Applications of Quantum
Physics
Module:2
Applications of Quantum Physics
Particle in a 1-D box (Eigen Value and Eigen Function), 3-D
Analysis (Qualitative), Tunneling Effect (Qualitative) (AB 205),
Scanning Tunneling Microscope (STM).
Arpan Kumar Nayak (Ph.D)
School of Advanced Sciences (SAS)
Particle in a 1-Dimensional Box
Time Independent Schrödinger Equation
  2 d 2
 V ( x)  E
2
2m dx
Region I
Region II
Region III
KE
PE
TE
Wave function is dependent on time and position function:
V(x)=∞
K.E=0
0
V(x)=0
V(x)=∞
K.E=0
L
0
V(x)=0 for L>x>0
V(x)=∞ for x≥L, x≤0
Classical Physics: The particle can
exist anywhere in the box and follow
a path in accordance to Newton’s
Laws.
Quantum Physics: The particle is
expressed by a wave function and
there are certain areas more likely to
contain the particle within the box.
1
( x, t )  f (t ) ( x)
0
Time Independent Schrödinger Equation
x
  2 d 2 ( x)
 V ( x)  E
2
2m dx
Region I and III:
Applying boundary conditions:
  2 d 2 ( x)
  *  E
2
2m dx
Region II:
  2 d 2 ( x)
 E
2m dx2
 0
2
Finding the Wave Function
  2 d 2 ( x)
 E
2m dx2
d 2 ( x) 2m

 2 E
dx2

This is similar to the general differential equation:
d 2 ( x)

 k 2
2
dx
  A sin kx  B cos kx
So we can start applying boundary conditions:
x=0 ψ=0
0  A sin 0k  B cos0k
x=L ψ=0
0  Asin kL
A0
where n=
Calculating Energy Levels:
k 2 2
E
2m
2mE
k  2

2
E
n
h
2
L 2m4 2
2
2
2
h

2
E
2
k 2h2
E
2m4 2
2
nh
8mL2
 II  A sin
nx
L
But what is ‘A’?
Normalizing wave function:
L
 ( A sin nx / L)
2
dx  1
0
L
 x sin 2(nx / L) 
A  
 1
2
4
(
n

/
L
)

0
n 

sin
2
L
2L
L
A  
1
n 
2


4
L


2
0  0  B *1 B  0
kL  n
Our new wave function:
*
Since n=
2 L 
A   1
2
*
A
2
L
Our normalized wave function is:
 II 
2
nx
sin
L
L
Case-1
n=1
E1 
2
h
2
8mL
 E0
 1  min, x  0, L
 1  max, x  L / 2
1 
2
x
sin
L
L
Case-2
n=2
E2 
2
4h
2
8mL
 4 E1
2 
 2  min, x  0, L / 2, L
 2  max, x  L / 4,3L / 4
2
2x
sin
L
L
Particle in a 1-Dimensional Box
 II 
2
nx
sin
L
L
Applying the
Born Interpretation
 II
2
2  nx 
  sin

L
L 
n=4
n=3
E
x/L
2
n=4
E
n=3
n=2
n=2
n=1
n=1
x/L
Particle in a 3-Dimensional Box
2
nxx
X ( x) 
sin
Lx
Lx
2
nyy
Y ( y) 
sin
Ly
Ly
2
nzz
Z ( z) 
sin
Lz
Lz
8
nxx
nyy
nzz
 (r ) 
sin(
) sin(
) sin(
)
V
Lx
Ly
Lz
2
2
2
2
h nx
ny
nz
Enx , ny , nz 
( 2  2  2)
8m Lx
Ly
Lz
Operator
• A rule that transforms a given function into
another function
Identifying the operators
Linear Operator
• A linear operator has the following properties



A f 1  f 2   A f 1  A f 2


Acf   c A f
Eigen Function and Eigen value
In any measurement of the observable associated
with the operator , the only values that will ever be
observed are the eigenvalues ‘a’ which satisfy the
eigenvalue equation:
This is the postulate that the values of dynamical
variables are quantized in quantum mechanics.

A f ( x)  kf ( x)

f(x) is eigenfunct ion of A with eigen valu e k
Eigen function and eigen value
f x   e
ikx
Is it eigen function of momentum operator ?
What is eigen value ?
Eigenvalue equation
Eigenvalue equation
(Operator)(function) = (constant factor)*(same function)
^

Example: eikx is an eigenfunction of a operator Px = -ih
F(x) = eikx
= -i h  eikx
x
= -i2 hk2eikx
= h k2eikx
Thus eikx is an eigenfunction

x
Tunneling Effect
Tunneling Effect
The quantum particle exhibits wave-like nature, it can reflect and transmit (tunnel) through the potential
barrier
Barriers and Tunneling
(Tunnel Effect)
Consider a particle of energy E approaching a potential barrier of height
V0, and the potential everywhere else is zero.
First consider the case of the energy greater than the potential barrier.
In regions I and III the wave numbers are:
In the barrier region we have
Reflection and Transmission
The wave function will consist of an incident wave, a reflected wave, and
a transmitted wave.
The potentials and the Schrödinger wave equation for the three regions
are as follows:
The corresponding solutions are:
As the wave moves from left to right, we can simplify the wave functions
to:
Probability of Reflection and
Transmission
The probability of the particles being reflected R or transmitted T is:
Because the particles must be either reflected or transmitted we have: R + T
= 1.
By applying the boundary conditions
x → ±∞, x = 0, and x = L, we arrive at
the transmission probability:
Tunneling
Now we consider the situation where
classically the particle doesn’t have
enough energy to surmount the
potential barrier, E < V0.
The quantum mechanical result is one of the most remarkable features of
modern physics. There is a finite probability that the particle can penetrate the
barrier and even emerge on the other side!
The wave function
in region II becomes:
The transmission probability that
describes the phenomenon of tunneling is:
Tunneling wave function
Consider when κL >> 1 then the transmission probability becomes:
This violation of classical physics is allowed by the uncertainty principle. The
particle can violate classical physics by DE for a short time, Dt ~ ħ / DE.
Scanning Tunneling
Microscopy
The
scanning
tunneling microscope
was developed at
IBM Zürich in 1981
by Gerd Binning and
Heinrich Rohrer who
shared the Nobel
Prize for physics in
1986 because of the
microscope.
Iron atoms on the surface of Cu(111)
Scanning Tunneling Microscopy
Developed on the principle of quantum mechanical tunneling
Components
1 metal tip
2 Piezoelectric scanner
3 current amplifier
4 Bipotentiostat
5 feedback loop
6 detector
Tip and sample must be electrically conductive (metals)
Advanced STM
Ultra high vacuum and low temperature STM set-up
Graphene
Graphene
STM working Principle
STM works on the principle of quantum mechanical tunneling
Tunneling current
I  A  eV  e
2
2 m
2
d
A is constant
e is electron charge
V is voltage
m is mass of electron
Φ is work function of metal tip
d is distance between tip and sample
STM working procedure
Constant current mode
 Image the surface with
constant tunnel current and
variable height
 Feed back loop help to
maintain constant current
 Surface (height) structure can
detect
Constant height mode
 Image the surface with
constant height and
variable tunnel current
 Electron density on the
surface can detect
Surface of different structures
Resolution : 0.1 nm
It is resolve individual atoms in a materials
STM images of gold and Si crystal
STM Applications
Widely used in nanotechnology
 Image the surface structure
 Estimate surface roughness
 3D images of the surface
 Locate the defect on the surface of crystal
 Understand electric structure of materials