3 Crucially important Experimentslaid the foundation of QUANTUM THEORY
• ATOMIC AND MOLECULAR SPECTRA
►ENERGY TRANSFERRED, i.e., EMITTED OR ABSORBED,
WAS DONE ONLY IN DISCRETE QUANTITIES
• PHOTOELECTRIC EFFECT• PHOTOELECTRIC EFFECT
►ELECTROMAGNETIC RADIATION (earlier considered to be
a wave) BEHAVED LIKE A STREAM OF PARTICLES.
• ELECTRON DIFFRACTION.►ELECTRONS( which were believed to behave like particles
since their discovery) BEHAVED LIKE WAVE.
Atomic and molecular spectra
�Radiation is emitted and absorbed at a series of discrete
frequencies
�This supports the discrete values of energy of atoms and molecules
� Then energy can be discarded or accepted only� Then energy can be discarded or accepted onlyin packets
Conclusion:Internal modes of atoms and
molecules can possess only
certain energies
These modes are quantized
A typical atomic emission spectrumA typical atomic emission spectrum
A typical molecular absorption spectrum
Shape is due to the combination of electronic and vibrationalTransitions of a molecule
Photoelectric effect
�we can think radiation as a stream of particles, each
having an energy hν
�Particles of electromagnetic radiation are called photons�Particles of electromagnetic radiation are called photons
�Photoelectric effect confirmed that radiation can be
interpreted as a stream of particles
�No electrons are ejected, unless the frequency exceeds�No electrons are ejected, unless the frequency exceeds
a threshold value
�The kinetic energy of the ejected electrons varies linearly
with the frequency of the incident radiation
�Even at low light intensities, electrons are ejected
immediately if the frequency is above the threshold value
φν −= hvme
2
2
1
φ is the work function of the metal
�When photoejection cannot occur as
photon supplies insufficient energy to expel electronφν <h
�Kinetic energy of an ejected electron should increase
linearly with the frequency
�When a photon collides with an electron, it gives up all
its energy, so electrons are expected to appear as soon
as the collisions begin
The diffraction of electrons
�Diffraction is a typical characteristic of wave
•Diffraction is the interference between waves caused by
an object on their path
•Series of bright and dark fringes
�Davisson-Germer experiment showed the diffraction of
electrons by a crystal
�This experiment shows that
wave character is expected
for the particles
de Broglie relation
p
h=λ
p Linear momentum of the travelling particle
λ Wave length of that particle
�Wavelength of a particle should decrease as
its speed increases
�For a given speed, heavy particles should have�For a given speed, heavy particles should have
Shorter wavelengths than lighter particles
Wave-Particle duality
�Particles have wave-like properties and waves have particle-like properties
�When examined on an atomic scale
�the concepts of particle and wave melt together�the concepts of particle and wave melt together�particle taking on the characteristic of waves and waves the characteristics of particles
This joint wave-particle character of matter and radiation
Is called wave-particle duality
•A particle is spread through space like a wave
•There are regions where the particle is more likely to be
found than others
Dynamics of microscopic systems
•To describe this distribution the concept of wavefunction ψ
is introduced, in place of trajectory
•A wavefunction is the modern term for de Broglie’s matter
wave
• According to classical mechanics a particle may have a
well defined trajectory with precise position and momentum
•In quantum mechanics a particle cannot have a precise
trajectory, there is only a probability
•The wavefunction that•The wavefunction that
determines its probability
distribution is a kind of
blurred version of trajectory
The Schrödinger equation
Schrödinger Equation
ψψ ExVdx
d
m=+
Ψ− )(
22
22h
ψψ EH =ˆor
Hamiltonian
Schrödinger equation for a single particle of massM and energy E (In one dimension)V Potential energy
hπ2
h=1.054 x 10-34 J .S
We can justify the form of Schrödinger equation(in case of a freely moving particle) V = 0 everywhere
ψEdx
d
m=
Ψ−
2
22
2
h
h
2
1
)2( mEk =Sinkx=ψA solution is where
[ can be verified by putting in (1)]Sinkx=ψ
--------------(1)
Comparing
λ
πxSin
2
With the standard form of aharmonic wave of length λ, which is
Sinkx
k
πλ
2=we get
[ can be verified by putting in (1)]Sinkx=ψ
Energy E = ( )
m
p
m
mvmv
222
122
2 ==
But E =m
k
2
22h
∴π hh
kp =×==2
h∴λπλ
π hhkp =×==
2
2h
This is de Broglie’s relation.So Schrödinger equation has led to anexperimentally verified conclusion
The Born interpretation
Probability of finding a particle in a small region
of space of volume δV is proportional to ψ2 δV
� ψ2 is probability density
Wherever ψ is large, there is high probability� Wherever ψ2 is large, there is high probabilityof finding particle
� Wherever ψ2 is small, there is small chance of finding particle
Probabilistic interpretation
(a)Wavefunction
No direct physical interpretation
(b)Its square (its square modulus if
if it is complex)if it is complex)
probability of finding a particle
(c)The probability density
density of shading
�Infinite number of solutions are allowed mathematically
�Solutions obeying certain constraints called
boundary conditions are only acceptable
�Each solution correspond to a characteristic value of
E. Implies-
• Only certain values of Energy are acceptable.
• Energy is quantized
The uncertainty Principle
It is impossible to specify simultaneously, witharbitrary precision, both the momentum and theposition of a particle
�If we know the position of a particle exactly,we can say nothing about its momentum.
�Similarly if the particle momentum is exactlyknown then its position will be uncertain
•A sharply localized wavefunction by
�adding wavefunctions of many wavelengths
�therefore, by de Broglie relation, of many different
linear momenta
�Number of function increases
� wavefunction becomes sharper
�Perfectly localized particle is
obtained
� discarding all information about
momentum
Quantitative version of Uncertainty Principle
h2
1≥∆∆ xp
p∆ Uncertainty in the linear momentum
x∆ Uncertainty in positionx∆ Uncertainty in position
Smaller the value of ,x∆greater the uncertainty in its momentum (the largervalue of )p∆and vice versa
Variable 1
Variable 2 x y z px py pz
x
y
z
px
py
pz
Observables that cannot be determined simultaneously with arbitrary precision are marked with a grey rectangle; all others are unrestricted
Applications of quantum mechanics
Translation: a particle in a box
•A particle in a one-dimensional region
•Impenetrable Walls at either end
•Its potential energy is zero between x=0 and x=L
•It rises abruptly to infinity as the Particle touches wall
Boundary conditions
�The wave function must be zero where V isinfinite, at x<0 and x>L
�The continuity of the wavefunction then requiresit to vanish just inside the well at x=0 and x=L
�The boundary conditions for this system are therequirement that each acceptable wavefunctionmust fit inside the box exactly
,2
,......3
2,,2
n
LorLLL == λλ with n=1,2,3…
•Each wavefunction is a sine wave with one of these
wavelengths
λ
πx2sin
2 22 , , ,......
3
LL L L or
nλ λ= =
permitted wavefunctions are
• sine wave has the form
permitted wavefunctions are
L
xnNn
πψ sin=
�N is the normalization constant
The total probability of finding the particle betweenx =0 and x =L is 1
(the particle is certainly in the range somewhere)
1
0
2 =∫ dx
L
ψ
SubstitutingSubstituting1sin
0
22 =∫ dxL
xnN
Lπ
12
12 =× LN and hence2
1
2
=
LN∴
Permitted Energies of the particle
�The particle has only kinetic energy
m
p
2
2
�The potential energy is zero everywhere insidethe box
�de Broglie relation showsnhh
p == ,....2,1=n�de Broglie relation showsL
p2
==λ
,....2,1=n
Permitted energies of the particle
2
22
8mL
hnEn = ,..2,1=n
n is the quantum number
Zero Point Energy
�Quantum number n cannot be zero (for this system)
�The lowest energy that the particle possess is not zero
2
2
8mL
h2
8mL
This lowest irremovable energy is called thezero point energy
The energy difference between adjacent levels is
2
2
18
)12(mL
hnEEE nn +=−=∆ +
1.Greater the size of the system
Less important are the effects
of quantization
2.Greater the mass of the particle
Less important are the effects
of quantization
Vibration: the harmonic oscillator
Hooke’s Law: Restoring force = kx−
k is the force constant and
x is the displacementx is the displacement
Potential energy 2
2
1)( kxV =
After solving Schrödinger equation
The only allowed energies are
νυυ hE )2
1( +=
,......2,1,0=υ1
2
1
2
1
=
m
k
πν
is the vibrational quantum
number
is the frequency (in cycles
per second or hertz, Hz)
υ
ν