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Lecture 17: Simple Harmonic Oscillator: Average Values for Position and Momentum - YouTube
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[Music]
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come back to the last part of this
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elementary lecture on harmonic
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oscillator in this part let me do a
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simple calculation and demonstrate how
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to do elementary integrals so this part
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is on the average values for position
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and momentum operators in quantum
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mechanics for the harmonic oscillator in
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fact it's extremely simple if I have to
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use the wave functions as given here I
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don't think this lecture should be there
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in the first place because the average
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value for the position and the momentum
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for a harmonic oscillator centered at X
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is equal to zero is actually zero
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therefore what are we talking about okay
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we do talk about the average value for
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the energy partition if we discussed
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that the harmonic oscillator Hamiltonian
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has two non-trivial parts the kinetic
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energy as well as the potential energy
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part the kinetic energy is given by the
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momentum squared operator divided by
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twice the mass of the oscillator and the
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potential energy is given in terms of
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the harmonic oscillator force constant
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half KX square therefore some integral
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calculations involving the hermite
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polynomials and the Gaussian functions
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can become unwieldy as we say when the
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higher-order functions are involved and
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there are better ways of handling
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harmonic oscillator using what is known
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as the operator representation or it's
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also called occupation number
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representation by some physicists and
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the others call it as the harmonic
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oscillator raising and lowering
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operators so there are many different
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ways by which we can study them however
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let's stay with the statement that the
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position and momentum the average values
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are zero how do we show that it's very
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easy I told you that if we have a
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an integral of an odd function f of X is
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minus F of minus X then this is zero now
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remember the wave functions for the
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harmonic oscillators are given in terms
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of say let us take say zero of X
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it's quite obvious since the
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probabilities of finding the oscillator
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on the plus X site for any given X is
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the same as the probability for finding
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the oscillator for the minus X that X
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the probabilities are evenly distributed
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you can easily see that the positions
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with the value of a minus x on the
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negative side and a plus X on the
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positive side multiplied by identical
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probabilities cancel out therefore if
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you were to do this the integral is X in
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north squared e to the minus alpha x
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squared because it's a square of the
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wave function from minus infinity to
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plus infinity and you know x DX and you
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know X e to the minus alpha x squared is
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an odd function therefore this integral
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is 0 this is true for any wave function
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this is at sy 0 the expectation value is
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calculated please remember the
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expectation value of any operator in the
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states I is given by psy star a acting
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on psy.d Tahoe divided by the integral
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psy star psy dito so since this is a
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normalized wave function for us this is
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equal to 1 and here we have put in a as
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the position operator which is the X
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itself and the D tau and the limits are
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from minus infinity to plus infinity DX
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so this is what we had done therefore if
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you calculate this for any state say en
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please remember that
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it's going to involve this integral
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namely in n square the normalization
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constant minus infinity to plus infinity
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X e to the minus Alpha X square but now
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it will involve the hermite polynomial H
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en route all for X times
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in Brooke Alpha X DX okay
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therefore you see that if the hermite
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polynomial is odd for any given odd in
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then the two odd functions multiplied
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together gives you an even function and
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therefore you see exponential is already
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an even function the product of the two
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have great polynomials is an even
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function because they have the same
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hermite polynomials of order M and X is
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odd and therefore this is an integral of
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an odd function between symmetric limits
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minus infinity to infinity or function
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of X DX and therefore this is you know
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therefore the average value for the
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position of the harmonic oscillator
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independent of what state the harmonic
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oscillator is in is always the midpoint
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the point where the harmonic oscillator
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is at equilibrium and the potential
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energy is zero at that point
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now if the harmonic oscillator for
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example is not centered at X but we have
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a slightly different coordinate system
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such that we represent the harmonic
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oscillator Y is a sigh of XA of Y let us
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do that as exponential minus y minus y
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naught whole square by two where Y
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naught is the center because you see
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this function will have a maximum at Y
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is equal to Y naught and therefore this
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is a Gaussian shifted from Y is equal to
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0 to a Gaussian shifted at Y is equal to
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Y naught so if you have it at 0 this is
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now the Gaussian shifted at y 0 and this
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point is the midpoint which is y is
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equal to y 0 therefore if you calculate
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what is the average value for this
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function
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for the position namely what is the
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average why if you do that you can
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easily show that y times e to the minus
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y minus y naught whole square dy between
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the limits minus infinity to plus
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infinity and with the normalization
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constant M square some some
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normalization constant square you can
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show that this will give you y naught
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which is the value at which the function
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is on the average as zero potential
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energy and it's the next point what
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about the momentum please remember the
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momentum operator is minus IH bar d by
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DX it should be obvious that the
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derivative operator is something like an
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odd character because it changes an even
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function d by DX of an even function
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will immediately become an odd function
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or d by DX of an odd function will
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become an even function
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for example if you do the derivative D
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by DX of X an odd function because it
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changes sign is going to be one which is
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even independent of the sign of X in
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this case of course independent of X as
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well but what about D by DX of X square
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it gives you two the X this is even this
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is odd
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in a sense you can see this because the
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derivative is has the odd character
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therefore you can see immediately that
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when we talk about the average values
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for the momentum at any given wave
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functions I n if we have to calculate
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the average value of the harmonic
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oscillator in the state for the momentum
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operator then the integral is in n
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square the normalization constant
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between minus infinity to plus infinity
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please remember no momentum being a
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derivative operator minus IH bar Li by
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DX you need to put the wave function
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star here and the wave function itself
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this is a real function therefore you
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have e to the minus Alpha X square by
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two okay
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H en route alpha X that's the sy n star
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on the side this is the operator P and
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acting on the wave function Alpha X
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square by 2 H n root Alpha X D X now
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please remember this is odd or even
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depending on whether n is odd or even
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okay therefore if you take the
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derivative of an odd function you will
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get an even function but please remember
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if H n is even
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then the derivative of H M will give you
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an odd function therefore the product of
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the two is odd if H n is odd the
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derivative of the same H in here which
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is the odd function will give you an
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even function and therefore the product
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is again or therefore the integral for
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any state sy M of the average value for
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the momentum is also zero so I mean it
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looks like it's a trivial result but
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again it's very easy to imagine that if
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the harmonic oscillator has forward
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momentum in this direction and if it has
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a backward momentum in this direction
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because momentum is a vector and
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therefore you can always say forward in
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one direction means backward in the
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other direction since the probability is
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for the value the absolute value of the
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momentum for any given X the property
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density is the same for whether it is
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plus X or minus X the averages add with
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the vectorial sign of P in the plus the
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X direction the probability remains the
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same but the value of the momentum is
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positive in the negative x direction the
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probability density is the same for that
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value of x but the momentum is negative
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because it has a negative sign and
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therefore the moment I cancel each other
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for every such value of xn minus X X and
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minus X therefore the integral should be
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visualized as being close going to zero
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because it has this odd character the
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last important point for the harmonic
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oscillator has something to do with the
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average values for the kinetic energy
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and the potential energy which I would
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want you to calculate but they are not
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zero average values for
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kinetic energy of the harmonic
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oscillator and the potential energy of
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the harmonic oscillator so the kinetic
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energy term is given by minus H bar
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square by 2m d squared by the x squared
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this is the operator for the kinetic
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energy the potential energy operator is
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of course half K x squared X being the
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operator so when you talk about the
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average value of kinetic energy at sy n
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you discussed this quantity namely n n
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square minus infinity to plus infinity e
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to the minus Alpha X square by 2 H in
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root Alpha X this is the sy in with the
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normalization constant n and then you
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have the operator which is minus H bar
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square by 2 m d square by d x squared
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again acting on the wave function Alpha
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X square by 2 H in root Alpha X DX this
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integral is not zero because if H n is
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odd H n is odd and therefore it's an R
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times the odd function this is a second
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derivative the second derivative does
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not change the ordinates or the evenness
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of the function if it has that character
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an odd function it remains an odd
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function and even function remains an
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even function therefore the kinetic
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energy the average value of the kinetic
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energy for the harmonic oscillator
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after all it's a square of the momentum
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it doesn't depend on the direction of
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the momentum
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therefore for positive x and for
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negative x they keep adding the momentum
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for each value of the position so this
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is nonzero please calculate to this and
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I would suggest that you do this for n
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equal to zero or this will be part of
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one of the quizzes that you will find
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and in a similar way the potential
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energy
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average value for sigh n is given by
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again from minus infinity to plus
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infinity but since it is x squared you
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can write 1/2 K X see X does not change
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except to multiply then you can write
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the wave function sine squared X DX and
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again this is not equal to 0 for the
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ground state the harmonic oscillator
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average value for the kinetic energy for
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sy 0 is equal to the average value for
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the potential energy size 0 and that's
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equal to H bar Omega by 4 and please
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remember the e 0 is H bar Omega by 2
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therefore the average values for the
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kinetic energy and the potential energy
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are exactly distributed as equal
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contributions to the total energy but a
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similar expression can be calculated for
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various values of the wave functions and
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the various values of the kinetic energy
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and those exercises I believe it as
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exercises for you to work out in detail
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the harmonic oscillator is an extremely
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important problem as far as the chemists
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are concerned in the sense that if you
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want to study the vibrational
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spectroscopy if you want to study
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vibrational Raman spectroscopy infrared
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or Raman spectroscopy and if you want to
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study electronic spectroscopy with the
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vibrational coordinate changes and so on
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these are all there in the spectroscopy
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applications in chemistry the harmonic
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oscillator model is crucial and the fact
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that the average value of the position
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goes to 0 has something to do with the
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the transition probability connecting
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two different harmonic oscillator
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elements we will see more of that when
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it comes to the study of molecular
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spectroscopy and and when we study the
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infrared spectroscopy until then this is
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a sort of a
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very elementary summary for the
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solutions of the harmonic oscillator and
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how they behave and what can be learned
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from them and this can be used to build
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the next level of study of harmonic
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oscillator using raising and lowering
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operator formalism that will form a part
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of a much more advanced course later
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that I will be offering until then thank
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you
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