3.20 Sspd_chapter 1_part 9_pauli-exclusion principle

SSPD_Chapter 1_Part 9_ THE FOUR QUANTUM NUMBERS OF ATOMIC ELECTRONS AND PAULI-EXCLUSION PRINCIPLE.

Till now we have solved the Schrodinger Equation for 1-D Potential Well. If the same equation is solved for a real life 3-D Potential Well associated with a Hydrogen Atom then we get the following equation:

The solution of this Equation gives the full quantum mechanical description of electron bound to the host atom in this case hydrogen atom.

The potential well of a hydrogen atom or any real atom has a spherical symmetry hence we use spherical coordinates r, Ө, φ, while solving Eq.(1.74)

By taking the nonrelativistic Hamiltonian, we get three quantum numbers n, l, and m.

n is the principal quantum number or the total quantum number and specifies the K,L,M,N……Shells of the electron orbits It gives the quantized values of energies associated with these principal Shells.

In Hydrogen Atom , the principal energy levels are:

Table 1.5. Energy Levels of principal quantum numbers.

The energies also known as eigen energies associated with the principal quantum number are negative since we have taken the energy of an electron removed far away from the atom as zero and we are considering the energy of bound electrons. Therefore the eigen energies are negative. There are infinite such bound states becoming very closely spaced for large values of n.

The second quantum number ‘l’ is azimuthial or angular momentum quantum number. It gives the quantization of orbital angular momentum L as well as the shape of the orbital sub-shell belonging a given shell n.

L = Iω = Moment of inertia of the orbiting electron × angular orbital velocity;

For a given n, l = 0, 1 2 …….(n-1)

i.e. n =1, l =0 ; only one value of l;

n=2, l =0, 1;

n=3; l =0, 1, 2;

The magnitude of orbital angular momentum |L| = [√{l (l +1)}]×ћ

And |L| _max = (l +Δ)×ћ;

Therefore l = 0 implies zero orbital angular momentum and a perfectly spherical sub-shell belonging to a given shell n.

l =0 constrains the orbital sub-shell to be have a spherical symmetry and therefore all possible orbital angular momentum vectors cancel one another to give a resultant |L| = 0.

l = 1 implies |L| = [√2]×ћ. This implies a resultant orbital angular momentum and hence an ellipsoidal sub-shell.

l = 2 implies |L| =[√{2(2+1)}]×ћ = [√6]×ћ. This implies a higher oblateness of the ellipsoidal sub-shell.

Figure 1.33. The possible permitted subshells of electrons belonging to a host atom.

From nonrelativistic treatment there is n-fold degeneracy for every value of principal quantum number ‘n’. This means the electrons in given shell ‘n’ have different values of ‘l’ ranging from 0 to n-1 and all these have same energy level corresponding to ‘n’. This is termed as n-fold degeneracy.

For n=1, l = 0 , there is no degeneracy

For n=2, l = 0, 1 and there is two fold degeneracy;

For n = 3, l = 0, 1 , 2 and there is three fold degeneracy;

Next we have magnetic quantum number ‘m’. This implies the spatial quantization of orbital angular momentum. For a given value of l we have a given value of Orbital Angular Momentum.

Say we take l = 2. This has L= [√6]×ћ. Now when a magnetic field is applied then L vector can be permitted to assume only those spatial directions in which the z axis projections of L i.e. L _Z = 0ћ, ±1ћ, ±2ћ, as shown in Figure(1.34).

Figure 1.34. In the presence of a magnetic field, the spatial quantization of L = [√6]×ћ;

Therefore l will range from 0 to (n-1) and each value of l between 0 and n-1 has 2l +1 eigen states.

In Fig.(1.34) L corresponding to l = 2 can be parallel or anti-parallel giving a projection of ±2ћ on z axis. L can be inclined to z-axis to give a projection of ±ћ or L can be transverse to z-axis giving a projection of 0ћ.

L corresponding to l = 0 is transverse to the z-axis. L corresponding to l = 1 is obliquely aligned with respect z-axis so as to give a projection of ±1ћ on the z-axis and L is also transverse to z-axis.

Therefore a given value of n has altogether ∑(2 l +1) degeneracy with l ranging from 0 to (n-1) . The total number of eigen states associated with principal quantum number n is n ² . These eigen state energies are not explicitly determined by l or m. Hence these n ² eigen states have the same energy level –(13.6/ n ² ). Therefore a given principal quantum number n has n ² -fold degeneracy. Energy states for a hydrogenic atom is given Fig.(1.35).

As seen in Figure 1.35. for n =1, l = 0; This has 1-fold degeneracy.

For n =2 , l = 0 and 1; Here there is 1-fold degeneracy for l = 0 and 3 fold degeneracy for l = 1;

For n= 3, l = 0,1 and 2; Here there is 1-fold degeneracy for l = 0 and 3 fold degeneracy for l = 1 and 5-fold degeneracy for l =2;

So we see that if in an atom the electronic shells are occupied upto n=3 then it has (1+3+5) =9 electrons. For every quantum state characterized by n, l and m there is further possibility of ±(1/2)ћ spin angular momentum.

for n =1, l = 0; This has 1-fold degeneracy. But l =0 has 2-fold degeneracy due to spin quantum number namely ±(1/2)ћ.

For n =1 there will be two electrons permitted.

For n= 2 there will be eight electrons permitted.

For n= 3 there will be eighteen electrons permitted.

And for n in general there will be 2n ² electrons permitted. This sets the rule by which the elements are built up in a Periodic Table.

Figure 1.35. Energy Level of Hydrogenic Atom from nonrelativistic Hamiltonian.

From the discussion above and from the discussion which will follow in the sequel part, we can conclude Pauli-Exclusion Principle namely:

No two electrons can have all the four quantum numbers to be the same.