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QUANTUM MECHANICS, Eigen, Classical physics - Coggle Diagram
QUANTUM MECHANICS
Experiments leading to it
Particel nature of radiation
Black body radiation
Explanation for the energy density per unit frequency vs frequency graph
Weins
Empirical fitted a formula
Rayleigh
Continous energy exchange(any amount)
Planck
Enregy exchange Integral multiple of hv
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
photoelectric effect
idea light / radiation is made of photons
electrons can only absorb one photns or zero
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
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
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
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
little bit of relativistic mechanics is involved
Pair production and anhilation
core is the energy conversion to mass
Production
Conditions
Not in free space (WHY)
A massive object like nucleus needs to absorb momentum
Absorb ?
Why
Highly energetic photon
charge amd momentum conservation
energy of photon = energy of electron + positron + recoil nuclues
anhilation
mometum conservation
two gamma photon
energy mass conserved
can occur in vacuum
positronium is produced
𝛾=0.51𝑀𝑒𝑉+ half kinetic energy of particle relative to their center of mass.
Wave nature of microscopic particles
Davisson and germer experiment
thomsons expermeint
double slit experiment
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
CONCLUSIONS
Indeterminisim
measurement disturbing state
Probalilitic nature
particle wave complemtary
Relativistic quantum mechanics
Dirac extended non relativistic to relativistic QM
Dirac notation
What is dual vector ?
Non relativistic quantum mechanics
Developed by schrodinger and hiesenberg
Idea of wave function
Operator on wave function
commutator
genralised uncertainty principle
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
interpretation
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
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
Expextation vwlue
Average of measurement on identically prepare systems doubt
Remember a measurement lead to wave function collapse
Hermitain operator
How to check rember the the inner product realtion
Use IBP VERY IMP
A equal Adagger
Conjugate transpose
Hermitian arjoint
Eigen vectors of two different eigen state orthonormal
Expextation value are real
Obeys relation .....
real eigen value
physical observable
linear operator
follow the distributive law
projection operator
outer product
idempotent
hermitian
normalised wave functio only
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
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
First derivative continous
Amplitude of. Matter wave
Square integrable
Normalisation
Probability density positive
why wave packet
localised wave
TISE
Stationary states
Definite energy
Standard deviation is zero
Probability density is independent of time
Expectation values do not change with time
Potential is only function x
Aplication
Infinite square well or particle in a box
CONFINED SYSTEM
DISCRETE ENERGIES
CONTINOUS WAVE FYNCTION
orthognal states
Free particle
poetntial zero everywhere
still localised
solution is not a plane wave
wave packet
what does the above wave packet solution tell us?
group veolcity
not able to find definite momentum
Note
group veolcity is different from thephase velocity only in non despersive medium
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
continous energy
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
PECICULAR EXAPLE WHERE THE CLASSICAL PHYSICS FAIL
Microscopic particle crosses a poetical barrier even though its energy is lesser than the barrier poetical
Quantum mechanics explanation
Wave nature
Transmission
Reflection
A aribitary
General approach
Divide the region according to where the potential changes
In each region apply TISE
AS boundary conditions
wave function must be normalisable
PSI AND ITS FIRST DERIVATIVE SHOULD BE CONTINOUS
TDse
uncertainty
energy time uncertainity
interpretation
usew in decay process
where delta t mean time c
why
delta e energyu of excitec state
tips for solving
writing eikx is sineand cosine
Hilbert space
Dual vector space doubt
Any constants you want to find assume complex
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
Eigen value
expectation value if particle in that eigen state
Eigen vector
Dictionary meaning proper , characteristic
If a particle at
Classical physics
Limitation
Fail with particle at very high speed