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Concluding Part 9 describes the wave nature of distinguishable and undistinguishable particles and the resulting Fermi-Dirac distribution and Bose-Einstein distribution respectively. This part describes the multiple fine structure in spectrum is caused due to the removal of degeneracy.


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
s K
2 0 0 +1/2
1 +1 ±1/2
0 ±1/2
-1 ±1/2
p L
3 0 0 +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
d M

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] .


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.

Questions & Answers

anyone know any internet site where one can find nanotechnology papers?
Damian Reply
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
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what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
many many of nanotubes
what is the k.e before it land
what is the function of carbon nanotubes?
I'm interested in nanotube
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
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Source:  OpenStax, Solid state physics and devices-the harbinger of third wave of civilization. OpenStax CNX. Sep 15, 2014 Download for free at http://legacy.cnx.org/content/col11170/1.89
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