point of view of quantum numbers, following are the permissible shells and subshells:
Table 1.7. Permissible Shells and Subshells
n | l | m | s | subshell | Shell |
1 | 0 | 0 | +1/2 |
-1/2 |
2 | 0 | 0 | +1/2 |
-1/2 |
1 | +1 | ±1/2 | |
0 | ±1/2 | ||
-1 | ±1/2 |
3 | 0 | 0 | +1/2 |
-1/2 |
1 | +1 | ±1/2 | |
0 | ±1/2 | ||
-1 | ±1/2 |
2 | +2 | ±1/2 | |
+1 | ±1/2 | ||
0 | ±1/2 | ||
-1 | ±1/2 | ||
-2 | ±1/2 |
Thus we see that the permissible electrons are subdivided in Major Shells:
K, L, M, N, O, P… where n = 1, 2, 3, 4, 5, 6 ……respectively.
Every Shell has Subshells.
n = 1, K has only s subshell having two electrons differentiated by spin quantum number ±1/2ћ.
n= 2 , L shell has two subshells, s and p subshell. s shell can accommodate at most 2 electrons .
Within L shell, p subshell can accommodate 6 electrons.
Therefore L shell can accommodate 8 electrons altogether.
n=3, M shell has 3 subshells, s. p and d subshells.
s shell can accommodate 2 elecrons,
p shell can accommodate 6 electrons.
d subshell can accommodate at most 10 electrons.
Therefore M shell can accommodate 18 electrons altogether.
Accordingly nth shell can accommodate 2n ^{2} electrons at most and these 2n ^{2} electrons are subdivided in s, p, d, f, subshells.
As seen from Table(1.7), no two electrons have the same set of four quantum numbers. This is known as Pauli-Exclusion Principle [Appendix XXXIV] .
SSPD_Chapter 1_Part 9_conclusions_1.8.2_FAR REACHING IMPLICATIONS OF THE REMOVAL OF DEGENERACY.
In Chapter 1_Part 7_Sec(1.6.3.) we had derived the formula for determining the energy of the electron in shell having n principal quantum number. According to Eq.(1.35):
E _{n} = -13.6eV/n ^{2}
According to this formula 2n ^{2} electrons at n quantum number will all have the same energy. This implies that all the n ^{2} electrons are at the same energy level. If this was to be true then we would say that all n ^{2} electrons ^{} are in a degenerate ^{} state ^{} but in fact this is not true. If relativistic considerations and spin considerations are applied then the degeneracy is removed. This has far reaching consequences in terms of the type of particles we are considering.
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