Transcript unit-4 - snist

Quantum Mechanics
WAVES
A wave is nothing but disturbance which is occurred
in a medium and it is specified by its frequency,
wavelength, phase, amplitude and intensity.
PARTICLES
A particle or matter has mass and it is located at a
some definite point and it is specified by its mass,
velocity, momentum and energy.
• The physical values or motion of a macroscopic
particles can be observed directly. Classical
mechanics can be applied to explain that motion.
• But when we consider the motion of Microscopic
particles such as electrons, protons……etc., classical
mechanics fails to explain that motion.
• Quantum
mechanics deals with motion
microscopic particles or quantum particles.
of
de Broglie hypothesis
• In 1924 the scientist named de Broglie introduced
electromagnetic waves behaves like particles, and
the particles like electrons behave like waves called
matter waves.
• He derived an expression for the wavelength of
matter waves on the analogy of radiation.
• According to Planck’s radiation law
E  h
c
 h ............(1)

• Where ‘c’ is a velocity of light and ‘λ‘is a wave length.
• According to Einstein mass-energy relation E  mc2 ......(2)
mc  h
2
From 1 & 2
c

h
mc
h

p

Where p is momentum of a photon.
• The above relation is called de Broglie’ s matter wave
equation. This equation is applicable to all atomic
particles.
• If E is kinetic energy of a particle
• Hence the de Broglie’s wave length
1
E  mv 2
2
p2
E
2m
p  2 mE
h

2mE
de Broglie
electrons
wavelength
associated
with
Let us consider the case of an electron of rest mass m0 and
charge ‘ e ‘ being accelerated by a potential V volts.
If ‘v ‘ is the velocity attained by the electron due to
acceleration
The de Broglie wavelength  
h
 
m0 v
0
12.26

A
V
h
2eV
m0
m0
1
m0 v 2  eV
2
2eV
v
m0
Characteristics of Matter waves
• Lighter the particle, greater is the wavelength associated
with it.
• Lesser the velocity of the particle, longer the wavelength
associated with it.
• For V= 0, λ=∞ . This means that only with moving particle
matter wave is associated.
• Whether the particle is charged or not, matter wave is
associated with it. This reveals that these waves are not
electromagnetic but a new kind of waves .
It can be proved that the matter waves travel faster than light.
E  h
We know that
E  mc 2
mc 2
h  mc   
h
2
The wave velocity (ω) is given by w  
mc 2 h
w(
)( )
h mv
2
mc
h
As the particle velocity v cannot where 
& 
h
mv
exceed velocity of light c,
ω is greater than velocity of light.
c2
w
v
Experimental evidence for matter waves
1.Davisson and Germer ’s Experiment.
2.G.P. Thomson Experiment.
DAVISON & GERMER’S EXPERMENT
• Davison and Germer first detected electron waves in 1927.
• They have also measured de Broglie wave lengths of slow
electrons by using diffraction methods.
Principle:
• Based on the concept of wave nature of matter fast moving
electrons behave like waves. Hence accelerated electron
beam can be used for diffraction studies in crystals.
Experimental arrangement
• The electron gun G produces a fine beam of electrons.
• It consists of a heated filament F, which emits electrons due to thermo
ionic emission
• The accelerated electron beam of electrons are incident on a nickel
plate, called target T. The target crystal can be rotated about an axis
perpendicular to the direction of incident electron beam.
• The distribution of electrons is measured by using a detector called
faraday cylinder c and which is moving along a graduated
circular scale S.
• A sensitive galvanometer connected to the detector.
High voltage
Nickel
Target
Anode
filament
cathode
G
Faraday
cylinder
c
S
Circular scale
Galvanometer
G
Results
• When an electron beam accelerated by 54 volts was
directed to strike the nickel crystal, a sharp maximum in
the electron distribution occurred at scattered angle of 500
with the incident beam.
• For that scattered beam of electrons the diffracted angle
becomes 650 .
• For a nickel crystal the inter planer separation is
d = 0.091nm.
Incident electron beam
0
650 25
250
650
I
V = 54v
C
U
R
R
E
N
T
0
500
θ
Diffracte
d beam
• According to Bragg’s law
2d sin   n
2  0.091nm  sin 650  1 
  0.165nm
• For a 54 volts , the de Broglie wave
length associated with the electron is
given by
• This is in excellent agreement with the
experimental value.
• The Davison - Germer experiment
provides a direct verification of de
Broglie hypothesis of the wave nature
of moving particle.
0
12.26

A
V
0
12.26 A

54
  0.166nm
G.P THOMSON’S EXPERIMENT
• G.P Thomson's experiment proved that the diffraction pattern
observed was due to electrons but not due to electromagnetic
radiation produced by the fast moving charged particles.
EXPERIMENTAL ARRANGEMENT
• G.P Thomson experimental arrangement consists of
(a) Filament or cathode C
(b) Gold foil or gold plate
(c) Photographic plate
(d) Anode A.
• The whole apparatus is kept highly evacuated discharge tube.
• When we apply potential to cathode, the electrons
are emitted and those are further accelerated by
anode.
• When these electrons incident on a gold foil, those
are diffracted, and resulting diffraction pattern
getting on photographic film.
• After
developing the photographic plate a
symmetrical pattern consisting of concentric rings
about a central spot is obtained.
G.P THOMSON EXPERIMENT
cathode
Discharge tube
slit
Anode
Vacuum pump
Gold foil
Photo
graphic
plate
Photographic film
Diffraction pattern.
E
Incident
electron
beam
A
θ
Brage plane
r
Gold foil
B
radius
θ
θ
L
Tan 2θ = r / L
If θ is very small 2θ = r / L
2θ = r / L ………. (1)
o
c
According to Braggs law
2d sin   n (  and , n  1)
2d ( )  n


L
d
d 
d  
r
2
r
( )
L
According to de Broglie’ s wave
h
equation

2m0eV
m0 
m
2
v
1 2
c
Where m0 is a relativistic mass of an electron
n
from., eq (3)
L
L
h
d  d  (
)
r
r 2m0 eV
d  4.08 A
0
• The value of ‘d’ so obtained agreed well with the
values using X-ray techniques.
• In the case of gold foil the values of “d” obtained by
the x-ray diffraction method is 4.060A.
Heisenberg uncertainty principle
• This principle states that the product of uncertainties in
determining the both position and momentum of particle
is approximately equal to h / 4Π.
h
xp 
4
Where Δx is the uncertainty in determine the position
and Δp is the uncertainty in determining momentum.
• This relation shows that it is impossible to determine
simultaneously both the position and momentum of the
particle accurately.
• This relation is universal and holds for all canonically conjugate
physical quantities like
1. Angular momentum & angle
2. Time & energy
h
4
h
t E 
4
  j 
Consequences of uncertainty principle
• Explanation for absence of electrons in the nucleus.
• Existence of protons and neutrons inside nucleus.
• Uncertainty in the frequency of light emitted by an atom.
• Energy of an electron in an atom.
Physical significance of the wave function
• The wave function ‘Ψ’ has no direct physical meaning. It is
a complex quantity representing the variation of a Matter
wave.
• The wave function Ψ( r, t ) describes the position of a
particle with respect to time.
• It can be considered as ‘probability amplitude’ since it is
used to find the location of the particle.
ΨΨ* or ‫׀‬Ψ‫׀‬2 is the probability density function.
ΨΨ* dx dy dz gives the probability of finding the electron
in the region of space between x and x + dx, y and y + dy,
z and z + dz.

*

dxdydz  1

-


2
dxdydz  1
-
The above relation shows that’s a ‘normalization
condition’ of particle.
Schrödinger time independent wave equation
• Schrödinger wave equation is a basic principle of a
fundamental Quantum mechanics.
• Consider a particle of mass ‘m’ ,moving with velocity ‘v’ and
wavelength ‘λ’. According to de Broglie,
h

p
h

...........(1)
mv
• According to classical physics, the displacement for a
moving wave along X-direction is given by
S  A sin(
2

 x)
• Where ‘A’ is a amplitude ‘x’ is a position co-ordinate and
‘λ’ is a wave length.
• The displacement of de Broglie wave associated with a
moving particle along X-direction is given by
 (r , t )  A sin(
2

 x)
If ‘E’ is total energy of the system
E  P.E  K .E
h
( )2
p2
E V 
V  
2m
2m
2
h
E V 
2m2
h2
 E  V ......(2)
2
2 m
Periodic changes in ‘Ψ’ are responsible for the wave nature
of a moving particle
d ( ) d
2
 [ A sin
.x ]
dx
dx

d ( ) 2
2

A cos( .x )
dx


d 2
2 2
2
 [ ] A sin(
.x )
2
dx


d 2
4 2
2
  2 A sin(
.x )
2
dx


d 2
4 2
 2 
2
dx

1
1 d 2
 2
 (3)
2
2

4  dx
From equation 3……
h2
 E V
2
2m
h2
1
d 2
[
]  [E V ]
2
2
2m
4  dx
 h 2 d 2
 [E  V ]
2
2
8 m dx
d 2
8 2 m

[ E  V ]
2
2
dx
h
d 2
8 2 m

[ E  V ]  0
2
2
dx
h
This is Schrödinger time independent wave equation in one dimension.
In three dimensional way it becomes…..
 2  2  2 8 2 m
 2  2  2 [ E  V ]  0
2
x
y
z
h
Particle in a one dimensional potential box
• Consider an electron of mass ‘m’ in an infinitely deep one-
dimensional potential box with a width of a ‘ L’ units in
which potential is constant and zero.
v ( x )  0,0 x  L
v ( x )  , x  0 & x  L
V=0
X=0
X=L
V
Periodic positive ion cores
Inside metallic crystals.
V 
+ + + + + + +
+ + + + + + +
+ + + + + + +
+ + + + + + +
+ + + + + + +
X
One dimensional periodic
potential in crystal.
The motion of the electron in one dimensional box can
be described by the Schrödinger's equation.
d 2 2m
 2 [ E  V ]  0
2
dx

Inside the box the potential V =0
d 2 2m
 2 [ E ]  0
2
dx

d 2
2m
2
2
 k   0  where, , k  2 E
2
dx

The solution to above equation can be written as
 ( x)  A sin kx  B cos kx
Where A,B and K are unknown constants and to
calculate them, it is necessary to apply boundary
conditions.
• When X = 0 then Ψ = 0 i.e. |Ψ|2 = 0 ……. a
X=L
Ψ = 0 i.e. |Ψ|2 = 0 …… b
• Applying boundary condition ( a ) to equation ( 1 )
A Sin K(0) + B Cos K(0) = 0
• Substitute B value equation (1)
Ψ(x) = A Sin Kx
B=0
Applying second boundary condition for equation (1)
0  A sin kL  (0) cos kL
A sin kL  0
sin kL  0
kL  n
n
k
L
Substitute B & K value in equation (1)
(nx)
 ( x)  A sin
L
To calculate unknown constant A, consider normalization condition.
L
Normalization condition
  ( x)
2
dx  1
0
nx
o A sin [ L ]dx  1
L
2
2
1
2nx
A  [1  cos[
]dx  1
2
L
o
L
2
A2
L
2nx L
[x 
sin
]0  1
2
2n
L
A2
L 1
2
A  2/ L
The normalized wave function is
n
 n  2 / L sin
x
L
2mE
k 2 2
2
k 
E
2

2m
 n   h 

 

L   2 
E
2m
n
h
where,.k 
& 
L
2
n2h2
E
8mL2
2
2
The wave functions Ψn and the corresponding energies En
which are called Eigen functions and Eigen values, of the
quantum particle.
Normalized Wave function in three dimensions is given by
n3z
n1x
n2y
 n  (2 / L ) sin
sin
sin
L
L
L
3
The particle Wave functions & their energy Eigen values
in a one dimensional square well potential are shown in
figure.
nx
 n  2 / L sin
L
n2h2
En 
8mL2
E3=9h2 / 8mL2
n =3
L/3
2L / 3
E2=4h2/8mL2
n=2
L/2
√ (2 / L)
E1=h2 / 8mL2
n=1
X=0
L/2
X=L
Conclusions
1.The three integers n1,n2 and n3 called Quantum numbers
are required to specify completely each energy state.
2.The energy ‘ E ’ depends on the sum of the squares of the
quantum numbers n1,n2 and n3 but not on their individual
values.
3.Several combinations of the three quantum numbers may
give different wave functions, but not of the same energy
value. Such states and energy levels are said to be
degenerate.