3 Crucially important Experimentslaid the foundation of QUANTUM THEORY

ATOMIC AND MOLECULAR SPECTRAENERGY TRANSFERRED, i.e., EMITTED OR ABSORBED,

WAS DONE ONLY IN DISCRETE QUANTITIES PHOTOELECTRIC EFFECT PHOTOELECTRIC EFFECTELECTROMAGNETIC 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 spectraRadiation is emitted and absorbed at a series of discrete

frequenciesThis 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 only

in packetsConclusion:Internal modes of atoms and molecules can possess only certain energiesThese 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 h

Particles of electromagnetic radiation are called photonsParticles 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 exceedsNo electrons are ejected, unless the frequency exceedsa threshold value

The kinetic energy of the ejected electrons varies linearlywith the frequency of the incident radiation

Even at low light intensities, electrons are ejected immediately if the frequency is above the threshold value

= hvme 221

is the work function of the metal

When photoejection cannot occur asphoton supplies insufficient energy to expel electron

The diffraction of electrons

Diffraction is a typical characteristic of wave

Diffraction is the interference between waves caused byan object on their pathSeries of bright and dark fringes

Davisson-Germer experiment showed the diffraction ofelectrons by a crystal

This experiment shows thatwave character is expectedfor the particles

de Broglie relation

ph

=p Linear momentum of the travelling particle Wave length of that particle

Wavelength of a particle should decrease as its speed increases

For a given speed, heavy particles should haveFor a given speed, heavy particles should haveShorter 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 togetherthe concepts of particle and wave melt togetherparticle taking on the characteristic of waves and waves the characteristics of particles

This joint wave-particle character of matter and radiationIs 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

found than others

A wavefunction is the modern term for de Broglies matterwave

According to classical mechanics a particle may have a well defined trajectory with precise position and momentumIn quantum mechanics a particle cannot have a precisetrajectory, there is only a probability

The wavefunction thatThe wavefunction thatdetermines its probabilitydistribution is a kind ofblurred version of trajectory

The Schrdinger equationSchrdinger Equation

ExVdxd

m=+

)(

2 222

h EH =or

Schrdinger equation for a single particle of massM and energy E (In one dimension)V Potential energy

hpi2h

=1.054 x 10-34 J .S

We can justify the form of Schrdinger equation(in case of a freely moving particle) V = 0 everywhere

Edxd

m=

2

22

2h

Sinkx=A solution is

--------------(1)

Comparing

pixSin 2

With the standard form of aharmonic wave of length , which is

Sinkx

kpi 2=we get

Energy E = ( )

m

pm

mvmv

2221 222

==

But E =m

k2

22h

pi hhkp === 2h pipi hhkp ===

22

h

This is de Broglies relation.So Schrdinger equation has led to anexperimentally verified conclusion

The Born interpretationProbability of finding a particle in a small regionof space of volume V is proportional to 2 V

2 is probability density

Wherever is large, there is high probability Wherever 2 is large, there is high probabilityof finding particle

Wherever 2 is small, there is small chance of finding particle

Probabilistic interpretation

(a)Wavefunction No direct physical interpretation

(b)Its square (its square modulus ifif it is complex)if it is complex)probability of finding a particle

(c)The probability densitydensity of shading

Infinite number of solutions are allowed mathematically

Solutions obeying certain constraints calledboundary conditions are only acceptable

Each solution correspond to a characteristic value ofE. 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 locationIts wavefunction nonzero there and zeroeverywhere else

A sharply localized wavefunction byadding wavefunctions of many wavelengthstherefore, by de Broglie relation, of many differentlinear momenta

Number of function increases wavefunction becomes sharper

Perfectly localized particle isobtained

discarding all information aboutmomentum

Quantitative version of Uncertainty Principle

h21 xp

p Uncertainty in the linear momentumx Uncertainty in positionx Uncertainty in position

Smaller the value of ,xgreater the uncertainty in its momentum (the largervalue of )pand 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 mechanicsTranslation: a particle in a box

A particle in a one-dimensional regionImpenetrable Walls at either endIts 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 xL

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,......

32

,,2n

LorLLL == with n=1,2,3

Each wavefunction is a sine wave with one of thesewavelengths

pix2

sin

2 22 , , ,......3

LL L L orn

= =

permitted wavefunctions are sine wave has the form

permitted wavefunctions are

LxnNn

pi 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

SubstitutingSubstituting1sin

0

22= dxL

xnNL pi

1212

= LN and hence21

2

=

LN

Permitted Energies of the particle

The particle has only kinetic energy

m

p2

2

The potential energy is zero everywhere insidethe box

de Broglie relation shows nhhp == ,....2,1=nde Broglie relation shows Lp 2== ,....2,1=n

Permitted energies of the particle

2

22

8mLhnEn = ,..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

8mLh

28mL

This lowest irremovable energy is called thezero point energy

The energy difference between adjacent levels is

2

2

1 8)12(

mLh

nEEE nn +== +

1.Greater the size of the systemLess important are the effects of quantization

2.Greater the mass of the particleLess important are the effects of quantization

Motion in Two-dimensions

From separation of variables

Note: See Derivation 12.3

Degeneracy