Quantum mechanics for Engineering Students

Engineering Physics Course Materials by Praveen N Vaidya, SDMCET Dharwad.
Quantum Mechanics
Quantum mechanics (QM, or quantum theory) is
a branch of physical science based on the
Quantum theory of radiation and de-Broblie
concept of matter waves and deals with the
behavior of matter and energy on the scale of
atoms and subatomic particles / matter waves.
Within the field of engineering, Quantum
mechanics is the foundation for several related
disciplines, including crystal structure,
conductivity in material and band structure
determination. Quantum mechanics plays an
important in the research and development of
material technology (Material engineering,
nanotechnology and smart materials)
Comparison between de-Broglie waves
and monochromatic waves:
de-Broglie waves
• Formed by group of waves of slight different
wavelengths. Group velocity concept is used.
• The localization of matter waves is explained
by taking concept of standing wave pattern
given by the superposition of no. of waves.
• Wavelength and momentum (λ = h/p) of
de-Broglie waves are not well defined as
wave length of de-Broglie wave is given by
the average value wavelengths of number of
waves superposed with each other.
• As de-Broglie waves are localized the position
of the particle can be identified with more
accuracy, hence position is well defined.
On other hand, Monochromatic waves are
• Single wave or wave train.
• Not localized but progressive or continuous.
• Hence wave length and momentum are well
defined.
• Giving poor indication of identifying the exact
position of particle due to progressive.
From the above discussion it is clear that,
superposition of large number of waves leads to
more localization and better identification of
position of particle but poor identification of wave
length and momentum.
Fever the number of waves leads to less localization
and poor identification of position of particle and
better identification of momentum and
wavelength.
Therefore it needs certain amount of compromise in
wave representation of matter in terms of defining
the momentum and position of the particle
In other words with some amount of uncertainty
only we can simultaneously measure both the
momentum and position of particle.
Heisenberg’s Uncertainty Principle (HUP)
The uncertainty in finding the momentum (p) and
position (x) of particle is properly explained by the
theory developed by German Physicist Werner
1

Heisenberg and is termed as Heisenberg
Uncertainty Principle
Statement: “It is impossible to determine both
position and momentum of a particle
simultaneously and accurately. The product of
uncertainty involved in the determination of
position and momentum simultaneously is
greater or equal to h/2π”
Explanation: The measurement of both position
and momentum of a moving particle
simultaneously cannot be done without involving
some error. If accuracy in the measurement of
position increases (decrease in error) in turn,
accuracy in the measurement of momentum
decreases (increase in error) and vice versa. This
uncertainty in the measurement is due to inherent
wave property of matter but not due to error in
measurement or foulty instrument. If ∆x is error
(uncertainty) involved in the position and ∆px is
the error involved in the measurement of
momentum then according to Heisenberg’s
Uncertainty Principle (HUP)
∆ x ∆ px ≥ h / 2π
Significance of HUP
We know that according to classical mechanics,
if we know the position and momentum of a
particle at some instant of time, its position and
momentum at any later instant of time can be
determined accurately. But according to HUP, if
we try to measure position accurately, error in
the measurement of momentum increases and
vise-versa. Thus in quantum mechanics there is no
place for the word ‘exactness’, and is replaced by
the word ‘probability’. HUP implies that an event,
which is impossible to occur according to classical
physics, has finite probability of occurrence
according to quantum mechanics.
The relation is universal and holds for all the
canonically conjugate physical quantities like
position and momentum (∆x & ∆p), energy and time
(∆E & ∆t), anguler momentum and angle (∆L &
∆Ө) etc.
Τhe uncertainty between energy and time is,
(∆E x ∆t) > h/2π
Τhe uncertainty between angular momentum
and angular displacement.
∆L x ∆ θ > h/2π
Illustrations of Heisenberg Uncertainty
Principle.
1. The non-existence of electron in the
nucleus.
The non existence of electron in nucleus is proved
by comparing the energy needed to an electron to
exist in nucleus and energy of β-rays (electrons)
emitted during radioactivity.
The diameter of nucleus of any atom is of the order
of 10-14
m. If any electron is confined within the
nucleus then the heighest uncertainty in the
measurement of its position (∆x) must not greater
than 10-14
m.
i.e., ∆ x ≤ 10 -14
m.
2

According to Heisenberg’s uncertainty principle,
equation
∆ x ∆p > h / 2π
Then its uncertainty in the momentum is given
by
This is the uncertainty in the momentum of
electron.
Non-Relativistic method:
The kinetic energy of the electron is given by,
2m/pE 2
=
Momentum of the electron =1.055X10-20
kg.m/s
m is the mass of the electron = 9.11X10-31
kg
∴ E =
)(9.11X102
)(1.055X10
31-
2-20
×
= 0.0610X10-9
J
= 3.8X108
eV
This is kinetic energy given by the required by
electron to stay inside the nucleus.
But the experimental results on β decay show
that the maximum kinetic an electron can have
when it is confined with in the nucleus is of the
order of 3 – 4 Mev. Therefore the free electrons
cannot exist within the nucleus.
Relativistic method:
According to the theory of relativity the energy
of a particle is given by
E = mc2
=
)/cv-(1
cm
22
2
0
Where m0 is the particle’s rest mass and m is the
mass of the particle with velocity v.
Squaring the above equation we get,
Or
2
2
4
2
1
c
v
cm
E o
−
=
or
22
62
2
vc
cm
E o
−
= --- 1
Similarly Momentum of the particle
== vmp
)/cv-(1
vm
22
0
22
422
22
vc
cvm
cp o
−
= ------------------------------2
Eqn (1) – eqn (2) gives
)(
)(
22
22
42
222
vc
vc
cm
cpE o
−
−
=−
)( 22222
cmpcE o+= ---------------3
the momentum p =1.055X10-20
kg.m/s
p2
= 1.113x10-40
and the rest mass of electron mo = 9.11X10-31
kg
mo
2
c2
= (9.1 x 10-31
)2
(3X108
)2
= 0.0007x10-40
Substitute in equation (3) above
E2
> (3X108
)2
(1.113 X 10-40
+ 0.0007X 10-40
)
E > 3.164 x 10-12
J
Or E > 20 MeV
But the experimental results on β-decay shows that
the maximum kinetic an electron can have when it
is confined with in the nucleus is of the order of 3- 4
Mev. Therefore the electrons cannot exist within the
nucleus.
3

2. Width of spectral lines (Natural
Broadening)
In the study of transitions in atomic spectra, and
indeed in any type of spectroscopy, one must be
aware that those transitions are not precisely
"sharp". There is always a finite width to the
observed spectral lines.
One source of broadening is the "natural line
width" which arises from the uncertainty in
energy of the states involved in the transition.
The uncertainty principle provides a tool for
characterizing the very short-lived transitions
produced in high energy collisions in
accelerators.
The uncertainty principle in the form
suggests that, for particles with extremely short
lifetime, there will be a significant uncertainty in
the measured energy.
A typical lifetime for an atomic energy state is
about 10-8
seconds, corresponding to a natural
line width of about 6.6 x 10-8
eV.
From above discussion it is clear that the if the
life time of an atom in excited state is very small
spectral width is large or spreaded or there
involve certain uncertainty in measuring the
wavelength of spectral line. Similarly if the spectral
lines go more and more sharp there exists
uncertainty in the measurement of time
Wave function (ψ)
A wave function is a mathematical model, which
depicts the behavior of the electron like microscopic
bodies in moton depending on its position and time.
Just like a periodically variable quantity
(Electric and magnetic vectors in Electromagnetic
wave) along direction of vibration of wave in
mechanical or electromagnetic waves, the wave
function in de-Broglie matter waves represents
finding the position and motion of particle in wave
packet. More over the wave function is a complex
quantity it contains both real and imaginary parts.
The wave function is represented by ψ (sigh) and is
given by,
[ ])/(exp vxtio −−= ϖψψ
or [ ])(exp kxtio −−= ϖψψ
But wave function can not determine the exact
location of the particle but it represents only
probability of finding particle.
Physical significance of ψ
Although ψ represents a tool for probability of
finding the particle, it can not do this job
individually because of following reason.
The probability of finding a particle some where at
a given time has two extream limits, 1 for presence
of particle and 0 that of absence. But if amplitude
of wave function is negative, then finding the
4

probability is meaningless; and it is a complex
number of form (a+ib). Hence ψ has no direct
physical significance.
But square of the wave function i.e. [ψ]2
has
definite some meaning. It is just like square of
amplitude of monochromatic wave which is
equals to intensity. By this analogy we may
interpret that [ψ]2
as the probability density of
the particle associated with the de broglie wave
Probability density
it is known that ψ represents probability of
finding the particle at a point x, y, z and at an
instant t is proportion al to [ψ]2
Since ψ is complex quantity it has both real and
imaginary part i.e.
ψ(x,y,z.t) = (a+ib),
where a and b are the real functions of variables
(x,y,z,t).
One can get the probability density by
multiplying it with its complex conjugate i.e.
ψ*
(x,y,z.t) = (a-ib)
Then, ψ(x,y,z.t) ψ*
(x,y,z.t) = (a+ib) (a-ib)
i.e. P = [ψ(x,y,z.t)]2
= a2
+ b2
which is real and positive and represents the
probability density
The probability density is probability of finding
the particle in an elemental volume in three
dimensions is given by
P dV= [ψ]2
dV
Similarly probability of finding particle in de-
Broglie linear wave (along length of x) is given by
P dx= [ψ]2
dx
probability of finding particle between two points x1
and x2 is given by
dxx
x
x
2
2
1
)(∫ψ
Normalization of wave function
If the probability of finding a particle is maximum
i.e. 1 any where in whole space (x1 = - ∞ and x2 = +
∞) in three dimension and any where along the
length in one dimension, wave function is said to be
normalized.
Along one dimension the probability of finding a
particle any where in whole space is given by
1)(
2
=∫
+∞
∞−
dxxψ
the above condition is said to be normalized
Properties of wave function:
• Wave function ψ is single valued. i.e. it
represents the characteristic properties of one
particle.
• Wave function ψ is continuous value either
positive or negative.
• ψ is finite for all values of dx, dy, dz.
• ψ also called as probability amplitude and have
no direct physical significance.
• If ψ multiplied with its complex conjugate ψ*
gives the probability density have the direct
significance.
• ψ Vanish at he boundaries
5

One Dimensional Time independent
Schrödinger wave equation
Because particles could be described as
waves, later in 1925 Erwin Schrödinger
analyzed what an electron would look like as a
wave around the nucleus of the atom. Using this
model, he formulated his equation for particle
waves. He treated everything as waves whereby
each electron has its own unique wave function.
Time independent Schrödinger wave
equation
Consider a moving particle, then de-Broglie
wave traveling along x-axis can be represented
by the equation
Where Ψ(x,t) is called wave function. The
equation 2 can be written as
Differentiating equation 2 w.r.t. x twice we get.
Differentiating equation 2 w.r.t. t twice we get
General wave equation is given by
Substituet (3) and (4) in eqn.(5)
The total energy of the particle is given by
Where V is the potential energy
-------------- 6
Consider a of m is mass of particle, v is its velocity
along the +ve x-axis. Then according to de-
Broglie’s hypothesis, wavelength of the wave
associated with the particle is given by
or
Substitute equation (7) in (6) we have,
Substituting in equation 6 we get
The above equation is called one-dimensional
Schrödinger’s wave equation.
6
______ 4
______ 5
______1
______ 2
______ 3
7
h
−−−−−=
λ
mv

In three dimensions the Schrödinger wave
equation becomes
Nature of Eigenvalues and
Eigenfunctions
Once the set up of Schrödinger wave equation is
done, one can get a wave function of a system by
solving the equation. Any physical system that
characterized by its own position, energy,
momentum etc. can be completely described
with the help of the wave function ψ of the
system.
But Schrödinger wave equation, which is a
second order differential equation, has multiple
solutions. All solutions may not represent the
physical system under consideration. Those
wave function, which represent the physical
system under consideration are acceptable, are
called Eigenfunctions.
A wavefunction ψ can be acceptable as eigen
function if it satisfies the following conditions.
1. ψ should be single valued and finite every
where.
2. ψ and its first derivatives with respect to its
variables are continuous everywhere.
The solution of the Schrödinger wave equation
gives the wavefunction ψ of a system. With the
knowledge of ψ we can determine the Energy of
the given system. Since all wave functions are not
acceptable, corresponding values of energies are
also not acceptable. Only those values of energy
corresponding to the Eigenfunctions are acceptable,
and are called Eigenvalues.
Applications of Schrodinger wave equation:
1. Eigen Values and Eigen Functions of a
Particle in a potential well of infinite height
(Particle in box)
The potential well is an imaginary concept, and
represents the motion of a particle trapped under the
strong potential.
For example the for electron moving around
the nucleus has constant potential (equipotential
surface) and motion across the energy bands is
influence by potential energy of the respective
orbits and is represented by potential well.
Consider an electron of mass m, moving
along positive x-axis between two walls of infinite
height, one located at x=0 and another at x=a. Let
potential energy of the electron is assumed to be
zero in the region in-between the two walls and
infinity in the region beyond the walls.
Region beyond the walls:
The Schrödinger’s wave equation representing the
motion of the particle in the region beyond the two
walls is given by,
7

The only possible solution for the above
equation is ψ=0. Hence the probability of
finding the particle in the region x<0 and x>a is
Zero . i.e., particle cannot be found in region
beyond the walls.
Region between the two walls:
The Schrödinger’s wave equation representing
the motion of the particle in the region between
the two walls is given by
Solution of the equation 1 is of the form
ψ = Asinαx + Bcosαx -------- 2
Where A and B are constants to be determined.
Since particle cannot be found inside the walls
The equations are called boundary conditions.
Using the I boundary condition in equation 2, we
get
Therefore equation 2 becomes
-----------------3
Using condition II in equation 3 we get
Therefore correct solution of the equation 1 can be
written as
-------------------4
The above equation represents Eigenfunctions.
Where n=1,2,3,..(n=0 is not acceptable because, for
n = 0 the wave function ψ becomes zero for all
values of x. (Then particle cannot be found
anywhere). Substituting for ‘a’ in equation 1a we
get
8

Therefore energy Eigenvalues are represented by
the equation
Where n=1, 2, 3, . . .,
It is clear from the above equation that particle
can have only desecrated values of energies. The
lowest energy that particle can have corresponds
to n=o, and is called zero-point energy. It is
given by (Eo = Ezero point)
Normalization of wave function:
We know that particle is definitely found
somewhere in space
Therefore Normalized wave function is given by
Electrons are accelerated to a potential of 100V.
Calculate their energy, group velocity and
phase velocity.
Wave function, Probability density and
energy of the particle in different at
different eigen states.
For n = 1, Energy eigen value equation is 2
2
2
8ma
h
E =
then eigen function and probability density at
different value of x is given by the following table
x P = Ψn
2
0 0 0
a/4 a/1 1/a
a/2
a
2
a
2
3a/4 - a/1 1/a
a 0 0
Simillarly the for n=2, 2
2
2
8
4
ma
h
E = = 4Eo
then eigen function and probability density at
different value of x is given by the following table
9

Above potential well diagram the solid lines represents the
values eigen functions (ψ) and dotted lines represents the
probability denisty (ψ2
) for energy levels n=1 and n=2
Free Particle
Consider a particle of mass m moving along
positive x-axis. Particle is said to be free if it is
not under the influence of any field or force.
Therefore for a free particle potential energy can
be considered to be constant or zero. The
Schrödinger wave equation for a free particle is
given by.
Equation (1) is also written as,
Or 0
8 2
2
2
2
=+ ψ
ψ
π
E
dx
d
m
h
Or ψ
ψ
π
E
dx
d
m
h
=− 2
2
2
2
8
or Hψ = Eψ where
H= 2
2
2
2
8 dx
d
m
h
π
− it is called as Hamiltonian, which
is an energy operator.
The solution of the equation 1 is of the form
ψ = Asinαx + Bcosαx 3
Where A and B are unknown constants to be
determined. Since there are no boundary conditions
A, B and a can have any values.
Energy of the particle is given by
Since there is no restriction on E. Therefore energy
of the free particle is not quantized. i.e., free particle
can have any value of energy.
Or
Here there is no restriction for values of n and a
hence both will take infinite values. There fore
energy value gives indeterminate result by the use
of quantum mechanics. Hence to solve free particle
problem classical mechanics is sufficient.
1. Suppose visible light of wavelength = 5 x 10- 7
m is used to determine the position of an electron to
within the wavelength of the light. What is the
minimum uncertainty in the electron's speed?
Solution:
We shall use the Heisenberg uncertainty relation
10

x p = h/4π.
With p = mv or dp = m x dv,
assuming negligible uncertainty in the mass of
the electron, we find
v = h/4πmdx =
= 115.8 m/s.
is minimum uncertainty in finding the speed of
electron.
2. An electron has a speed of 300m/s accurate to
0.01%, with what fundamental accuracy can we
locate the position of the electron.
3. The maximum uncertainty in locating an
electron in its orbit is 4 Ǻ. Determine the
minimum accuracy with which its velocity is
determined.
4. The bohr radius of an electron orbit in
hydrogen atom is 0.053 nm. Calculate of K.E.
energy of Electron in that orbit.
5. The average time that an electron remains in
excited state is 10-8
s. Calculate the limit of
accuracy with which the excitation energy and
wavelength of the emitted radiation can be
determined.
6. Find the probability that a particle trapped in a
box of length L wide can be found to be 0.45L
and 0.55L for the ground state, first and second
excited state. If length of well is 0.5 mm
calculate the first four eigen values (energy of
first four energy levels) in eV
6. Suppose that the initial position of a 1kg object
and an electron are known to within a nuclear
radius 10-14
m and its initial velocity measured
to the accuracy allowed by the uncertainty
principle. Assume that no force acts on the
object and find its position uncertainty one
year later.
Now suppose that the initial position of an electron
(mass = 9.1×10-31
kg) is known to within a
nuclear radius and its initial velocity is
measured to the accuracy allowed by the
uncertainty principle. Assume that no force
acts on the electron and find its position
uncertainty one year later.
1. Explain the Uncertainty in the measurement of
position and momentum of particle in de-
Broglie wave.
2. State Heisenberg Uncertainty principle. What is
the significance of uncertainty principle?
Mention the different expressions of
Uncertainty principle.
3. Show that an electron cannot exist in a nucleus
by using Heisenberg Uncertainty principle.
11

4. Give the evidence for the broadening to
spectral line, incase of sources of light using
Heisenberg Uncertainty principle.
5. What is wave function and give its
significance. Mention characteristic
properties of wave function.
6. What is probability density? Briefly explain
the Normalization condition of the Wave
function.
7. Set up the Time independent Schrodinger
Wave equation in one dimension.
8. What are Eigen values and Eigen functions?
Write the characteristic properties of Eigen
values and Eigen functions.
9. What is a potential well? Give the
normalized equations of Energy Eigen value
and Eigen function of an electron moving in
a one dimensional potential well.
10. Show that a gives a series of de-Broglie
standing waves when moving in a potential
well in a given direction (one dimension).
11. What is a free particle? Explain the Eigen
value and Eigen function problem for free
particle, is that fit into Schrodinger wave
equation?
12. The uncertainty is inherent property of ________.
13. Uncertainty principle proposed by __________.
14. If certainty in the measurement of position
increases then uncertainty in the measurement
of momentum_________.
15. The uncertainty in measurement of energy is
high then uncertainty in the measurement of
time is___________.
16. There is no physical wave function, because___
17. Probability density has physical significance
because________
18. The product of wave function and its complex
conjugate is given by______________.
19. The probability density of a particle in infinite
space is________.
20. Probability of finding particle in space is_____.
12

21. Eignen (wave) function of for a particle
trapped in one dimension potential well
is_________.
22. The energy (eigen) value at first excited state
for the electron trapped in potential well of
1A width is_________.
13

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