Quantum mechanics

1. Quantum Mechanics

2. Introduction
1.Stability of an atom
2.Spectral series of Hydrogen atom
3.Black body radiation
There are a few phenomenon which the classical mechanics
failed to explain.
Max Planck in 1900 at a meeting of German Physical
Society read his paper “On the theory of the Energy
distribution law of the Normal Spectrum”. This was the start
of the revolution of Physics i.e. the start of Quantum
Mechanics.

3. Quantum Mechanics
Quantum Physics extends that range to the region of small
dimensions.
It is a generalization of Classical Physics that includes
classical laws as special cases.
Just as ‘c’ the velocity of light signifies universal constant, the
Planck's constant characterizes Quantum Physics.
sec.10625.6
sec.1065.6
34
27
Jouleh
ergh
-
-
´=
´=

4. Quantum Mechanics
1.Photo electric effect
2.Black body radiation
3.Compton effect
4.Emission of line spectra
It is able to explain
The most outstanding development in modern science was
the conception of Quantum Mechanics in 1925. This new
approach was highly successful in explaining about the
behavior of atoms, molecules and nuclei.

5. Photo Electric Effect
The emission of electrons from a metal plate when illuminated
by light or any other radiation of any wavelength or frequency
(suitable) is called photoelectric effect. The emitted electrons
are called ‘photo electrons’.
V
Evacuated
Quartz
tube Metal
plate
Collecting
plate
Light
^^^^^^^^
A
+
_

6. Photo Electric Effect
Experimental findings of the photoelectric effect
1.There is no time lag between the arrival of light at the metal
surface and the emission of photoelectrons.
2.When the voltage is increased to a certain value say V
o
, the
photocurrent reduces to zero.
3.Increase in intensity increase the number of the
photoelectrons but the electron energy remains the same.
3I
Voltage
Photo
Current
2I
I
V
o

7. Photo Electric Effect
4. Increase in frequency of light increases the energy of the
electrons. At frequencies below a certain critical frequency
(characteristics of each particular metal), no electron is
emitted.
Voltage
Photo Current
v
1
v
2
v
3

8. Einstein’s Photo Electric Explanation
The energy of a incident photon is utilized in two ways
1.A part of energy is used to free the electron from the atom
known as photoelectric workfunction (W
o
).
2.Other part is used in providing kinetic energy to the emitted
electron .
÷
ø
ö
ç
è
æ 2
2
1
mv
2
2
1
mvWh
o
+=n
This is called Einstein’s photoelectric equation.

9. If , no photoelectric effect
max
KEWh
o
+=n
max
KEhh
o
+=nn
)(
max o
hKE nn-=
o
nn<
o
oo
hc
hW
l
n==
o
oo
o
A
eVWW
hc
)(
12400
==l

10. It is in form of . The graph with on y-axis
and on x-axis will be a straight line with slope
oo
hheV nn-=
o
V If is the stopping potential, then
)(
max o
hKE nn-=
e
h
e
h
V
o
o
nn
-=
cmxy +=
eh
o
V
n

11. Photons
Einstein postulated the existence of a particle called a photon,
to explain detailed results of photoelectric experiment.
l
n
hc
hE
p
==
Photon has zero rest mass, travels at speed of light
Explains “instantaneous” emission of electrons in photoelectric
effect, frequency dependence.

12. Compton Effect
When a monochromatic beam of X-rays is scattered from a
material then both the wavelength of primary radiation
(unmodified radiation) and the radiation of higher wavelength
(modified radiation) are found to be present in the scattered
radiation. Presence of modified radiation in scattered X-rays is
called Compton effect.
electron
scattered
photon
recoiled electron
nhE=
c
h
p
n
=
''nhE=
v
q
f fcosmv
fsinmv
incident
photon
q
n
cos
'
c
h
q
n
sin
'
c
h

13. From Theory of Relativity, total energy of the recoiled electron
with v ~ c is
22
cmKmcE
o+==
Similarly, momentum of recoiled electron is
22
cmmcK
o-=
2
22
2
1
cm
cv
cm
K
o
o
-
-
=
ú
ú
û
ù
ê
ê
ë
é
-
-
= 1
1
1
22
2
cv
cmK
o
22
1 cv
vm
mv
o
-
=

14. Now from Energy Conversation
fq
nn
cos
1
cos
'
22
cv
vm
c
h
c
h
o
-
+=
ú
ú
û
ù
ê
ê
ë
é
-
-
+= 1
1
1
'
22
2
cv
cmhh
onn (i)
From Momentum Conversation
(ii)along x-axis
fq
n
sin
1
sin
'
0
22
cv
vm
c
h
o
-
-= (iii)along y-axis
and

15. Rearranging (ii) and squaring both sides
fq
nn
2
22
22
2
cos
1
cos
'
cv
vm
c
h
c
h
o
-
=÷
ø
ö
ç
è
æ
- (iv)
fq
n
2
22
22
2
sin
1
sin
'
cv
vm
c
h
o
-
=÷
ø
ö
ç
è
æ
(v)
Rearranging (iii) and squaring both sides
Adding (iv) and (v)
22
22
2
2
22
1
cos
'2'
cv
vm
c
h
c
h
c
h
o
-
=-÷
ø
ö
ç
è
æ
+÷
ø
ö
ç
è
æ
q
nnnn
(vi)
From equation (i)
22
1
'
cv
cm
cm
c
h
c
h
o
o
-
=+-
nn

16. On squaring, we get
Subtracting (vi) from (vii)
(vii)
22
22
2
2
22
22
1
)'(2
'2'
cv
cm
hm
c
h
cm
c
h
c
h
o
oo
-
=-+-+÷
ø
ö
ç
è
æ
+÷
ø
ö
ç
è
æ
nn
nnnn
0)'(2)cos1(
'2
2
2
=-+-- nnq
nn
ohm
c
h
)cos1(
'2
)'(2
2
2
q
nn
nn -=-
c
h
hm
o
)cos1(
'
)'(
2
q
nn
nn -=-
c
h
m
o

17. But
is the Compton Shift.
l
n
c
=
)cos1(
''
11
q
llll
-=÷
ø
ö
ç
è
æ
-
h
cm
o
and
'
'
l
n
c
= So,
)cos1(
''
'
q
llll
ll
-=÷
ø
ö
ç
è
æ- h
cm
o
)cos1(' qlll -=D=-
cm
h
o
lD
q
It neither depends on the incident wavelength nor on the
scattering material. It only on the scattering angle i.e.
is called the Compton wavelength of the electron
and its value is 0.0243 Å.cm
h
o

18. Experimental Verification
Monochromatic
X-ray Source
photon
θ
Graphite
target
Bragg’s X-ray
Spectrometer
1. One peak is found at same
position. This is unmodified radiation
2. Other peak is found at higher
wavelength. This is modified signal of
low energy.
3. increases with increase in .lD q
lD

19. 0.0243 (1- cosθ) Å=-=D )cos1(ql
cm
h
o
=D
max
l
lD
So Compton effect can be observed only for radiation having
wavelength of few Å.
Compton effect can’t observed in Visible Light
is maximum when (1- cosθ) is maximum i.e. 2.lD
0.05 Å
For 1Å ~ 1%
lD=l
=l
For 5000Å ~ 0.001% (undetectable)

20. Pair Production
When a photon (electromagnetic energy) of sufficient
energy passes near the field of nucleus, it materializes into
an electron and positron. This phenomenon is known as pair
production.
In this process charge, energy and momentum remains
conserved prior and after the production of pair.
Photon
Nucleus (+ve)
-
e
+
e

21. The rest mass energy of an electron or positron is 0.51
MeV (according to E = mc
2
).
The minimum energy required for pair production is 1.02
MeV.
Any additional photon energy becomes the kinetic energy
of the electron and positron.
The corresponding maximum photon wavelength is 1.2 pm.
Electromagnetic waves with such wavelengths are called
gamma rays . )(g

22. Pair Annihilation
When an electron and positron interact with each other due
to their opposite charge, both the particle can annihilate
converting their mass into electromagnetic energy in the
form of two - rays photon.g
gg+®+
+-
ee
Charge, energy and momentum are again conversed. Two
- photons are produced (each of energy 0.51
MeV plus half the K.E. of the particles) to conserve the
momentum.
g

23. From conservation of energy
gn
2
2cmh
o
=
Pair production cannot occur in empty space
In the direction of motion of the photon, the momentum is
conserved if
q
n
cos2p
c
h
=
q
qchn
qcosp
qcosp
p
p
-
e
+
e
here m
o
is the rest mass and
22
11 cv-=g

24. gvmp
o=
Momentum of electron and positron is
(i)
1cos£q1<
c
vBut
qn cos2cph=
Equation (i) now becomes
qgn cos2cvmh
o
=
qgn cos2
2
÷
ø
ö
ç
è
æ
=
c
v
cmh
o
and
gn
2
2cmh
o
<

25. But conservation of energy requires that
gn
2
2cmh
o
=
Hence it is impossible for pair production to conserve both
the energy and momentum unless some other object is
involved in the process to carry away part of the initial
photon momentum. Therefore pair production cannot occur
in empty space.

26. Wave Particle Duality
Light can exhibit both kind of nature of waves and particles
so the light shows wave-particle dual nature.
In some cases like interference, diffraction and polarization
it behaves as wave while in other cases like photoelectric
and compton effect it behaves as particles (photon).

27. De Broglie Waves
Not only the light but every materialistic particle such as
electron, proton or even the heavier object exhibits wave-
particle dual nature.
De-Broglie proposed that a moving particle, whatever its
nature, has waves associated with it. These waves are
called “matter waves”.
Energy of a photon is
nhE=
For a particle, say photon of mass, m
2
mcE=

28. Suppose a particle of mass, m is moving with velocity, v then
the wavelength associated with it can be given by
hvmc=
2
l
hc
mc=
2
mc
h
=l
mv
h
=l
p
h
=l or
(i) If means that waves are associated with
moving material particles only.
¥=Þ=l0v
(ii) De-Broglie wave does not depend on whether the moving
particle is charged or uncharged. It means matter waves are
not electromagnetic in nature.

29. Wave Velocity or Phase Velocity
When a monochromatic wave travels through a medium,
its velocity of advancement in the medium is called the
wave velocity or phase velocity (V
p
).
k
V
p
w
=
where is the angular frequency

and is the wave number.
pnw2=
l
p2
=k

30. Group Velocity
So, the group velocity is the velocity with which the energy
in the group is transmitted (V
g
).
dk
d
V
g
w
=
The individual waves travel “inside” the group with their
phase velocities.
In practice, we came across pulses rather than
monochromatic waves. A pulse consists of a number of
waves differing slightly from one another in frequency.
The observed velocity is, however, the velocity with which
the maximum amplitude of the group advances in a
medium.

31. Relation between Phase and Group Velocity
dk
d
V
g
w
= )(
p
kV
dk
d
=
dk
dV
kVV
p
pg +=
( )lpl
p
2
2
d
dV
VV
p
pg +=
( )ll1
1
d
dV
VV
p
pg +=

32. l
l
d
dV
VV
p
pg -=
÷
ø
ö
ç
è
æ
-
+=
l
l
l
d
dV
VV
p
pg
2
1
1
l
l
d
dV
p
So, is positive generally (not always).
pg
VV<Þ
In a Dispersive medium V
p
depends on frequency
¹
k
w
generally
i.e. constant

33. l
l
d
dV
VV
p
pg -=
0=Þ
ld
dV
p
pg
VV=Þ
In a non-dispersive medium ( such as empty space)
=
k
w
constant p
V=

34. Phase Velocity of De-Broglie’s waves
According to De-Broglie’s hypothesis of matter waves
mv
h
=l
wave number
h
mv
k
p
l
p22
== (i)
If a particle has energy E, then corresponding wave will
have frequency
h
E
=n
then angular frequency will be
h
Ep
pnw
2
2==

35. Dividing (ii) by (i)
(ii)
h
mc
2
2p
w=
mv
h
h
mc
k p
pw
2
2
2
´=
v
c
V
p
2
=
But v is always < c (velocity of light)
(i) Velocity of De-Broglie’s waves (not acceptable) cV
p>
(ii) De-Broglie’s waves will move faster than the
particle velocity (v) and hence the waves would left the
particle behind.
)(
pV

36. Group Velocity of De-Broglie’s waves
The discrepancy is resolved by postulating that a moving
particle is associated with a “wave packet” or “wave
group”, rather than a single wave-train.
A wave group having wavelength λ is composed of a
number of component waves with slightly different
wavelengths in the neighborhood of λ.
Suppose a particle of rest mass m
o
moving with velocity v
then associated matter wave will have
h
mc
2
2p
w= and
h
mv
k
p2
= where
22
1 cv
m
m
o
-
=

37. and
On differentiating w.r.t. velocity, v
22
2
1
2
cvh
cm
o
-
=
p
w
22
1
2
cvh
vm
k
o
-
=
p
( )
2
3
22
1
2
cvh
vm
dv
d
o
-
=
pw
(i)
( )
2
3
22
1
2
cvh
m
dv
dk
o
-
=
p
(ii)

38. Wave group associated with a moving particle also
moves with the velocity of the particle.
o
o
m
vm
dk
dv
dv
d
p
pw
2
2
.=
Dividing (i) by (ii)
gVv
dk
d
==
w
Moving particle wave packet or wave groupº

39. Davisson & Germer experiment of electron
diffraction
•If particles have a wave nature, then under appropriate
conditions, they should exhibit diffraction
•Davisson & Germer measured the wavelength of electrons
•This provided experimental confirmation of the matter waves
proposed by de Broglie

40. Davisson and Germer Experiment

41. 0f=
0
90f=
Current vs accelerating voltage has a maximum (a bump or
kink noticed in the graph), i.e. the highest number of electrons
is scattered in a specific direction.
The bump becomes most prominent for 54 V at φ ~ 50°
I
n
c
id
e
n
t

B
e
a
m

42. According to de Broglie, the wavelength associated with an
electron accelerated through V volts is
o
A
V
28.12
=l
Hence the wavelength for 54 V electron
o
A67.1
54
28.12
==l
From X-ray analysis we know that the nickel crystal acts as a
plane diffraction grating with grating space d = 0.91 Å

43. o
oo
65
2
50180
=
÷
÷
ø
ö
ç
ç
è
æ -
=q
Here the diffraction angle, φ ~ 50°

The angle of incidence relative to the family of Bragg’s plane

44. From the Bragg’s equation
which is equivalent to the λ calculated by de-Broglie’s
hypothesis.
ql sin2d=
o
o
o
AA 65.165sin)91.0(2 =´´=l
It confirms the wavelike nature of electrons

45. Electron Microscope: Instrumental Application
of Matter Waves

46. Resolving power of any optical instrument is proportional to the
wavelength of whatever (radiation or particle) is used to
illuminate the sample.
An optical microscope uses visible light and gives 500x
magnification/200 nm resolution.

Fast electron in electron microscope, however, have much
shorter wavelength than those of visible light and hence a
resolution of ~0.1 nm/magnification 1,000,000x can be achieved
in an Electron Microscope.

47. Heisenberg Uncertainty Principle
It states that only one of the “position” or “momentum” can be
measured accurately at a single moment within the instrumental
limit.
It is impossible to measure both the position and momentum
simultaneously with unlimited accuracy.
or
uncertainty in position
uncertainty in momentum
®Dx
®D
x
p
then
2

³DD
xpx
p2
h
=\
The product of & of an object is greater than or equal to
2

x
pDxD

48. If is measured accurately i.e. xD 0®Dx ¥®DÞ
x
p
Like, energy E and time t.
2

³DDtE
2

³DDqL
The principle applies to all canonically conjugate pairs of quantities in
which measurement of one quantity affects the capacity to measure
the other.
and angular momentum L and angular position θ

49. Determination of the position of a particle by a microscope
i
Incident
Photon
Scattered
Photon
Recoiled electron
Suppose we want to determine accurately the position and
momentum of an electron along x-axis using an ideal microscope
free from all mechanical and optical defects.
The limit of resolution of the
microscope is
i
x
sin2
l
=D
here i is semi-vertex angle of the
cone of rays entering the objective
lens of the microscope.
is the order of uncertainty in the
x-component of the position of the
electron.
xD

50. The scattered photon can enter the microscope anywhere between
the angular range +i to –i.
We can’t measure the momentum of the electron prior to illumination.
So there is uncertainty in the measurement of momentum of the
electron.
The momentum of the scattered photon is (according to de-Broglie)
l
h
p=
Its x-component can be given as
i
h
p
x sin
2
l
=D
The x-component of the momentum of the recoiling electron has the
same uncertainty, (conservation of momentum)
x
pD

51. The product of the uncertainties in the x-components of position and
momentum for the electron is
i
h
i
px
x sin
2
sin2
.
l
l
´=DD
This is in agreement with the uncertainty relation.
2
.

>=DD hpx
x

52. Order of radius of an atom ~ 5 x10
-15
m
then
If electron exist in the nucleus then
Applications of Heisenberg Uncertainty Principle
(i) Non-existence of electron in nucleus
mx
15
max
105)(
-
´=D
2

³DD
xpx
2
)()(
minmax

=DD
xpx
120
min ..101.1
2
)(
--
´=
D
=D smkg
x
p
x

MeVpcE 20== \ relativistic

53. Thus the kinetic energy of an electron must be greater than 20
MeV to be a part of nucleus
Thus we can conclude that the electrons cannot be present
within nuclei.
Experiments show that the electrons emitted by certain unstable
nuclei don’t have energy greater than 3-4 MeV.

54. But
Concept of Bohr Orbit violates Uncertainty Principle
m
p
E
2
2
=
2
.

³DDpx
m
pp
E
D
=D
m
pmvD
= p
t
x
D
D
D
=
pxtE DD=DD ..
2
.

³DDtE

55. According to the concept of Bohr orbit, energy of an electron in a
orbit is constant i.e. ΔE = 0.
2
.

³DDtE
¥®DÞt
All energy states of the atom must have an infinite life-time.
But the excited states of the atom have life–time ~ 10
-8
sec.
The finite life-time Δt gives a finite width (uncertainty) to the energy
levels.

56. Two-slit Interference Experiment
Laser
Source
Slit
Slit
Detector
Rate of photon arrival = 2 x 10
6
/sec
Time lag = 0.5 x 10
-6
sec
Spatial separation between photons = 0.5 x 10
-6
c = 150 m
1 meter

57. –Taylor’s experiment (1908): double slit experiment with very dim
light: interference pattern emerged after waiting for few weeks
–interference cannot be due to interaction between photons, i.e.
cannot be outcome of destructive or constructive combination of
photons
Þ interference pattern is due to some inherent property of each
photon - it “interferes with itself” while passing from source to
screen
–photons don’t “split” –
light detectors always show signals of same intensity
– slits open alternatingly: get two overlapping single-slit diffraction
patterns – no two-slit interference
–add detector to determine through which slit photon goes:
Þ no interference
– interference pattern only appears when experiment provides
no means of determining through which slit photon passes

58. Double slit experiment – QM interpretation
– patterns on screen are result of distribution of photons
– no way of anticipating where particular photon will strike
– impossible to tell which path photon took – cannot assign
specific trajectory to photon
– cannot suppose that half went through one slit and half through
other
– can only predict how photons will be distributed on screen (or
over detector(s))
– interference and diffraction are statistical phenomena
associated with probability that, in a given experimental setup, a
photon will strike a certain point
– high probability Þ bright fringes
– low probability Þ dark fringes

59. Double slit expt. -- wave vs quantum
• pattern of fringes:
–Intensity bands due to
variations in square of
amplitude, A
2
, of resultant
wave on each point on
screen
• role of the slits:
– to provide two coherent
sources of the secondary
waves that interfere on the
screen
• pattern of fringes:
–Intensity bands due to
variations in probability, P,
of a photon striking points
on screen
• role of the slits:
– to present two potential
routes by which photon can
pass from source to screen
wave theory quantum theory

60. Wave function
yyy *||
2
=
The quantity with which Quantum Mechanics is concerned is the
wave function of a body.
|Ψ|
2
is proportional to the probability of finding a particle at a
particular point at a particular time. It is the probability density.
Wave function, ψ is a quantity associated with a moving particle. It
is a complex quantity.
Thus if iBA+=y iBA-=*y
222222
*|| BABiA +=-==Þ yyy
then
ψ is the probability amplitude.

61. Normalization
td
|Ψ|
2
is the probability density.
The probability of finding the particle within an element of volume
tyd
2
||
Since the particle is definitely be somewhere, so
1||
2
=ò
¥
¥-
tyd
A wave function that obeys this equation is said to be normalized.
\Normalization

62. Properties of wave function
1.It must be finite everywhere.
If ψ is infinite for a particular point, it mean an infinite large
probability of finding the particles at that point. This would
violates the uncertainty principle.
2. It must be single valued.
If ψ has more than one value at any point, it mean more than
one value of probability of finding the particle at that point
which is obviously ridiculous.
3. It must be continuous and have a continuous first derivative
everywhere.
zyx ¶
¶
¶
¶
¶
¶ yyy
,, must be continuous
4. It must be normalizable.

63. Schrodinger’s time independent wave equation
One dimensional wave equation for the waves associated with a
moving particle is
From (i)
ψ is the wave amplitude for a given x.
where
A is the maximum amplitude.
λ is the wavelength
y
l
py
2
2
2
2
4
-=
¶
¶
x
(ii)
)0,( and ),(
)(
2
)( x
i
pxEt
i
AetxAetx
l
p
yy ===
-
-


64. vm
h
o
=l
2
22
2
1
h
vm
o
=Þ
l
2
2
2
1
2
h
vmm
oo ÷
ø
ö
ç
è
æ
=
22
21
h
Km
o
=
l
where K is the K.E. for the non-relativistic case
(iii)
Suppose E is the total energy of the particle
and V is the potential energy of the particle
)(
21
22
VE
h
m
o
-=
l

65. This is the time independent (steady state) Schrodinger’s wave
equation for a particle of mass m
o
, total energy E, potential
energy V, moving along the x-axis.
If the particle is moving in 3-dimensional space then
Equation (ii) now becomes
y
py
)(2
4
2
2
2
2
VEm
hx
o--=
¶
¶
0)(
2
22
2
=-+
¶
¶
y
y
VE
m
x
o

0)(
2
22
2
2
2
2
2
=-+
¶
¶
+
¶
¶
+
¶
¶
y
yyy
VE
m
zyx
o


66. For a free particle V = 0, so the Schrodinger equation for a
free particle
0
2
2
2
=+Ñ yy E
m
o

0)(
2
2
2
=-+Ñ yy VE
m
o

This is the time independent (steady state) Schrodinger’s wave
equation for a particle in 3-dimensional space.

67. Schrodinger’s time dependent wave equation
)(pxEt
i
Ae
-
-
=

y
Wave equation for a free particle moving in +x direction is
(iii)
y
y
2
2
2
2

p
x
-=
¶
¶
where E is the total energy and p is the momentum of the particle
Differentiating (i) twice w.r.t. x
(i)
2
2
22
x
p
¶
¶
-=Þ
y
y (ii)
Differentiating (i) w.r.t. t
y
y

iE
t
-=
¶
¶
t
iE
¶
¶
=Þ
y
y

68. For non-relativistic case
Using (ii) and (iii) in (iv)
(iv)
y
yy
V
xmt
i +
¶
¶
-=
¶
¶
2
22
2


E = K.E. + Potential Energy
txV
m
p
E
,
2
2
+=
yyy V
m
p
E +=Þ
2
2
This is the time dependent Schrodinger’s wave equation for a
particle in one dimension.

69. Linearity and Superposition
2211 yyy aa+=
If ψ
1
and ψ
2
are two solutions of any Schrodinger equation of a
system, then linear combination of ψ
1
and ψ
2
will also be a solution
of the equation..
Here are constants
Above equation suggests:
21&aa
is also a solution
(i) The linear property of Schrodinger equation
(ii) ψ
1
and ψ
2
follow the superposition principle

70. 21yyy +® Then
Total probability will be
2
21
2
|||| yyy +==P
due to superposition principle
)()(
21
*
21 yyyy ++=
))((
21
*
2
*
1 yyyy ++=
1
*
22
*
12
*
21
*
1 yyyyyyyy +++=
1
*
22
*
121 yyyy+++= PPP
21PPP +¹
Probability density can’t be added linearly
If P
1
is the probability density corresponding to ψ
1
and P
2
is the
probability density corresponding to ψ
2

71. Expectation values
dxxfò
¥
¥-
=
2
||)(y
Expectation value of any quantity which is a function of ‘x’ ,say f(x)
is given by
for normalized ψ
Thus expectation value for position ‘x’ is
><)(xf
dxxò
¥
¥-
=
2
||y><x
Expectation value is the value of ‘x’ we would obtain if we
measured the positions of a large number of particles described by
the same function at some instant ‘t’ and then averaged the
results.

72. dxxò
=
1
0
2
||y
Solution
><x
1
0
4
2
4
ú
û
ù
ê
ë
é
=
x
a
Q. Find the expectation value of position of a particle having wave
function ψ = ax between x = 0 & 1, ψ = 0 elsewhere.
dxxaò
=
1
0
32
><x
4
2
a
=

73. Operators
y
y
p
i
x
=
¶
¶
(Another way of finding the expectation value)
For a free particle
An operator is a rule by means of which, from a given function
we can find another function.
)(pxEt
i
Ae
-
-
=

y
Then
Here
xi
p
¶
¶
=

^
is called the momentum operator
(i)

74. y
y
E
i
t
-=
¶
¶
Similarly
Here
t
iE
¶
¶
=
^
is called the Total Energy operator
(ii)
Equation (i) and (ii) are general results and their validity is the
same as that of the Schrodinger equation.

75. U
ximt
i +÷
ø
ö
ç
è
æ
¶
¶
=
¶
¶
2
2
1

If a particle is not free then
This is the time dependent Schrodinger equation
^^^
..UEKE +=
^
^
2
^
2
U
m
p
E
o
+=Þ
UU=\
^
U
xmt
i +
¶
¶
-=
¶
¶
2
22
2


y
yy
U
xmt
i +
¶
¶
-=
¶
¶
2
22
2



76. If Operator is Hamiltonian
Then time dependent Schrodinger equation can be written as
U
xm
H +
¶
¶
-=
2
22
^
2

yyEH=
^
This is time dependent Schrodinger equation in Hamiltonian
form.

77. Eigen values and Eigen function
Schrodinger equation can be solved for some specific values of
energy i.e. Energy Quantization.
yyaa=
^
Suppose a wave function (ψ) is operated by an operator ‘α’ such
that the result is the product of a constant say ‘a’ and the wave
function itself i.e.
The energy values for which Schrodinger equation can be solved
are called ‘Eigen values’ and the corresponding wave function are
called ‘Eigen function’.
then
ψ is the eigen function of
a is the eigen value of
^
a
^
a

78. Q. Suppose is eigen function of operator then find
the eigen value.
The eigen value is 4.
Solution.

x
e
2
=y
2
2
dx
d
2
2
^
dx
d
G=
2
2
^
dx
d
G
y
y= )(
2
2
2
x
e
dx
d
=
x
eG
2
^
4=y
yy4
^
=G

79. Particle in a Box
Consider a particle of rest mass m
o
enclosed in a one-dimensional
box (infinite potential well).
Thus for a particle inside the box Schrodinger equation is
Boundary conditions for Potential
V(x)=
0 for 0 < x < L
¥
{
for 0 > x > L
Boundary conditions for ψ
Ψ =
0 for x = 0
{
0 for x = L
0
2
22
2
=+
¶
¶
y
y
E
m
x
o

x = 0 x = L
¥=V ¥=V
particle
0=V
0=\V inside (i)

80. Equation (i) becomes
p
h
=l
k
p2
=

p
k=Þ

Em
o2
=
2
22

Em
k
o
=Þ
(k is the propagation constant)
(ii)
0
2
2
2
=+
¶
¶
y
y
k
x
(iii)
General solution of equation (iii) is
kxBkxAx cossin)( +=y (iv)

81. Equation (iv) reduces to
Boundary condition says ψ = 0 when x = 0
(v)
0.cos0.sin)0( kBkA +=y
1.00 B+= 0=ÞB
kxAx sin)(=y
Boundary condition says ψ = 0 when x = L
LkAL .sin)(=y
LkA .sin0=
0¹A 0.sin=Þ Lk
pnLksin.sin=Þ

82. Put this in Equation (v)
(vi)
L
n
k
p
=
L
xn
Ax
p
y sin)(=
When n # 0 i.e. n = 1, 2, 3…., this gives ψ = 0 everywhere.
pnkL=
Put value of k from (vi) in (ii)
2
22

Em
k
o
=
2
2
2

Em
L
n
o
=÷
ø
ö
ç
è
æp

83. Where n = 1, 2, 3….
Equation (vii) concludes
o
m
k
E
2
22

=Þ
2
22
8Lm
hn
o
= (vii)
1. Energy of the particle inside the box can’t be equal to zero.
The minimum energy of the particle is obtained for n = 1
2
2
1
8Lm
h
E
o
= (Zero Point Energy)
If momentum i.e.0
1®E 0® 0®Dp
¥®DÞx
But since the particle is confined in the box of
dimension L.
Lx=D
max

84. Thus zero value of zero point energy violates the
Heisenberg’s uncertainty principle and hence zero value is
not acceptable.
2. All the energy values are not possible for a particle in
potential well.
Energy is Quantized
3. E
n
are the eigen values and ‘n’ is the quantum number.
4. Energy levels (E
n
) are not equally spaced.
n = 1
n = 3
n = 2
3
E
1E
2E

85. Using Normalization condition
L
xn
Ax
n
p
y sin)(=
1sin
0
22
=ò
dx
L
xn
A
L
p
1|)(|
2
=ò
¥
¥-
dxx
ny
1
2
2
=÷
ø
ö
ç
è
æL
A
L
A
2
=Þ
The normalized eigen function of the particle are
L
nx
L
x
n
p
y sin
2
)(=

86. Probability density figure suggest that:
1. There are some positions (nodes) in the box that will never be
occupied by the particle.
2. For different energy levels the points of maximum probability
are found at different positions in the box.
|ψ
1
|
2
is maximum at L/2 (middle of the box)
|ψ
2
|
2
is zero L/2.

87. Eigen function
zyx
yyyy=
L
zn
L
yn
L
xn
AAA
zyx
zyx
ppp
y sinsinsin=
L
zn
L
yn
L
xn
L
zyx ppp
y sinsinsin
2
3
÷
÷
ø
ö
ç
ç
è
æ
=
2
2
222
8
)(
mL
h
nnnE
zyx ++=
Particle in a Three Dimensional Box
zyx
EEEE ++= Eigen energy