Why Quantum Mechanics?
• Inadequacy of classical mechanics in explaining certain
experimental observations [stability of atoms, blackbody
radiation, etc.]
• Need for a new theory to explain phenomena on the
atomic scale
• Quantum mechanics was able to successfully explain such
events
• Useful for explaining the structure of atoms and molecules
• Reduces to classical mechanics when we deal with large
dimensions compared to atomic distances i.e., classical
mechanics is a special case of a more general principle of
quantum mechanics
• Difficult to visualize the quantum behavior unlike the
classical events
Blackbody Radiation
• Max Planck (1901)
• Black body: absorbs all incident
radiation
• Spectrum of the emitted
radiation approximately the
same for all solids at a given
temperature and approaches
black body spectrum
• Planck’s assumptions:
– Energy of an oscillator, E = nh
– Transition from one level to other
associated with the emission or
absorption of energy
– Energy lost or gained in the
transitions is discrete
Photoelectric Effect
• Einstein (1905)
• Light incident on a metal surface causes ejection of
electrons from the metal surface
• Classical explanation:
– Electrons leave the metal surface only if they have sufficient
energy to overcome the potential barrier at the surface, W called
the work function. W is a constant for a given chemically clean
metal surface
– Kinetic energy imparted to the electron was a function of the
intensity of the radiation.
– As such, a lower intensity limit would exist below which no
electron emission could take place
– Could not explain the observed behavior
Photoelectric Effect
• Einstein’s assumptions:
– Incident radiation made up of discrete pulses
called photons
– Photons are absorbed or given off in discrete
amounts
– Energy of photon is an integral multiple of h
– Photons behave like waves of corresponding
frequency
Em
o

W
A
I2
I1
A
+
V
Vs
V
h = W + Em
Em = h - W
Hydrogen Spectra
• Niels Bohr (1913)
• Model to explain and predict the hydrogen spectra
• Experimental observations:
– Every atom has a unique spectra characteristic of the atom
– Plot of intensity of the emitted light as a function of the
wavelength gives a series of sharp lines rather than a continuous
distribution of wavelengths
– These lines appear in several groups called Lyman, Balmer and
Paschen etc.
– The various series in the spectrum were found to follow the
following empirical forms where R is a constant called the
Rydberg constant.
1 1 
Lyman :   cR  2  2 , n  2,3,4,...
1 n 
1 
1
Balmer :   cR  2  2 , n  3,4,5,...
n 
2
1 
1
Paschen :   cR  2  2 , n  4,5,6,....
n 
3
Bohr’s Model
• Bohr’s postulates:
– Electrons exist in certain stable, circular orbits about the nucleus
– Electron may shift to an orbit of higher or lower energy by
absorption or emission of a photon of energy h
– Angular momentum of the electron in an orbit is always an integral
multiple of Planck’s constant
• Bohr’s radius
– For an electron in an orbit of radius r around the proton in the
nucleus,
q2
mv 2


2
4 0 r
r
From the third postulate,
mvr  n
This gives,
4 0 n 2 
rn 
mq 2
2
Bohr’s Model
• Expression for energy in Bohr’s orbits:
Total energy of the electron in the nth orbit,
2
1
q
E n  K.E.  P.E.  mv 2 
2
4 0 rn
where v  n / rn
mq 4
mq 4


2 2 2
2(4 0 ) n 
(4 0 ) 2 n 2  2
mq 4

2(4 0 ) 2 n 2  2
Now, the energy difference between different levels n1 and n2 ,
 1
mq 4
1 
 2  2 
En  En 
2 2 
2(4 0 )   n 1
n2 
2
1
Principles of Quantum Mechanics
• Probability and the Uncertainty Principle
– Probabilistic nature of events involving atoms and electrons
– Impossible to describe with absolute precision events involving
particles on the atomic scale
– Uncertainty in quantum calculation not result of any shortcoming
of the theory
– The magnitude of the inherent uncertainty described by the
Heisenberg uncertainty principle:
• (x)(p)  
• (t)(E)  
– We cannot speak of finding a particle at a particular position
– Expected values of position, momentum, energy etc. are relevant
• Wave nature of atomic particles
– de Broglie equation (1924)
– Relationship between wavelength and momentum,  = h/p
Schrodinger Equation
• Postulates:
– Each particle represented by a wave function (x,y,z,t)
– Classical quantities like momentum, energy etc. replaced by
quantum mechanical operators as follows:
Location, x  x
Momentum, p( x ) 
Energy, E  
 
j x

j t
– Probability of finding a particle in a volume dx dy dz is * dx dy
dz. The total probability of finding the particle anywhere should be
one. So, the integral of * dx dy dz over the entire space is
normalized such that



*
dxdydz  1
– The expectation value <Q> of any variable Q is calculated as,
Q 



 *Q op  dx dy dz
Schrodinger Equation
From classical mechanics, the energy of a particle may be written as,
E = K.E + P.E.
= p2/2m + V
Substituting for the relevant operators, for a particle in one dimension,
 2  2  ( x, t )
  ( x , t )


V
(
x
)

(
x
,
t
)


2m x 2
j
t
In three dimensions, the equation becomes
2 2
 

   V  
2m
j t
2
2
2






where  2  


x 2
y 2
z 2
Thus, the problem of Quantum mechanics reduces to the solution of the
Schrodinger under the given boundary conditions.
Time Independent Schrodinger Equation
•
•
The wave function  includes both space and time dependencies.
It is easier to calculate these dependencies separately and then combine them
together
Suppose for one-dimensional Schrodinger equation,
(x,t) = (x)(t) then
 2  2 (x )

( t )


(
t
)

V
(
x
)

(
x
)

(
t
)



(
x
)
2m x 2
j
t
By separating the variables, we get the time-dependent equation
in one dimension, d( t ) jE
 ( t )  0
dt

and the time-independent equation,
•
•
d 2  ( x ) 2m
 2 [E  V( x )] ( x )  0
dx 2

Many problems are time independent and so we need only space dependent
solution
We use the time independent Schrodinger equation when the property of
atomic systems has to be calculated in stationary conditions
Schrodinger Equation for a Free Particle
We consider the case of a electron which moves freely in the positive x
direction. The potential energy is zero and so the Schrodinger equation
may be written as, 2
  2m

E  0
x 2  2
The solution of this equation is of the form, (x) = Aejkx + Be-jkx
where k  2mE / 
But, as the electron is moving in positive x direction, we only need to
take the first term which corresponds to the wave motion in positive
direction. So, (x) = Aejkx
2 2
p2
Now , E 
k and also E 
2m
2m
h
So, p  k  (from de Broglie equation )

2
k
and is called the wave vector.

As the wave vector, k can take any values, all values of energies are
allowed for a free electron as in the classical case.
Criteria for Solution of Schrodinger Equation
• Both  and d/dx must be continuous at the boundaries. If
it is not so, it would equivalent to creation or destruction of
particles as *  is proportional to the probability of
finding the particle
•  must be single-valued as otherwise this would lead to
more than one probability of finding the particle
•  should never be identical to zero as that would amount
to zero probability of finding the particle anywhere in the
system
Infinite Potential Well
• Simplest example
• One dimensional problem
• Potential defined as:
V=
– V(x) = 0 for 0 < x < L ( inside the well)
– V(x) =  elsewhere
I
• The Schrodinger equation for region II,
V=
II
III
x
0
L
 2  ( x ) 2m
 2 E ( x )  0
2
x

• The solution for this equation is of the form,
 ( x )  Ae jkx  Be  jkx
where
k  2mE / 
• As per the barrier is infinite, the particle cannot be found in
these regions. So, the wave function must be zero in the region
I and III as otherwise there will be a non-zero probability of
finding the particle in these regions
Infinite Potential Well
• As the wave function is zero in region I and III, for the
wave function to be continuous at x = 0 and x = L,
II (0) = 0 and II (L) = 0
• II (0) = 0  B = -A and so (x)= Aejkx - Ae-jkx = 2jAsinkx
• II (L) = 0  2jAsinkL = 0  k = n/L
where n = 1,2,3,… (Solution with n =0 is not a possible
solution as then (x) = 0
• This gives,
n 2 2 2
En 
2mL2
where n  1,2,3,...
• This means that the electron inside an infinite potential
well cannot have all energy values. It can only take
discrete values. So, the energy is quantized.
Infinite Potential Well
• The constant A in the wave
function can be found by
normalizing the wave function, 
• This gives  n  2 / L sin(n / L) x
• Also, the probability of finding
the particle inside the well is not
uniform
• When n becomes large, the
probability of finding the particle
becomes uniform as in the
classical case.
Finite Potential Barrier
Problem: A free electron with energy, E moving in the positive x
direction with a potential barrier of height V0 at x = 0 such that E < V0.
V
V0
I
II
x=0
x
We have to write two different Scrodinger equations for region I and
region II as the electron is free in the region I and the potential is V0 in
region II.
2
  2m
 2 E  0
x 2

 2  2m
and in region II,
 2 (E  V0 )  0
x 2

In region I,
The solution to the Schrodinger equation in region I is ,
 I  Ae jkx  Be  jkx
where   2mE /  2
Finite Potential Barrier
The solution to the Schrodinger equation in region II is ,
 I  Cex  De x
where   2m( V0  E ) /  2
Now, as E < V0, (V0-E) would be positive and so  is real. We
determine the constants A ,B ,C and D by means of the boundary
conditions. Now, at x= , II= . This means that the probability
density II* II would become infinity. So, C0 i.e., II = De x
At the boundary, x=0, the wave function and its derivative should be
continuous. So, at x=0, I= II and d II/dx=d II/dx. This gives,
A  B  D and Aj  Bj   D
Solving these two equations, we get
A
D

D

(1  j ) and B  (1  j )
2

2

So, we get A and B in terms of D. We can solve for D by normalizing
the wave function.
Finite Potential Barrier
The wave function in region II, II= De- x Thus, the wave
function decreases exponentially in region II but has a nonnegative value inside the barrier. This has following
implications:
– In classical case, the electron would never enter the barrier as the
energy of the electron is less than the potential barrier.
– Now, the electron will have a non-zero probability of being found
inside the barrier
– If the potential barrier is high, the wave function will decrease very
rapidly and will soon become zero
– And if the barrier is only moderately high and is relatively narrow, the
wave function will continue across the barrier on the other side (we
have taken a barrier of very large width)
– The quantum mechanical effect of penetration of a potential barrier by
the electron is called “tunneling” and has important implications in
solid state physics
Hydrogen Atom
• Solution of Schrodinger equation in three dimensions for
the coulombic potential
• The Coulomb potential is given by,
q2
V(r ) 
4 0 r
• Use of spherical coordinates required as the potential is
inversely proportional to the distance from the nucleus
• The Schrodinger equation can be written in the polar
coordinates as,
1   2  
1
 
 
1
 2  2m
 2 E  V r   0
r
 2
 sin 
 2 2
2
2
r r  r  r sin   
  r sin  

• Solution of the form (r,,) = R(r)()() is obtained
Hydrogen Atom
• After the separation of variables, separate solution is found
for the r-dependent, -dependent and -dependent
equations and then the total wave function is obtained from
the product.
• The -dependent equation is of the form,
• The solution to this
•
d 2
2

m
0
2
d
equation is,
1 jm
 m () 
e
2
As value of  varies between 0 and 2, the value of 
should repeat after every 2 for which m must be an
integer i.e.,
m = ….,-2,-1,0,+1,+2,….
Hydrogen Atom
• Similarly, solutions of r-dependent and -dependent
equations are quantized and so each state is represented by
three quantum numbers n, l and m.
• The interrelationship between the equations leads to
following restrictions:
– Principal quantum number, n = 1,2,3….
– Azimuthal quantum number, l = 0,1,..,n
– Magnetic quantum number, m = -l,-(l-1),..,-1,0,1,..,(l-1),l
• Another quantum number, called spin quantum number
required to explain the observed spectral behavior of
hydrogen atom in non-homogeneous magnetic field.
• This comes from quantization of intrinsic angular
momentum of the electron. Spin quantum number, s can take
values +1/2 and -1/2
Hydrogen Atom
• Each allowed state of the electron in the hydrogen atom is
uniquely described by the four quantum numbers n,l,m and s
and is represented by nlms .
• The principal quantum number, n corresponds to the orbit of
electron in the Bohr model
• The energy for the electron in the hydrogen atom is given by,
given by,
 mq 4
En 
322o2 2n 2
• Electron in the hydrogen atom can occupy any of the large
number of excited states in addition to the lowest energy
level 100
• Energy difference between various states properly accounts
for the observed lines in the hydrogen spectra
Periodic Table
• Energy levels obtained by solution of the Schrodinger
equation for the hydrogen atom cannot be extended to other
atoms without appropriate modifications
• Quantum number selection rules valid for complicated
spectra and adequately explain the electronic structure of
atoms
• Pauli’s exclusion principle restricts the occupancy of same
quantum state by more than one electron
• This decides the maximum number of electrons occcupying
a shell(orbit)
• Maximum number of electrons in a shell - 2n2
• The different values of l denote orbitals and are represented
as s (l = 0), p (l = 1), d (l = 2), f (l = 3) etc.