Transcript Document

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.
Waves and Particles : What do we mean by them?
Material Objects:
Ball, Car, person, or point like objects called particles.
They can be located at a space point at a given time.
They can be at rest, moving or accelerating.
Falling Ball
Ground level
Waves and Particles: What do we mean by them
Common types of waves:
Ripples, surf, ocean waves, sound waves, radio waves.
Need to see crests and troughs to define them.
Waves are oscillations in space and time.
Direction of travel, velocity
Up-down
oscillations
Wavelength ,frequency, velocity and oscillation size defines waves
Particles and Waves: Basic difference in behaviour
When particles collide they cannot pass through each other !
They can bounce or they can shatter
Before collision
Another after
collision state
shatter
After collision
Collision of truck with ladder on top with a
Car at rest ! Note the ladder continue its
Motion forward ….. Also the small care front
End gets smashed.
Head on collision of a car and truck
Collision is inelastic – the small car is dragged along
By the truck……
Waves and Particles Basic difference:
Waves can pass through each other !
As they pass through each other they can enhance or cancel
each other
Later they regain their original form !
Waves and Particles:
Wavelength
Frequency
Spread in space and time
Waves
Can be superposed – show
interference effects
Pass through each other
Localized in space and time
Particles
Cannot pass through each other they bounce or shatter.
A SUMMARY OF DUAL ITY OF NATURE
Wave particle duality of physical objects
LIGHT
Wave nature -EM wave
Particle nature -photons
Optical microscope
Convert light to electric current
Interference
Photo-electric effect
PARTICLES
Wave nature
Matter waves -electron
microscope
Discrete (Quantum) states of confined
systems, such as atoms.
Particle nature
Electric current
photon-electron collisions
• 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 of
microscopic particles or quantum particles.
Classical world is Deterministic:
Knowing the position and velocity of
all objects at a particular time
Future can be predicted using known laws of force
and Newton's laws of motion.
Quantum World is Probabilistic:
Impossible to know position and velocity
with certainty at a given time.
Only probability of future state can be predicted using
known laws of force and equations of quantum mechanics.
Tied together
Observer
Observed
Wave Nature of Matter
Louis de Broglie in 1923 proposed that
matter particles should exhibit wave
properties just as light waves exhibited
particle properties. These waves have
very small wavelengths in most situations
so that their presence was difficult to observe
These waves were observed a few years later by Davisson and
G.P. Thomson with high energy electrons. These electrons show
the same pattern when scattered from crystals as X-rays of simi
wave lengths.
Electron microscope
picture of a fly
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  mc ......(2)
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 Broglile’ s matter
wave equation. This equation is applicable to all
atomic particles.
• If E is kinetic energy of a particle
1
E  mv 2
2
p2
E
2m
p  2 mE
h
• Hence the de Broglie’s wave length  
2mE
• de Broglie wavelength associated with
electrons:
• 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
1
m0 v 2  eV
2
2eV
v
m0
h

 
m0 v
0
12.26

A
V
h
2eV
m0
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.
We know that
E  h
E  mc 2
h  mc 2
mc 2

h
The wave velocity (ω) is given by
Substituting for λ we get
• As the particle velocity v cannot
exceed velocity of light c, ω is
greater than
velocity of light.
w  
mc 2
w(
)
h
h

mv
mc 2 h
w(
)
h mv
c2
w
v
Experimental evidence
for matter waves
There was two experimental evidences
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.
High voltage
Nickel
Target
Anode
filament
cathode
G
Faraday
cylinder
c
S
Circular scale
Galvanometer
G
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
parallel 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.
Results:
• When the electron beam accelerated by 54 volts
was directed to strike the nickel crystal, a sharp
maximum in the electron distribution occurred at
an angle of 500 with the incident beam.
• For that incident beam the diffracted angle
becomes 650 .
• For a nickel crystal the inter planer separation
d = 0.091nm.
Incident electron
beam
I
650
V = 54v
650
C
U
R
R
E
N
T
0
250
250
500
θ
Diffracted
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
0
12.26

A
V
0
12.26 A
• This is in excellent agreement with  
54
the experimental value.
• The Davison - Germer experiment   0.166 nm
provides a direct verification of de
Broglie hypothesis of the wave
nature of moving particle.
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 (p)
(d) Anode A.
• The whole apparatus is kept highly evacuated discharge tube.
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
• 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.
According to Braggs law
2d sin   n
if ,  ..is..very.small
2d ( )  n
for., n  1

d
.......(2)
2
n
frm,.eq .....(1)
L
d   ..............(3)
r
According to de Broglie’ s wave equation
h

( for., electron)
2m0 eV
where., m0 ..is..a., relatavestic..mass
n
from., eq (3)
L
h
d (
)
r 2m0 eV
d  4.080 A
• 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
  j 
4
h
t E 
4
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.
• This equation arrives at the equation stating with de Broglie’s
idea of Matter wave.
• According to de Broglie, a particle of mass ‘m’ and moving with
velocity ‘v’ has a wavelength ‘λ’.
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 wave along X-direction is given by
 (r , t )  A sin(
2

 x)
• If ‘E’ is total energy of the system E = K.E + P.E ---------(4)
•
If ‘p’ is a momentum of a particle
K.E = ½ mv2
= p2 / 2m
• According to de Broglie’ s principle λ = h / p
p=h/λ
K.E = p2 / 2m
• The total energy
E  P.E  K .E
h2
E V 
2m2
h2
 E  V ..............(5)
2
2m
• Where ‘V’ is potential energy.
• Periodic changes in ‘Ψ’ are responsible for the wave
nature of a moving particle
d ( ) d
2
 [ A sin
.x ]
dx
dx

2
2

A cos( .x)

d 2

 [
2
2
] A sin(
2
.x )
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
2
n
frm...eq ,.5
2
h

E

V
2
2m
2
2
h
1 d
[
]  [E V ]
2
2
2m 4  dx
h
d
 [E  V ]
2
2
8 m dx
2
2
d
8 m

[ E  V ]
2
2
dx
h
2
2
d 2 8 2 m
 2 [ E  V ]  0
2
dx
h
• This is Schrödinger time independent wave
equation in one dimension.
In three dimensional way it becomes…..
      8 m
 2  2  2 [ E  V ]  0
2
x
y
z
h
2
2
2
2
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
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
V 
X
Nuclei
Periodic positive ion cores
Inside metallic crystals.
One dimensional periodic
potential in crystal.
The motion of the electron in one dimensional
box can be described by the Schrödinger's
equation.
d  2m
 2 [ E  V ]  0    1
2
dx

2
d
Inside the box the potential V =0
d  2m
 2 [ E ]  0    (2)
2
dx

2
d 2 2
 k   0    (3)
2
dx
where., k
2
2mE

2
The solution to equation (3) can be written as
 ( x)  A sin kx  B cos kx    (4)
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 ------1
x=L
Ψ = 0 i.e. |Ψ|2 = 0 ------2
• Applying boundary condition (1) to equation (4)
A sin k(0) + B cos K(0) = 0
B=0
• Substitute B value equation (4)
Ψ(x) = A sin kx
•
Applying second boundary condition for equation (4)
0  A sin kL  (0) cos kL
A sin kL  0
sin kL  0
kL  n
n
k 
L
• Substitute k value in equation (5)
(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
A
L
2nx L
[x 
sin
]0  1
2
2n
L
2
A
L 1
2
A  2/ L
2
The normalized wave function is
nx
 n  2 / L sin
L
n1x n2y n3z
3
 n  (2 / L ) sin
sin
sin
L
L
L
n 2  n12  n22  n32
•
The wave functions Ψn and the corresponding
energies En which are often called Eigen
functions and Eigen values, describe the
quantum state of the particle.
where., k
2
2mE

2
k 2 2
E
2m
where.,
n
h
k 
& 
L
2
 n   h 

 

L   2 

E
2m
2 2
n h
E
8mL2
2
2
• The electron wave functions ‘Ψn’ and the
corresponding probability density functions
‘ |Ψn|2 ’ for the ground and first two excited
states of an electron in a potential well are
shown in figure.
n 
2 / L sin
nx
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. since for a particle inside the
box, ‘ Ψ ’ cannot be zero, no quantum number
can be zero.
2.The energy ‘ E ’ depends on the sum of the
squares of the quantum numbers n1,n2 and n3
and no on their individual values.
3.Several combinations of the three quantum
numbers may give different wave functions, but
of the same energy value. such states and
energy levels are said to be degenerate.
A localized wave or wave packet:
A moving particle in quantum theory
Spread in position
Spread in momentum
Superposition of waves
of different wavelengths
to make a packet
Narrower the packet , more the spread in momentum
Basis of Uncertainty Principle
ILLUSTRATION OF MEASUREMENT OF ELECTRON
POSITION
Act of measurement
influences the electron
-gives it a kick and it
is no longer where it
was ! Essence of uncertainty
principle.