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The abrupt change in the refractive index or the dielectric constant is referred to as step index change. In the same way abrupt voltage change is referred to as step voltage change.

Matter wave behaves in an analogous fashion at a step voltage change as a light wave behaves at step index change.

Fig(1.26) Reflection and Transmission of electron matter wave across step voltage change from V 1 to V 2 .

For matter wave,

wave vector k=2π/λ = 2πp/h = [√{2m(E-V)}]/ћ;

Therefore the two wave vectors are:

k 1 = [√{2m(E-V 1 )}]/ћ and k 2 = [√{2m(E-V 2 )}]/ћ;

The incident forward electron matter wave:

ψ(z,t) incident = ψ 0incident Exp[j(ωt- k z1 .z)]

The reflected backward electron wave:

ψ(z,t) reflected = ψ 0reflected Exp[j(ωt+ k z1 .z)] 1.55

The transmitted forward electron wave:

ψ(z,t) transmitted = ψ 0transmitted Exp[j(ωt- k z2 .z)] 1.56

At the interface or at the step,

ψ 0incident - ψ 0reflected = ψ 0transmitted 1.57

Eq.(1.57) is the interface boundary condition.

Now we return to the original infinite potential well case.

Inside the potential well we have both the wave vectors ±i[√{2mE}]/ћ]hence we have both forward and backward traveling wave components.

Outside the well Schrodinger Equation is of the form:

2 ψ /∂x 2 + (2m(E-V)/ћ 2 )ψ = 0

But outside the well, V = ∞ therefore the wave equation reduces to:

2 ψ /∂x 2 - (2m∞/ћ 2 )ψ = 0

The characteristic equation has real roots hence the solution is hyperbolic.

Therefore ψ(z,t) = [AExp(-k 2 .z) + BExp(+k 2 .z)]Exp(iωt) 1.58

Where k 2 = [√{2m∞}]/ћ]

In Eq(1.58), the term Exp(+k 2 .z) is not admissible as it is exponentially growing function. Hence B=0 and the solution outside the well is:

ψ(z,t) = [AExp(-k 2 .z)]Exp(iωt) 1.59

But since k 2 = infinity implies the wave function does not exist outside the well. The transmitted wave outside the well almost instantaneously attenuates to zero since k 2 is the attenuation coefficient of the exponentially decaying wave and this coefficient is infinite.

Inside the potential well there are two waves one forward and the second backward and both have equal amplitude but opposite sign so that at the two boundaries of the potential well the sum of the two produce nodes.

ψ 0incident - ψ 0reflected = 0 at the boundaries 1.60

Therefore

ψ(z,t) incident = A Exp[j(ωt- k z1 .z)]

ψ(z,t) reflected = -AExp[j(ωt+ k z1 .z)] 1.61

The superposition of the two waves is:

ψ(z,t) = AExp[j(ωt)] [ Exp(- k z1 .z)- Exp(+ k z1 .z)]

ψ(z,t) = AExp[j(ωt)][Cos(k z1 .z)-iSin(k z1 .z) -Cos(k z1 .z)-iSin(k z1 .z)]

ψ(z,t) = -2iAExp[j(ωt)][Sin(k z1 .z)]

or ψ(z,t) = BExp[j(ωt)][Sin(k z1 .z)] 1.62

Eq.(1.62) is the mathematical formulation of the standing wave.

According to boundary conditions, this standing wave is suppose to have two nodes :

ψ(0,t)= ψ(W,t) =0

therefore k z1 .W = (2π/λ n )W = n π where n = 1, 2, 3,………….

Therefore W=n. λ n /2 1.63

This is the same restriction as was imposed in Eq.(1.51). This Eq(1.63) sheds a very important light on the quantization of energy levels permitted for an electron in an infinite potential well.

Inside the potential well there is the superposition of incident and reflected wave. There are only certain cases for which constructive interference takes place for the remaining cases there is destructive interference.

The cases for which the superposition leads to node formation at the boundaries z = 0 and W, only for those cases electron has an existence. For all the remaining cases electron goes out of existence by destructive interference.

Therefore we assert that in an infinite potential well , the electron can be in existence only for the energy levels:

E= E 0 , 4 E 0 , 9 E 0 , 16 E 0 ………

Where E 0 = h 2 /(8mW 2 )

The above result comes from Eq.(1.50) and its previous line.

An identical situation prevails in a hydrogen atom. The orbits which allow constructive interference to take place only those orbits are permitted. Remaining are forbidden. The orbital radii which support constructive interference are the radii where electron can exist as standing wave.

These are the radii exactly predicted by Bohr’s Law:

Orbital Angular Momentum = Integral Multiple of (h/(2π)) as already seen in Chapter 1_part 6.

Fig(1.27) An electron in 3 quantum states n=1,2,3 in an infinite potential well.

An electron in first three permissible quantum states in an infinite potential well are shown in Fig.(1.27

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
what is a peer
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
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Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
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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
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sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
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what does nano mean?
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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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