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This requires {(√(2mE))/ ћ}W = 0 radians or

{(√(2mE))/ ћ}W = π radians or

{(√(2mE))/ ћ}W = n π radians where n= 0,1,2,3,4…….

Squaring both sides we get:

{(2mE)/ ћ 2 }W 2 = (n π) 2

or E = n 2 h 2 /(8mW 2 ) 1.49

Eq.(1.49) implies that the permissible energy states of an electron in an infinite potential well are quantized.

Why electrons cannot occupy a continuum energy states as they do in free space? The answer is the following:

We had assumed at the beginning of the analysis that V(x) = 0. This implies that potential energy is zero and electron possesses only Kinetic Energy.

Therefore total energy E = p 2 /(2m) = n 2 h 2 /(8mW 2 )

Therefore p = (nh)/(2W) 1.50

From de Broglie postulate: λ = h/p = h/[(nh)/(2W)]

To satisfy the standing wave condition in bounded space which an infinite 1-D potential well is, following boundary condition must be satisfied

W = n λ/2 1.51

Eq.(1.51) is the necessary condition for Standing Wave pattern. This standing wave pattern requirement causes the quantization of energy states.

Here we digress briefly to the chapter of light to fully understand the behavior of matter wave.

Fig.(1.25) A plane wavefront light ray perpendicularly incident on an interface of two optical mediums.

Whenever light travels from one optical medium of refractive index n 1 to the other optical medium of refractive index n 2 , the incident wave Transverse Electromagnetic Wave (TEM) experiences partial reflection at the interface of the two media and partial transmission into the second medium.

Let us assume that medium 1 is absolute vacuum hence its refractive index = n 1 = 1 and medium 2 is a solid medium of refractive index n 2 = n. The mathematical form of the incident wave, reflected wave and the transmitted wave is given in Fig(1.25).

Wave vector in medium 1 is k 1 = 2π/λ 1 and wave vector in medium 2 is k 2 = 2π/λ 2 ;

And ν λ 1 = c, ν λ 2 = v; 1.52

Therefore c/v = λ 1 / λ 2 = n/1;

Therefore λ 1 = n. λ 2 1.53

We know that if the second medium is metal, the incident light is totally reflected and the reflected light experiences a phase change of 180°.

Also at the interface E incident + E reflected = 0;

But if the second medium is dielectric then we have partial reflection and partial transmission and at the interface we have: E incident + E reflected = E transmitted ;

The incident wave is the forward wave:

E(z,t)= E xoincident Exp[j(k z1 z – ω.t)];

The reflected wave is the backward wave:

E(z,t)= E xoreflected Exp[j(k z1 z + ω.t)];

The transmitted wave is also moving in forward direction therefore it is:

E(z,t)= E xotransmitted Exp[j(k z2 z – ω.t)]; 1.54

Wave vectors have been defined in Eq.(1.52).and Eq.(1.53).

The incident forward and reflected backward wave interfere to form Standing Wave as they do on a mismatched transmission line. If a transmission line is not terminated in Characteristic Impedance then partial reflection takes place at the load and a partial standing wave pattern is formed on the transmission line. Standing Wave implies there is no transmission of energy. In case of metal there is total reflection. Hence we have 100% standing wave in medium 1 and there is no penetration of light into the metallic medium 2. Hence no transmission of light energy. For dielectric medium 2 , we have partial reflection hence only partial standing wave.

Questions & Answers

Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
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LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
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Loga
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
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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