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

•Particle is at a definite location•Its wavefunction nonzero there and zeroeverywhere else

•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

•The allowed energy levels & (sine wave) functions.

• Number of nodes n-1

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

Motion in Two-dimensions

From separation of variables

Note: See Derivation 12.3

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

(a)The wavefunctions (b) the probability densities of the first three states of harmonic oscillator