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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

what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
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
Daniel
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
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
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.
Tarell
what is the actual application of fullerenes nowadays?
Damian
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.
Tarell
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.
Virgil
is Bucky paper clear?
CYNTHIA
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.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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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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