QUANTUM MECHANICS
Experiments leading to it
Particel nature of radiation
Wave nature of microscopic particles
Non relativistic quantum mechanics
Developed by schrodinger and hiesenberg
Black body radiation
Explanation for the energy density per unit frequency vs frequency graph
Weins
Rayleigh
Planck
Enregy exchange Integral multiple of hv
Continous energy exchange(any amount)
Empirical fitted a formula
photoelectric effect
idea light / radiation is made of photons
electrons can only absorb one photns or zero
Compton effect particle nature of X rays
x ray wavelegth shift when incident on electrons at rest
unexpected because if radiation was continuous then electron would oscillate with same frequency (doubt ) (not deflect ) hence no shift
black body need not be black
also note color of abody is not simply the color it reflects rather it is the radiations emitted by it which can be due to reflection or from inside the material
max kinetic energy
work function
stopping potential minimum potential to stop the most energetic electron
two defination of the intensity very useful
energy
no. of photon string per unit area per unit time
doubt why electron be at rest
why elastic collision
why classically the wavelength should be same
why electron not absorb and why oscillate ?
wavelength shift depends only on the scattering angle
little bit of relativistic mechanics is involved
When we use Einstein energy masss equation and when we don't?
Better to find in terms of requency no approximations just square and cancel
Relativistic quantum mechanics
Dirac extended non relativistic to relativistic QM
Pair production and anhilation
core is the energy conversion to mass
Davisson and germer experiment
thomsons expermeint
double slit experiment
CONCLUSIONS
Indeterminisim
measurement disturbing state
Probalilitic nature
particle wave complemtary
Idea of wave function
Production
Conditions
Not in free space (WHY)
A massive object like nucleus needs to absorb momentum
Absorb ?
Why
Highly energetic photon
Operator on wave function
Think of it this way that the wave function has all the information about the particle when it in a state characterized by that particular wave function
9nce we know the wave function
Using different operators we can get the required info about the particle when it is jin a particular state
Use energy operator to find energy
Use momentum operator to find the momentum
Eigen
Eigen states
One eigen state cannot be constructed by using one or more eigen state
Doubt what to we mean by constructing a state from other states
why wave packet
localised wave
Eigen value
Eigen vector
Dictionary meaning proper , characteristic
Classical physics
Limitation
Fail with particle at very high speed
Why continous
What if not
Simple maybe is that whenever wave function in general as any wave function always has continoues like in em wave
Amplitude of. Matter wave
Dirac notation
What is dual vector ?
Hilbert space
Dual vector space doubt
commutator
Observable has corresponding qm opreator
Opreator equation trnaform one wave function to another wave function
In particular if new wave function is constant times old wave then eigen value equation
If a particle at
Expextation vwlue
Average of measurement on identically prepare systems doubt
Remember a measurement lead to wave function collapse
Any constants you want to find assume complex
Hermitain operator
How to check rember the the inner product realtion
Use IBP VERY IMP
TISE
Stationary states
Aplication
Definite energy
Standard deviation is zero
Probability density is independent of time
Expectation values do not change with time
Potential is only function x
Infinite square well or particle in a box
TDse
Free particle
Finite square well problem
Symmetry theorem in QM
Continuity of Psi
Simplify solution by just eliminating not possible scenarios
Taking cases and break domain
Why we need unitless quantities for drawing graph
See limiting cases
Potential tend infinity
Potential than to zero
Graphical approach to solutions
Note that in these cases we are generally given the graph of the potential vs position
TUNEELING EFFECT
AS boundary conditions
wave function must be normalisable
PSI AND ITS FIRST DERIVATIVE SHOULD BE CONTINOUS
PECICULAR EXAPLE WHERE THE CLASSICAL PHYSICS FAIL
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anhilation
mometum conservation
two gamma photon
energy mass conserved
can occur in vacuum
positronium is produced
energy of photon = energy of electron + positron + recoil nuclues
charge amd momentum conservation
𝛾=0.51𝑀𝑒𝑉+ half kinetic energy of particle relative to their center of mass.
macroscopic particle
intensity add
number of particles ot ticks add
no interferne I = I1+12
MICROSCOPIC PARTTCILES
AMPLITUDES ADD
INTERFERNCE SIMILAR TO WAVES
SLIT 1 VS SLIT2 VS BOTH
First derivative continous
Square integrable
Probability density positive
Normalisation
A equal Adagger
Conjugate transpose
Hermitian arjoint
Eigen vectors of two different eigen state orthonormal
Expextation value are real
Obeys relation .....
real eigen value
expectation value if particle in that eigen state
linear operator
follow the distributive law
projection operator
genralised uncertainty principle
physical observable
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outer product
idempotent
hermitian
normalised wave functio only
if two operators commute
it means they can have same eigen function
why
how to find the commutator of two operators
i think its better to apply on a wave function
assume psi a wave function of the system
if psi x,t the partial
mostly incompatible observables this relation holds
commutator is not zero
commuator zero then compatible
but doubt in case of the exampler foe the s state
uncertainty
energy time uncertainity
interpretation
interpretation
usew in decay process
where delta t mean time c
why
delta e energyu of excitec state
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CONFINED SYSTEM
DISCRETE ENERGIES
CONTINOUS WAVE FYNCTION
orthognal states
poetntial zero everywhere
still localised
solution is not a plane wave
wave packet
continous energy
stationary state propagating plane wave
what paradox we faced
probability density is independent of time and position
speed of wave
not normalizable
free particle cannot exist in astaionary state
it cannot have definite enrgy and momentum
what does the above wave packet solution tell us?
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group veolcity
not able to find definite momentum
Note
group veolcity is different from thephase velocity only in non despersive medium
tips for solving
writing eikx is sineand cosine
Microscopic particle crosses a poetical barrier even though its energy is lesser than the barrier poetical
Quantum mechanics explanation
Wave nature
Transmission
Reflection
General approach
Divide the region according to where the potential changes
In each region apply TISE
A aribitary
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