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ya welcome back to the lectures
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the purpose of
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today's or this lecture is to introduce
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elementary mathematical functions a few
of them
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that you will need
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time and again during this course either
us
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solutions for the quantum problems that
you study
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or functions which you will need
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in order to understand the behaviour
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the mathematical and the spectroscopic outcomes
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of experiments and so on so let
me start with something very
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very elementary and this lecture is
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titled elementary mathematical functions
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used in our course it's not
exhaust you in 20 minutes
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I cannot say too many things the
first
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function that we will look at
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are the two sets
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of function exponential e
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to the plus or minus kx
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k is
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a real constant if k is
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imaginary or complex it has its own
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different set of properties k is real constant
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let's look at what the exponential
actually means I think
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most of you remember the plot
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when we picture the exponential
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as a function of the variable
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x and you write this the y-axis
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as exponential k of x for a given value of k
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if you plot this function
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obviously
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at x is equal to 0 this function has a
value 1
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we will start somewhere here some scale
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and then you can see that if k is
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positive k is
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greater than 0 then this
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is a growth function
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ok growth
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meaning that
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the function increases in its value as
x increases
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now thats for one value of k
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now let me call that k as k0
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some constant now suppose I have
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a different value of k
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the function may again start from one
but it may go something like this
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ok or it may be slower for another
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value of k or
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it may be really fast ok
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so let us do it by making this as k0
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k1, k2, k3
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yes some different values of k
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what's the relation between these it's
quite
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obvious that this goes much faster
for a given value of x then any
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other function obviously k3
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is larger than k2 than k1
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than k0 ok that's a
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pictorial representation of the function
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that's not the understanding of the
function understanding of the function
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slightly different
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I mean if we know that the constants are
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in this order the function when it is plotted
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look like that what's the understanding of the exponential growth
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let us try and understand this function
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now
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the way to see this is to consider this
particular case
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namely for x we start with one
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when x is 0
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at a time
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sorry at a particular value of x ok
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the function reaches
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some value here when
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x becomes this is x1
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2x1
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one, two, four.... two
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one at 2x1
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so at x1
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we have here and
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at 2x1 the function has the value
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4 for example some units and
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at 3x1 it reaches the value
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8 that is
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every increment identical increment
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if the value of the function
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doubles its previous value
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such a behavior is
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an exponential growth such a behaviour
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exponential this nothing special about
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being doubling being a double or
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doubling the function may start with
some value
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in the first interval whatever value that
it becomes from one it may become three
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but in the next same amount of interval
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the function 3 becomes three square
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in the next interval three square becomes three cube
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such growths are call
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exponential growths if we do it for three
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quite obviously it is even steeper or in this picture itself
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if you do it for 3 you were somewhere here
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and then 9 you were somewhere here so the point is
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you have here and then
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for 2x1 you are here
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and you see that the function goes even steeper
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this is what is meant by k
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this is what is implied by k. k tells
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how fast, in what ratio
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that the function grows
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with respect to the variable x, ok. this is
for
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exponential growth now what about k
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less than zero, that is negative which we call as exponential
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decay ok. e to the power kx
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k greater than zero, growth.
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k
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less than 0 it is decay, of course in both cases
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k real so
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you have e to the minus some value of k whatever is the number
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of x. ok
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if you plot this
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tests exactly similar but
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and an image
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kind of a picture, we will start
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with x
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and e to the (power) kx
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is zero that is, its one
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ok and suppose
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for a value x1 the function
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becomes one half, one by two
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then that's the value, for the same
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interval x1, that is 2x1
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the function
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decreases by the same fraction
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one half becomes one fourth, is that
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one fourth in the next interval
identically it will
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3x1 becomes one
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8 such a behavior
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if you connect is exponential
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decay, that's
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also exponential that's the nature of
the exponential function
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the ratio for
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of the function for any given period
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the ratio is the same from
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that is the ratio of the value before
the value after, if you take that value
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that ratio
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that ratio means for one particular
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interval ratio so here the, this is
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what is called the half life if you are
interested in
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decay processes and the
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number becomes one half at a particular time t1
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if you write the function exponential
kt
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where t is time
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if you do that instead of x you use t then you have t1
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2t1, 3t1 and so on so the
exponential is an extremely important
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function
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having this specific characteristics
and
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the derivative of an exponential d by
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dx of e to the (power) kx
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is k e to the (power) kx that you should know
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and the integral of zero
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to some constant c1, finite value
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of an exponential kx dx
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is obviously, you can calculate that
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if you
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don't put the limits you know that is
going to be
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one by k
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times the e to the (power) kx
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now for we have to be careful that this
integral
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is for a finite limit if you go from and
if k is positive
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if you go from zero to infinity this is
infinite
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the function is unbounded the integral
is
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infinite, if it is negative you know that
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0 to infinity
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if you have, exponential -kx
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you know what the answer is, 1/k
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so the properties of integration the
properties on
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differentiation and the simple nature of exponential is
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one extremely important function for you
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the second function that you need to worry about
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is, also an exponential but it's not
called an exponential
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it's called
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Gaussian if it is minus
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we usually call it a Gaussian function
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this is
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again important in all the quantum and spectroscopic studies that you have
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what's the nature of this function, unlike
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what you saw here, it's not
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increasing forever, its in fact is
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decreasing forever because if k is real
and positive
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this whole thing is decreasing
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as x either increases from 0 to infinity
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or x decreases from zero to minus infinity
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because the function is dependent on the square of x
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this is also known as an even function, and
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the shape of this function when you plot it
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are, x this is + infinity
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and this is - (minus) infinity
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you do that at x is equal to 0
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this whole thing is exponential 0, it is one
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and for all other values of
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x positive
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and x negative, it is symmetric
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about the line and this is
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obviously bell shaped, even function
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ok, now
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again, if e to the minus k x square for
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one value k1, suppose I want to
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plot this for another value k2
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where k2 is
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less than k1, it's quite clear
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that for any given x
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this will be smaller because k1 is
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more than k2, this one is smaller, this one is slightly larger
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and therefore you can see that the function that if k2 is less than k1
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you will have a
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more
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elaborate a wider function this is k2
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ok, k2, if
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you have k0, sorry
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yeah we have k2
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is less than k1 and you have
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k0 now less than
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k2 less than k1, if you do that then the
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function is even that, k0
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ok, so the smaller the value
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of the exponent
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the wider, the
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more extended the function is or the
opposite the
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larger the value of these case the more
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narrow the narrower the function is, yo go from
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in the reverse direction this is another
function which is extremely important
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for
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your calculations
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in spectroscopy and quantum
mechanics
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and again you must know that the derivative
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of this function e to the minus(-) k x square
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is minus 2kx
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e to the minus(-) k x square
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and the integral of this function
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from 0 to infinity e to the minus(-)
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k x square, dx is
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given by 1/2
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root pi by k
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this is a property
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and this being an even function
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you can also do the integral of the same function between
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the entire x-coordinate
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e to the minus(-) k x square dx and
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that is exactly twice this internal it is root pi over k
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so these are
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standard integrals known as Gaussian integral
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this is another function that
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you would need in studying the
properties of harmonic oscillators and
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quite a lot
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in the in understanding spectroscopic
line shapes and so on so
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basic properties you should be familiar with
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similar functions that we will
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see which are slightly modified from
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these functions namely multiplied by a
polynomial
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instead of e to the minus(-) k x
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we may have x multiplying e to the minus(-) k x
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f1(x), some function we may have
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x square e to the minus(-) k x, and so on
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many many such functions
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and also for the Gaussian we will have e to the minus(-) k x square
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and we will have x e to the minus(-) k x square
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we will have x square e to the minus(-) k x square
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so on, these are functions which we will see
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time and again in the limited 6 to 8 week course
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and the properties and the shapes of
these things
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should be known to you go back and draw some of these things
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let me draw two of them before I
conclude
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this small introduction to
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the mathematical ideas, suppose we want to plot
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x e to the minus(-) kx
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for some value of k
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and we will do that for
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the positive segment, please remember
we can't try
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to do this in the negative segment that is for the negative values of x
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you see the exponent, this whole thing becomes positive and therefore
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e to the positive number keeps on increasing, therefore on the negative side this
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function increases
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beyond limit for very large values, therefore we will
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stay from zero to some positive values
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and you can see that at x is equal to 0 this is 1, and this is 0
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therefore the function is 0, and
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for any, for any other x as
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x increases this increases e to the minus(-) kx decreases
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and therefore there is a competition between x and e to the minus(-) kx
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up to a point and that point is
obviously call the
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maximum of that function and after that point the exponential minus(-) kx
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drops of so much more quickly than
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x increasing that the competition is lost
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the function decreases forever and therefore
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there's a maximum and then the function goes to zero
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and how do we determine this maximum, we take the derivative
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of this function e to the minus(-) kx
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x and then set
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that equal to zero then you will find out
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that the function has a maximum, the
derivative of this is
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clearly it's a uv so you can do that and when you set
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the derivative to be 0 you will get a value for
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the maximum, so that is the maximum here
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that is an exercise calculate the maximum
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and similarly when you go to x square
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you would see that x square
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increases again and exponential minus(-) kx decreases
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since x square increases for larger values of x much more than
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x itself the competition is taken over for a little longer
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or a little larger value of x, and
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after that again the exponential wins over
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in fact the exponential wins over
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for all power of the polynomials of x
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if you go sufficiently far enough on the x
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eventually it's the exponential that will kill the whole thing
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it's very very important therefore if
you think about x square e to the minus
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kx
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I can only say that it would be
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or, somewhere else
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00:20:51,570 --> 00:20:56,950
the maximum would be some where else, farther away
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and the value of this will also be
different
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so these polynomials
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multiplied by the exponential are extremely important in understanding the
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00:21:07,620 --> 00:21:11,150
wavefunction and the properties of the wavefunction for
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hydrogen atom the polynomials involving
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the Gaussian and
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00:21:18,640 --> 00:21:22,180
the polynomials in front of them x and x
square and so on
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00:21:22,180 --> 00:21:25,650
are important in understanding the
harmonic oscillator
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00:21:25,650 --> 00:21:29,520
and other elementary models in quantum mechanics, therefore please keep this in
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mind
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00:21:30,100 --> 00:21:33,330
and please attempt some of the
exercises
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00:21:33,330 --> 00:21:37,370
given at the end of this lecture and
until we meet next time
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thank you
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