determination of fr
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EEE 313: Electrical Properties ofMaterial
Textbook: Principles of Electronic
Materials and Devices (3rdEdition)
- S.O. Kasap
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Lecture-11
Elementary Quantum Physics(contd.)
Course Instructor:Lecturer, EEE, Faculty of E&T
Eastern University
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Topics:
Electron as a wave: De Broglie relationship
Time independent Schrodinger equation
Infinite potential well: A confined electron
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The electron as a wave
DE BROGLIE relationship
De Broglie relationship relates the wave-like
properties (e.g. wavelength ) of matter to its particle-
like properties (e.g. momentum p) via = h/p
Problem-1: A 50 gram golf ball travelling at a
velocity of 20 ms-1 , calculate the wave length.
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Under certain conditions , electron exhibits wave like
properties. The general equation that describes the wave like
behavior and with the appropriate potential energy and
boundary conditions will predict the result of the
experiments, is called the Schrdinger equation.
The wave function can be written as (x,t).
In three dimension (x,y,z,t) represent the wave property of
electron.
To obtain the wavefunction for electron we need to know how
electron interact with its environments.
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Time-Independent Schrodinger Equation
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The time independent Schrdinger equation describes the wave
behavior of a particle under steady state conditions, i.e.when the
PEis time independent.The potential energy function is V=V(x,t) ,because the net force the
electron experiences is F=-dV/dx
Electrostatic potential energy given by V (r)=
If potential energy of electron is time independent, the total
wavefunction of the electron can be written as
2
04
e
r
, exp jEt
x t x
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Where is the wavefunction of the electron that
describes only the spatial behavior and E is the total
energy of the electron. The fundamental equationthat describes the electrons behavior by determining
is called time independent Schrdinger equation.
2
2 2
2
0
d x m
E V x xd x
2
, , ,x y z t is the probability of finding the electron
per unit volume at x,y,z at time t
2
, , ,x y z t dxdydz is the probability of finding the electronin a small elemental volume dx dy dz at
x,y,z at time t
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What will be the Schrodingers
equation for three dimensions??
[equation 3.16b, page-209]
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Boundary conditions to solve Schrodingers equation
Two important boundary conditions are often
used to solve the Schrodinger equation
(a) (x) must be continuous, single valued &smooth.
(b) d /dx must be continuous.
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Unacceptable forms of (x)
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Problems
Example-3.5: Solve the Schrodinger equation for a
free electron whose energy is E. What is the
uncertainty in the position of the electron & the
uncertainty in the momentum of the electron?[Page-
211] Example-3.6: Electrons are accelerated through a 100
V potential difference to strike a polycrystalline
sample. It has a diffraction angle of 30.4. From X-
ray studies, the separation of planes is 0.234nm. Whatis the wavelength of the electron and how does it
compare with that from the de Broglie relationship?
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Infinite Potential Well: A confined electron
Consider the behaviour of the electron when it is confined to a
certain region, 0
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= 0, when x 0 and x a, and is determined by
the Schrodinger equation in 0 < x < a with V = 0.
Therefore, in the region 0 < x < a
d2/ dx2+ (2m E ) /2= 0----- (1)
This is a 2ndorder linear differential equation. As a
general solution,(x) = A ejkx + B e -jkxwhere k is some constant(to be determined) and substitute this in equation (1)
to find k. (0) = 0, therefore, B = -A, so that
(x) = A [ ejkx - e jkx ] = 2Aj sin (kx) So, equation (1) becomes-
-2Ajk2(sin kx) + (2m/ 2) E (2Aj sin (kx)) = 0
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Energy of the electron :E = (2k2)/2m
Wave-function in infinite PE well,
n(x) = 2Aj sin (nx/a) Electron energy in infinite PE well is,
En = h2n2/ (8ma2)where n = 1,2,3
are called eigen-energies of the system.
What is the value of A???
To find the value of A, we will use normalization
condition. Applying this condition, we get
A = (1/2a) The resulting wave-function for the electron is,
n(x) = j (2/a)1/2sin (nx/a)
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Energy separation in infinite PE well
Energy differences between two consecutive
energy levels is ,
E = En+1
- En
= h2(2n + 1)/ (8ma2)
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Problems
Example-3.7: Consider an electron in an infinite
potential well of size 0.1 nm. What is the groundenergy of the electron? What is the energy required to
put the electron at the third energy level? How can
this energy be provided?[Page-216]
Example-3.8: Consider a macroscopic object 100
grams (say, an apple) confined to move between two
rigid walls separated by 1 m. What is the minimum
speed of the object? What should the quantumnumber n be if the object is moving with a speed 1
ms-1? What is the separation of the energy levels of
the object moving with that speed?[Page-217]
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Thank You
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