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welcome back to the lecture we continue
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from what was there in the last lecture
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on the Heisenberg's uncertainty
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principle and I introduced a simple
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quantity called the average value or the
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expectation value so in this part of the
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lecture we will consider the formal
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definition for expectation values in
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quantum mechanics and if time permits I
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shall talk more about the postulate
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early basis that is basis with which
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mathematically we can start that these
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are the starting points and then quantum
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mechanics we can build that's called the
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postulated a basis and the postulated
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basis in quantum mechanics will also be
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stated in very simple terms the
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postulates are mathematical in nature
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but we will see simple explanations
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hopefully okay first one is the
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expectation value it is denoted by the
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average value bracket for any quantity
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the average value is in general
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calculated according to the standard
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prescription that if there are M
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measurements and these things happen
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with different outcomes for the measured
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quantity a with values a 1 a 2 a m4
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different measurements then you know
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that the average is nothing but the sum
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over i is equal to one TN a i divided by
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n now on the other hand suppose you have
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a one occurring n1 times in an
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experiment repeated many many times a 2
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occurring in two times let me change n 1
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enter to something else which is
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standard
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p 1 x p2 times and likewise a n and
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these are the only possible values let's
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see these are the only outcomes that you
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have occurs pn x then the average is
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calculated by adding all the a1 so the
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average is calculated by adding all the
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a-1 p 1 x and adding a 2 which has
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happened p 2 times and likewise adding
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all the AMS which have occurred p.m.
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times dividing by p1 p2 p3 up to PN okay
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this is also the standard way in which
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you can calculate the averages if some
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values repeat many times then you want
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to find out how many times that it has
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repeated what's the probability that
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that value is repeated and so on now the
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same thing can now be written by writing
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a 1 into p 1 by p okay where p is the
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sum of all of the experiments p IE and
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therefore p 1 by p gives you the
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probability that you got a one for the
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measurement of a and likewise a 2 into p
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2 by p which is the probability that you
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have the outcome a 2 and so on therefore
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you have a NP
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in ye p okay so this is a probability
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within brackets that a given value
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occurs and then what's the average when
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you do this experiment many many times
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this is standard way of representing
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probabilities and infant mechanics the
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you remember psystar sign represents the
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probability density for the system at a
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given coordinate or at a given momentum
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the variable X in particle in the one
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dimensional box you talk about the size
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star psy DX as the probability that the
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system is in the space between X and X
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plus DX and in two dimensions sigh star
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psy DX dy talks about the probability
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that the system is in the area DX dy
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which is enclosed between X and X plus
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DX + YN y plus dy that's what it is and
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therefore sigh star psy is a sort of a
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probability and then what you have is
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the measured value whatever that you
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measure you measure the energy or you
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measure the position you measure the
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momentum does not matter some
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experimentally observable quantity for
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which there is an operator associated
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with it in quantum mechanics the
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measured value gives you the value with
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that probability and then the average
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value is the sum of all of those things
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the measured value times the probability
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that it happens summed over all such
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possible measured values therefore
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technically if you are looking at a as a
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function of X because please remember
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this is a continuous function that for a
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is defined for each and every value of x
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so what you think is it's like psystar
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sighing which is the probability x the
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value a X that happens x DX provider
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sigh starts I represent
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so probability density which means this
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integral psystar CDX should be equal to
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one okay should be equal to one so if
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you represent this by probability
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density in quantum mechanics the average
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value a is the probability x the value
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that happens with that probability
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summed over but with one small technical
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difference namely that the operator
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corresponding to a acting on sighing
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giving you the measured value and
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therefore the measured value times the
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size star psy is represented by this
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quantity divided by integral sigh star
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psy DX which of course is set to 1 if we
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think of psy star psy as the probability
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so this is the formal definition for the
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expectation value and this a is the
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operator associated with the measured
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quantity the physical property called e8
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this is the physical property and this
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quantity is the mathematical
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representation or at quantum mechanical
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representation quantum mechanical
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representation of that physical property
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you already know because the case of
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momentum for example the operator
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associated with P is minus IH bar d by
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DX for one dimension and we what is the
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operator for the position it is just the
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X itself what's the operator for the
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energy you have already seen that it is
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the Hamiltonian operator what's he
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operated associated with angular
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momentum it's a vector and has three
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components in three dimensions so if you
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write to that in say three dimensions
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you have three components and each one
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of them is represented by
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sponding operator which is slightly
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different from the the notation that we
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have here it will involve the
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derivatives so the point is every
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measured quantity has a mathematical
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representation in quantum mechanics and
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the average value that we expect by
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definition the average being the average
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of an infinitely large number of
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measurements the average value that we
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expect that system that you see here the
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average value is thus I star psy psy
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star operator sigh DX this is a
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fundamentally important thing to
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remember and again when we introduce the
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postulates of quantum mechanics this
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will be introduced as one of the
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postulates of quantum mechanics itself
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therefore in the last lecture when I
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said that the average value of the
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position logically turns out to be
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somewhere right in the middle of the box
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for a box of length L you can calculate
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for one dimension the average value X to
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be sy n of X if the state of the system
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is sy n then the average value in that
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state is sy n of X the position operator
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X and sine of X DX and sian being
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normalized to root 2 by L whatever you
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have that is root 2 by L times sine n pi
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x by l you have for the integral 0 to l
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sine n pi x by l x sine n pi x by l DX
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so this gives you when you do the
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integral this gives you the answer l by
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2 so very simple integral it is x sine
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squared X and the sine square axis of
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course you can write it as 1 minus cos
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2x x 2 and then you do you do the simple
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integral on x and x cos x it's very easy
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to do likewise the average value
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the momentum or the particle was also
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argued out to be zero based on the fact
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that the momentum is a vector and
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therefore it has a positive or a
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definite negative direction at any point
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in space if you do that the average
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value of the momentum will turn out to
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be zero for the particular one
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dimensional box and that's also easy to
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verify by writing this down as 2 by L
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sine n pi x by l now you remember to put
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the operator in the middle IH bar d by
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DX sine n pi x by l times DX now the
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derivative of the sine will give you a
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cosine n pi x by l you can see
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mathematically and the sine cosine will
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give you a sign to n pi x by l x 1 x 2
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but that integral between 0 to l is a
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full sign way and therefore that goes to
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0 ok DX so it's easy to verify simple
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relations like the expectation values
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for position and expectation values or
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momentum and these are the two things
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that you can think about and if you have
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the kinetic energy you already know that
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the particle in the box is only kinetic
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energy inside the box therefore the
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total energy is the same as that of the
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kinetic energy and you can see that the
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average value e for the particle in the
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state's I n is H squared n square by 8
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ml square that also comes out so these
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are simple prescriptions for doing
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calculations for the average values
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based on quantum mechanics no please
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remember these are average values
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expectation values that is these are
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what are expected when you do many many
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measurements but if I do a single
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experiment what value will i get is
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there a prescription in quantum
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mechanics that's what this equation
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tells you if the state of the system is
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in this function is in this state sy n
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for the particle in the one dimensional
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box it doesn't matter how many times I
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make measurements on that state for the
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energy
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for the energy doesn't matter all the
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times I will get only one answer namely
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e n sy n of X it's like the simple
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analogy you have a dive at six phases
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and you print only one dot on all the
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six phases therefore the die has only
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one state namely with an outcome of a
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single dot no matter how many times you
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throw the die you get only one dot as
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the answer because that's how you
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prepared the state of the system such
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states are called I en states in quantum
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mechanics in the case of a die you have
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six possible things that you have for a
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single die when dot-to-dot three dot
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four not five dot and six dots therefore
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you have six possible outcomes in the
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case of a particle in a box if I make a
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measurement and I do not know what the
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state of the system is what result can I
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expect for a single measurement I have
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already told you what result we can
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expect farl very large number of
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measurements and then what's the average
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that's what we did before what is it for
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a single measurement if you ask that
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question the answer is one of the
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eigenvalues of this is case of the dye
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which is a normal die or a regular die
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which has six different phases with one
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two three four five six dots there are
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six possible outcomes multiple outcomes
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therefore in a single experiment of
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throwing the dye we get a dot or two dot
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or three dot all with identical
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probability is 1 by 6 if the dye is a
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perfect cube because that I is not
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prepared in any other way likewise in
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quantum mechanics if the probabilities
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for all outcomes are uniform then in a
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single measurement one of these energies
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will be the outcome for the particle in
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the box if you measure the energy only
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one of the ians is possible which of the
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en statistics Einstein was very unhappy
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he said God doesn't play dice and needs
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more told him don't tell god what to do
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ok but there is an inherent statistical
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character built in the measurement
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outcomes according to what is called the
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Copenhagen school or the niels bohr
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school of quantum mechanics which is
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still practiced by most of us a single
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measurement will give you one of the
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eigenvalues and will result in the state
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of the system being one of that eigen
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state the eigenstate corresponding to
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that eigenvalue therefore if we make a
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measurement for a particle in the one
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dimensional box in an arbitrary state
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that they do not know what it is the
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result that we will get out is only one
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result and the that result the
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measurement will give you an eigenvalue
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en and the state of the system will
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become sy n this is fundamental in
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quantum mechanics and if the state is
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already an eigenstate then no matter how
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many times you make copies of that state
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and how many times you make the
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measurements you will always get the
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eigen value that is why I mean I wrote
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the average value for e in the last
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slide or a few minutes ago if I go back
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to the screen I've written not already
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here if the state of the system is sy n
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the measurement of energy every time
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will give you the same value H squared n
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square by 8 ml square and since it's the
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same value in all measurements the
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average is also the same as the single
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measurement if you know the state of the
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system very precisely that's what it is
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if you do not know the state of the
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system to be an eigenstate but an
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arbitrary sigh ok this is the result for
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an arbitrary sino let me write down the
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tab sigh here now if the system is in
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the state's I a measurement of a
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quantity physically will give you sighs
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talk SI DX integrated over the domains
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completely available to the system and
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for particle in the one dimensional box
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it's between 0 to L that's the whole
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space available to the system therefore
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you take the average by adding all the
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probabilities it's very easy to see that
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the same one is what you get because if
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you write e of sy n which is an
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eigenstate of the Hamiltonian operator
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then you see that this relation is sy n
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H sy n DX and you know between 0 to L
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you know that HCN is e n sy n and you
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know that sign is normalized therefore
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the answer is 0 to L say n say n DX and
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with Ian and this is equal to 1 and
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therefore the average value is the same
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as the eigen value for E and let me stop
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here and we will continue these
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discussions over the next few weeks on
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various aspects but it's important for
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us to remember that the expectation
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values are fundamentally
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in quantity and the fact that that
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inverse of a function and its complex
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conjugate is a very meaningful reason
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the very important reason why one is
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always interested in solving the
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Schrodinger equation to get the wave
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function first that the wave function
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has an interpretation due to probability
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is one thing but the wave function is
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extremely important in the actual
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calculation for the expectation values
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and the measurements and therefore you
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have a function which you cannot
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physically explain or visualize but it's
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very important and very useful for
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calculating average values as
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calculating other quantities called
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matrix elements calculating the average
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values through various processes and so
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on therefore the wave function has come
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to stay with all of us we'll continue
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00:21:03,669 --> 00:21:07,080
this in the next lecture thank
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you