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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