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welcome back to the lecture the earlier
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lecture talked about in the earlier
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lecture I talked about the particle in
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the one dimensional box and in the
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current one let's discuss the particle
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in a two dimensional two dimensional
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model or two degrees of freedom model
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the particles position coordinates are
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given x two x and y 2 coordinates in a
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plane orthogonal to each other and then
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we discuss the quantum problem the
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barriers are in finite therefore if you
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remember the problem p squared by 2 m
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plus v which is the energy term gets
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changed to or its read it on us p x
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square by 2 m plus p y squared by 2 m
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plus B and the PX is replaced in quantum
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mechanics by D minus H bar square by the
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term minus H bar square by 2 m the
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partial derivative now because we have
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the wave function as a function of two
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coordinates x and y and the momentum in
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the X Direction is given by the partial
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derivative and this is the square of the
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momentum so you have minus H bar square
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dou square by dou X square by 2 m and
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correspondingly 4p y squared you have
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those square by dou Y square this is the
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operator part for the kinetic energy of
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the Hamiltonian plus
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and the wave function is a function of x
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and y plus V some potential x SI xq mi y
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is equal to e phi x comma y this is the
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two dimensional Schrodinger equation in
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which you have got the H this term plus
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the we H acting on the psy giving you
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ESI and for the current problem of
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particle in the 2d box we consider thee
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to be infinite for all values of x other
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than from zero to L and all values of Y
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from 0 to some other say a or l1 or l2
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it doesn't matter if it's a rectangular
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box if it's a square box then
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essentially you are looking at the let's
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see if we can have a square
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something like that so 0 to L and why is
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also 0 to L only in this region we are
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looking at the particle properties and
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the particles behavior and for all
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others we have V is infinity for all
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values of x less than 0 or equal to and
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for all values of x greater than or
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equal to L and likewise for y less than
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or equal to 0 y greater than or equal to
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L so this is the infinite boundaries
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that you have it's not the single
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dimensional quantity but it's the
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surface in a sense that we protect the
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particle from escaping this region and
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inside V is 0 between x and y el between
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ylem and this is a square box ok so if
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we do that obviously the differential
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equation simplifies without the sturm
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and you have a derivative square in one
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direction your derivative square in
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another direction and then you have the
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sigh of XY a such a problem is easily
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solved by is written in terms of your
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product of a function of X alone and a
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function of white alone with this choice
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it is possible to separate this equation
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minus H bar square by 2 m dou square by
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dou X square plus del square by Del Y
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square Phi of X comma Y
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is equal to e times sine of X comma Y
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into two equations namely minus H bar
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square by 2 m d square by D X square X
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is equal to e 1 of X and minus H bar
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square by 2 m d square by D Y squared
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times y is equal to e 2 times y but
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these two constants a 1 and D 2 or
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constrained by e1 plus e2 is equal to e
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okay the actual separation of this is
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given in the notes that accompanies this
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video lecture therefore I would request
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you to look into that to see how this
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equation is separated into two one
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dimensional equation 14 x + 1 what why
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with the constraint that the energies
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for the two one-dimensional problems are
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related to the total energy as the sum
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v1 plus c2 now let us see the solutions
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that quantity which I have written on
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the board is namely
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this is the X equation and the
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corresponding Y equation is that
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obviously each one of them is like a
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one-dimensional part particle in a box
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therefore the solutions for each one of
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them will have a running quantum number
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for that particular equation the x
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component of the wave function will be
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given by the solution it's similar to
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the sigh of X that we wrote except that
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now we call it X of X and now this will
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have a quantum number going from one to
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three to some value which we call as n1
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in an exactly in an identical manner the
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Y equation will also have a free quantum
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number n 2 which will run from one to
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three to whatever that we take but
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please remember these two quantum
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numbers are not independent in the sense
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they are connected to the total energy
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the requirement that e1 + e2 is equal to
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e now remember the expression for e 1
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from the particle in a one dimensional
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box it is H square by 8 m l square m 1
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squared a free quantum number in the
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sense it's takes one two three integer
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values and e2 is also given by H square
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by 8 ml squared times n2 square such
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that this equation is satisfied
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therefore you have h square by 8 m l
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square times n1 square plus n2 square is
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equal to the total e so this is the only
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constraint that comes out in the
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separation of the two dimensional
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Schrodinger equation
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that the total energy is the sum of the
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two one-dimensional energies and that's
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possible because we don't have a
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potential which couples the two
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dimensions we put B is equal to zero and
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up for the the method of separation of
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variables separation of variables ok we
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have separated the x and y from the sigh
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of XY if you recall the sigh of XY we
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have separated that into the X equation
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and the y equation so that process is
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called the separation of the enables now
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how do these functions look like
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obviously you have the solutions for the
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quantum number n 1 in terms of the
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one-dimensional solution that you have
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seen in the previous lecture group 2 by
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L sine n 1 pi x by l and the energy is
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given by n 1 square and like phase for
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the y with the n2 square and with the
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constraint that the total energy e n 1
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plus en tu is e + 1 n 2 you have seen
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that okay what about the wave function
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the wave function now if you see this
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the wave function sigh of n1 n2 because
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it's obviously specified by the two
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quantum numbers n 1 and n 2 has the
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independent function x with the quantum
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number N 1 and Y with the quantum number
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n to each one is in a North organelle
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direction okay therefore you see this
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interesting thing next line when we have
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n 1 is 1 and n 2 is one when we have
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that case which is the starting point
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what is called the lowest energy for the
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particle in interdimensional box you can
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see that the wave function is given by
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say 11
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X comma Y and is given by the product of
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the two functions that you saw the X of
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X and Y of Y which gives u sine Phi X by
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M and sine PI Y by young let me repeat
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this when the quantum number is 11 the
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wave function is given by sy11 and it's
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given by the product of two by L sine pi
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x by l and sine PI Y by M and the energy
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is of course the sum of 1 square plus 1
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squared times the whole thing therefore
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the energy for this process e 11 is H
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square by 8 ml square times two what is
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interesting is the next choice you have
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sy n 1 n 2 is X of N 1 y of n 2 it's
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possible if n 1 is not equal to n 2 it's
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possible to have the wave function given
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by X of n 2 and y of n 1 because the
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energy is simply proportional to N 1
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square plus n 2 square times H square by
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of course 8 ml square which is the
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proportionality constant therefore you
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see that you have the same energy but
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you have to physically different states
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X of N 1 y of n 2 and X of n 2 y of n 1
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both states have the same energy this is
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what is called degenerate state
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degeneracy is too because there are two
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states which have the same quantum
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energy but we have different quantum
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states this is the introduction for the
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particle in a 2d box that the degeneracy
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is the additional factor now how do
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these things look like let us simplify
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this picture now I have a whole series
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of functions here with which you can
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fill up any number of pages if you wish
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you see that n 1 is 2 into equal to 1
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corresponds to the wavefunction sy21
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with sine 2pi x sine pi y by young and
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young 1 1 into 2 gives you the other
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function namely sine pi x by l sine 2pi
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x y by L and the energies are the same
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so if the quantum numbers are identical
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there is no degeneracy but if the
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quantum numbers are different for a
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square box because we have chosen the
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length L to be the same the square box
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gives you the solution that you have a
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minimum degeneracy of two if n1 is not
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the same as in two and you can see that
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for three and two that you have here the
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wave function sine three pi x by l and c
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2 pi by l and then two and three which
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is sine 2pi x by l sine 3 pi y by L so
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the axis choice the quantum number
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choice for a given axis determine see
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the functions state how do these things
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look like if we plot them I mean this
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plot looks fancy but actually doesn't
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have much interpretation or meaning but
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it's worth seeing the product wave
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function in two dimensions okay so you
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see the wave function you see the wave
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function sy11 using this picture
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it's a half-wave similar to what you had
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in your particle in a one dimensional
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box in the X direction and it's also a
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half way in the Y direction as you can
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see through the projection in the X
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Direction here of this rough and on the
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y direction also you have the same thing
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identical okay what about the sine
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square the size square which is
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associated with the probability that the
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particle be found not in a small length
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region DX but in a small area DX dy
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please remember sigh XY if you do that
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sighs square DX dy is the probability
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that the particle will be in the small
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rectangular region between X and X plus
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DX and y and y plus dy that's a small
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region and you can see that the size
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square is given like this therefore you
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can create I mean you can visualize what
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would be the probability exactly the
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same way that you have visualized the
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particle one dimensional box except that
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now we have emotional on the plane and
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now what is interesting is when you go
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to different quantum numbers where there
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is degeneracy sigh 12 if you look at
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this sigh 12 is quantum number one for
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the X direction and quantum number two
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for the y direction therefore this is
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the quantum number this is the quantum
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number for the X direction and you can
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see that it's a half wave which is
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either up or down it's either
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positive or negative the reason being
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the Y Direction wave is a full wave so
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in this direction what you have is if I
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may draw this the wave function looks
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like that in this direction the wave
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function looks like that therefore when
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you take the product of these two
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functions in negative side makes this
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wave function negative for half the
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length and therefore you see that for
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half the length you have either a
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positive wave function or you have a
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negative wave function that's only for
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the wave function we know that the wave
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function is not that important it's a
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square of the wave function which is
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important for probability interpretation
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and you can see that size square which
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removes this negative character of the
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function gives you now very beautifully
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the two n equal to one case for the
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x-axis and the n equal to 2 if you
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remember the graph that you've had for n
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equal to 2 or the y axis and this is the
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x axis therefore the features are
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captured the wave function features are
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captured when you're doing a surface
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plot and you can see that the pictures
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can be created for a larger number of
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them but there is a limit two dimensions
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and in three dimension we probably can
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use color at the most to distinguish the
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the function from the three axis but
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that's it you cannot visualize this for
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n dimensions so let us conclude this
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part of the particle in a two
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dimensional box with some examples of
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the wave functions and the squares of
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the wave function for different quantum
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numbers so here is a 2-1 as opposed to
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12 and you see all that happens is that
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for a 2-1 the wave function along the x
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axis is like this and the wave function
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along the y axis it's like that
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okay and you can see that actually sorry
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this is in the wrong direction so let me
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erase that because you are 0 starts from
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here therefore have not and this is the
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x axis that's the reason why part of it
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is negative and the other part is
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positive and the square of the wave
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function you can see that there are two
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humps along the x axis and along the way
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axis it's a quantum number one so you
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have only one similar to the one
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dimensional y axis and let's see one or
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two more examples and let me stop with
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that this is it I mean the exercise here
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what does this picture represent there
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is one here along the x axis and there
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are three peaks therefore you have this
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is it Y is 3 and X is 1 so it is sigh 13
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squared XY so the lecture notes give you
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many more such pictures but in the next
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part of this lecture we will see what do
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all these things mean in terms of
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probability calculations and in terms of
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a new idea called the expectation values
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will stop here for this particular part
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of the lecture thank
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you