WEBVTT
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so in the last lecture we came up to the point
of the bohr modal of the atom and we were
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able to see how this quantum mechanical principle
that he introduced to explain the atomic spectra
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of hydrogen atom in terms of line spectra
so this idea of energy quantisation was the
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critical one in terms of quantum mechanics
and modern approaches to quantum mechanics
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develop from the idea that this quantisation
is possible
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so a modern look at quantum mechanics can
be viewed from the fact that quantum mechanics
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is more like a linear algebra of variables
and hillbert spaces the most important part
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of quantum mechanics as quantum mechanics
lies in the fact that the physical interpretation
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of the results that are obtained so is the
physical interpretation which is very important
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in terms of the way of looking at quantum
mechanics which is otherwise mostly liner
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algebra hillbert spaces and some of it can
be simultaneously done by using mattresses
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matrices and others
so those are the tools that we have mathematical
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the more important part to understand in this
are the laws as in the newtonian picture also
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we knew that in classical mechanics we had
certain basics from where we started the problem
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here also there are a few laws which are known
as the postulate so the first postulate of
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quantum mechanics is basically to do with
the state of the system since we are all interested
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in the kind of physical systems that we have
which have photons conducting electrons in
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metal semi conductors atoms really the small
once microscopic once states of these rather
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than the diverse system represented by the
same type of functions are the state functions
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so we we are trying to look for fundamental
properties which can defined the principle
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of quantum mechanics to the first postulate
defines that every physically realisable state
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of the system is described in quantum mechanics
by a state function psi that contains all
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accessible physical information about the
system information about the system in that
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state so this definition of psi is roughly
the first postulate of quantum mechanics it
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leads to the principle of physically realisable
states that can be studied in the laboratory
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it has all the accessible information that
we can extend from the wave function and this
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is the state function is a function of position
momentum energy that is spatially localised
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now the principle of psi in general is mathematical
there are waves to make it observable and
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thats the part which will look into if psi
one and psi two represents two physically
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realisable states of the system then their
linear combination psi were as c one and c
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two are arbitrary complex constant represents
a third physically realisable state of the
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system
so this principle that any number of wave
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functions can combined to give raise to a
final wave function which is the principle
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of super position
is extremely important in quantum mechanics
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so the wave function psi x and t which is
the posi[tion] function of position and time
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probability amplitude is the one which can
be put together through super position of
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many many more wave functions which can come
together with certain amount of the contribution
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which can have arbitrary complex forms is
known as is what we are looking at so in some
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sense quantum mechanics is probability having
complex numbers associated with it so quantum
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mechanics is describing the outcome of an
ensemble of measurements or a large number
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of measurements where an ensemble is the measurement
consisting of very large number of identical
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experiments performed on identical non interacting
system all of which have been identically
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prepared so as to be in the same state so
if each of them could be measured they sum
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together is also the result which can be looked
at thats the basic premise of this entire
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picture however if a system is in a quantum
state represented by a wave function psi then
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there is a probability of its observation
which comes because of the observable in nature
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so the probability of a position measurement
at any time t of the particle will be detected
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with the volume of d v and that is why the
probability can be defined in terms of the
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square of the wave function so since the wave
functions are complex
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so the observable has to be a real value is
its a probability after all so its the complex
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conjugate measure is the product of the conjugate
with the respect to itself which gives raise
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to the mod psi squared value so whenever these
two coordinates are given position and time
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then this is the probability density associated
with both position and time one of the things
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to note here very importantly is that we are
only quantifying in terms of the coordinate
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space and time space and we are not mixing
momentum at the same time because thats what
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we had already started in the very beginning
showing that probability was measure of momentum
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and space cannot be done simultaneously in
the same dimension the importants of normalisation
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follows from the born interpretation
so this is the born interpretation of the
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state function as a position probability of
amplitude according to the second postulate
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of quantum mechanics which is what is it is
the integrated probability density can be
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represented can be interpreted as a probability
that in a position measurement that results
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in a position measurement at time t so since
the particle has to be always found you can
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we can always represent this with the fact
that if we take the measurement the total
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integral of that over the entire time then
we are going to always find the probability
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the total probability as one because where
at a given time or at the entire time period
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where the time has been integrated out then
the particle over the entire space will definitely
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give raise to its probability of find so the
total probability of finding a particle is
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absolute and that is what is represented
here so the normalisation condition which
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i just wrote for the wave function can be
written as total integral in terms of the
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entire volume space which becomes equal to
one so the limitations on the wave functions
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as a result of this requirement lies on the
fact that only normalisable functions can
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represent a quantum state and these are physically
admissible functions otherwise the particles
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existence become questionable and thats the
reason of this limitation which is required
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as per the postulate the state function must
be continuous and single valued state function
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must be smoothly varying function which means
that it should have a continuous derivative
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now these are requirements which make sure
that the way
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we are looking at this probability make sense
the third postulate of quantum mechanics lies
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in the fact that every observable in quantum
mechanics is represented by an operator which
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is used to obtain the physical information
about the observable from the state function
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for an observable that is represented in classical
physics by a function say q x and p the corresponding
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operator is q x cap p cap so for instance
proposition we write x with the cap t cap
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on top little hat which represents its operator
state similarly for an momentum operator we
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have t with hat which is h cross over i derivative
of space coordinate similarly for energy we
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have the hamiltonian operator which is essentially
a sum total of potential energy and kinetic
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energy and for a single coordinate space x
axis is just the double derivative is the
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momentum space p squared and p squared over
two m and the potential
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so this is how the operators are connected
to the observables and thats the third part
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postulate of quantum mechanics which tells
how to get the observable values from the
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state function so this principle of operative
math which is also be in put to use by many
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including dirac in his later develop into
quantum mechanics states that an operator
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is actually an instruction a symbol which
states or tells us to perform one or more
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mathematical acts and a function say f of
x the essential point is that they act on
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a function operators act on everything to
the right unless the action is constrain by
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brackets addition and subtraction rules for
the operator follow the same kind of principle
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as the associative low or the distributive
low as we can see at these two operators can
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act on the function based on these principles
and their action is determined by the principle
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as of mathematics as we are showing here the
product of two operators employees successive
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operations and the product of two operators
with the third operator can be great in this
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term
so basically two operators coming together
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this essentially a third operator now the
two operators commute if they obey the simple
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operator expression were this is shown by
this commutative bracket as they call it essentially
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the product of the two in one direction verses
the other if the two are equal then the commutative
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bracket gives rise to zero in the case where
this is not zero then you will get a situation
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where the operators do not commute the requirement
for the two operators to be commuting operator
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is a very important one in quantum mechanics
and it means that if we simultaneously measure
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the observable represented by these two operators
so that is essentially the principle behind
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finding out weather two operators can be simultaneously
measured or not the observable represented
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by them the non commutivity of the position
and the momentum operators that is the inability
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to simultaneously determine the particle position
and its momentum is represented with the heisenberg
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uncertainty principle which in mathematical
form is now expressed simply like this in
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terms of the commutative operator and this
can be journalised to any pair of observables
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to show that whenever there is commutator
and they are commuting then you can measure
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them simultaneously if they do not commute
then you cannot measure them simultaneously
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so that is the crux of this entire problem
so the development of the modern theory enables
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understanding of this quantum mechanics in
a very nice mathematical forms as i had mentioned
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initially the fourth postulate of quantum
a mechanics shows discusses about the evolution
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the equation of motion of these wave functions
so in nineteen twenty six so in the wave equation
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which represents a quantum equation of motion
and is of the form this which is for a single
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dimension x coordinate at a given time with
a hamiltonian which is a bohr kinetic and
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potential terms that is equated to the change
of the wave function with the respect to time
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this work of schrodinger was stimulated by
a nineteen twenty five paper by einstein on
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the quantum theory of ideal gas and the de
broglie theory of matter waves
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so on examining this time dependent schrodinger
wave equation one can also define an operator
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for the total energy as time evolution
operator like this and that can give raise
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to the principle of the fact that
so energy and time do not commute because
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this is a corrorali of the way of looking
at the time evolution so this schrodinger
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wave equation is sort of the principle behind
the idea of equation of motion of quantum
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states and that gives raise to the fact that
it is possible to get the equation of motion
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of the wave function with respect to the hamiltonian
which is what is written on the right hand
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side the left hand side so in other words
can be written in terms of h psi is equal
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to i h cross psi t t and this essentially
has led to the idea that there can be a definition
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of an total energy operator which can be of
this form because when you do a time integration
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of this particular schrodinger equation it
becomes h psi is equal to e psi
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so quantum mechanics has a new set of rules
promotion were the state of the system the
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wave function psi gives raise to an equation
of motion which is the schrodinger equation
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of motion h psi is of this kind were i is
the complex number which represents that its
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a negative square root of one the properties
of h have to be defined also in a certain
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way so that the results are observable so
the hamiltonian has to be hermitian so that
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a positive definite operator is the result
of these hermitian nature the time independent
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part is a result of this integration which
gives raise to the schrodinger equation which
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has the hamiltonian and the energy equated
in this form e psi is equal to h psi
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as a result of the fourth postulate of quantum
mechanics we just saw that the state function
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of an isolated quantum system is governed
by the time dependent schrodinger wave equation
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its i i h cross d psi by d t where h is as
we defined as the hemaltonian of the system
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on the isolated system is also means that
the time dependent schrodinger wave equation
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describes the evolution of a state provided
that there is no observation that have being
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made now this is the very important point
to remember because whenever you make measurements
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you part of the system and that is going to
change the way you look at the system and
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so it is just stated here on observation alters
the state of the observed systems and as it
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is the time dependent schrodinger in the wave
equation cannot describe such changes so that
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is built into the fourth postulate of quantum
mechanics
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so in order to make measurements one of the
other very important thing which is necessary
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is to be able to understand the basics of
laser matter interaction so in this case the
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interaction of radiation with matter often
is considered under the condition that the
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radiation is considered to be wave like accepting
totally well that the photo electric effect
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con does not consider the wavelike picture
alone of the photons but still in typical
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processes were we are considering the particle
or the matter as a practical which is quantised
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we consider the radiation as wavelike and
the matter as quantised so as long as the
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energy of the wavelike electromagnetic radiation
matches with the energy of the gap between
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the matter states say delta e transitions
or changes can occur
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so there are many different levels of this
interaction they can be very low energy waves
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which are micro wave radiation it can lead
to molecular rotation and torsions the next
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level up when it is of the region of infrared
radiation that would be leading to molecular
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vibrations and then there is this visible
region where it is often electronic and definitely
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the ultra violet region is changing the energy
of the electrons so beyond a certain point
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ultimately the electron will be separated
from the atom and that is basically the idea
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of ionization
so there have various different ways these
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interactions happen and depending on the energy
of the electromagnetic radiation that we are
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using we will be seeing different levels of
interactions so will deal with this a little
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bit more detail later this is just in terms
of ionization these are some examples somehow
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an x ray which is very high energy can actually
lead to a ionization it could also lead to
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other kinds of effects like crompton scattering
longer wavelength generations ionization in
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all of that as we just mentioned but in
most are the cases this is the case where
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it is the photoionization principle in the
case that we are looking at typically are
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the electronic transitions are ultra violet
and visible were the electronic champs which
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will different states that we are talking
about these are the quantised states and in
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the other cases were we look at by vibrations
and rotations they required much lesser energy
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and that s the case were be have different
places to look at now one of the popular models
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to test out quantum mechanics lies in the
principle of particle in a box
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so this is actually a very interesting
case because this is the first case where
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quantised systems come up if in this hamiltonian
for the simplest case which is kinetic
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part often represented by t or k e and the
potential part if the potential part goes
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to zero then its a free particle and if you
solve the schrodinger equation assuming that
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the v is never there which is basically meaning
the v equal to zero then it just becomes a
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wave equation and then continuous solution
are possible in this integrated form and the
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energy and everything else follows the continuous
picture however the movement this constrain
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that perhaps the potential goes to zero within
the certain range but beyond that
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so particle in a box is based on the idea
that you have infinite barriers at two ends
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of his own where thereof the particle will
have zero potential in a certain point in
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a certain space coordinate space and beyond
which it just goes to infinity so this principle
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of having constrain on the potential at two
edges beyond a certain length is the idea
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behind this principle of particle in a box
and the interesting part is that at the movement
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this particular problem is solved with
the idea that we have imposed free practical
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nature only within a certain region in space
leads to the situation were the energy available
24:47.870 --> 24:57.120
to the particle changes from being continuous
to discreet as if been shown here and correspondingly
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the wave functions will have discreet shapes
or forms as you are going through the different
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energy levels
so these energy states that we talk about
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n equal to one two three four as the energy
is keep on increasing these are known as quantum
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number so this is the energy quantum number
e of n where n e of n is the energy total
25:25.100 --> 25:33.470
energy and n is the quantum number essentially
telling us the state or the positions or the
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energy level that we are looking at were the
different energies are going to be there for
25:40.700 --> 25:49.350
a given wave function so this is the principle
behind looking at quantise system so practical
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in a box is one of the first cases of simplest
cases of looking at a quantised system at
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steady state which means time dependency is
integrated out and we get these results where
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the total energy of the system can always
be computed at any given quantum state and
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they keep on changing as he go to different
higher n values so as the n changes so it
26:14.710 --> 26:20.770
is quadratically increasing so this spacing
go up as we go higher in this particular case
26:20.770 --> 26:26.020
so that is the principle of particle in a
box which is the first example that is always
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taken in terms of quantum mechanics in the
next class will be going towards more few
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more of the examples of quantum mechanics
and then will take it from there further
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thank you