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