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# QUANTUM MECHANICS 1 PHY 6645

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This 2 page Study Guide was uploaded by Mrs. Linda Wiegand on Friday September 18, 2015. The Study Guide belongs to PHY 6645 at University of Florida taught by Staff in Fall. Since its upload, it has received 16 views. For similar materials see /class/206777/phy-6645-university-of-florida in Physics 2 at University of Florida.

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Date Created: 09/18/15

PHY 6645 K lngersent Study Guide OneDimensional Wave Mechanics General arguments and qualitative results will be presented in class concerning solutions to Schro39dinger7s wave equation for one dimensional systems In order to understand and apply this discussion you need to be able to carry out detailed calculations for a number of standard examples each described by a simple piecewise constant potential Potentials 14 below are listed in order of increasing complexity For each potential you should be able to 0 identify the range of energies E over which eigenstates of the time independent Hamil tonian will be continuously distributed in energy ii forbidden on general principle and iii not ruled out but if present distributed with discrete eigenvalues 0 determine the degeneracy with which any energy E would appear should it turn out to be an allowed eigenvalue 0 set up the form of a general solution wEx of the Schrodinger equation within each region of constant potential 0 apply the appropriate boundary conditions at each point where the potential changes 0 where relevant formulate the equation satis ed by bound state solutions It is not always straightforward to solve this equation 0 where relevant construct the wave function 7ERz describing a right moving wave coming from x 00 plus all scattered waves andor ii the wave function 7ELx corresponding to a left moving wave coming from x 00 plus all scattered waves 0 be able to gure out qualitatively at least the behavior when a wave packet approaches from either z 00 or z 00 0 calculate the probability current density j at any point x and calculate re ection and transmission coef cients R and T for waves incoming from x 00 or z 00 If you get stuck feel free to consult standard texts other students andor me 1 Step Potential Vz Vb m with Vb gt 0 See Shankar Sect 54 Merzbacher Sect 61 a Construct explicitly the states 7ERx and 7ELz where they exist and calcu late j R and T for each state b Verify that TE is independent of the direction of incidence as it must be for any one dimensional system described by a real Hamiltonian c In situations where both 7ELx and 7ERz exist are these two states orthog onal If not construct an orthogonalized pair of states to replace them d Check that you understand what happens when a Gaussian wave packet scatters from the step Shankar has a good discussion of this 2 Rectangular Potential Barrier V Vb a 7 with V66 gt 0 See Merzbacher Sect 62 a Construct the transfer matrix for this potential by combining the transfer matrices for two step potentials b Construct explicitly the states wERx and wELz where they exist and calcu late j R7 and T for each state Look for resonances7 ie7 peaks in the transmission as a function of E c Analyze the behavior of the eigensolutions in two special cases the in nite potential barrier7 Vb 00 ii the delta function barrier7 V 2lba6z d Understand the qualitative effects of perturbing the potential in various ways7 eg7 making V V1 for z gt 17 where 0 lt V1 ltlt Vb or making V vary linearly from Vb at z 7a to V1 at z a 3 Rectangular Potential Well Vz ilb a 7 with V541 gt 0 See Shankar Sect 527 Merzbacher Sect 64 a Construct the states wERx and wELx for arbitrary E gt 07 and calculate j R7 and T for each state Again7 look for resonances in Merzbacher describes this in some detail b Formulate the constraint on the allowed eigensolutions for E lt 0 c Check that you can nd the E lt 0 solutions explicitly in two special cases the particle in a box problem7 V 0 for lt 17 Vz 00 for 2 1 ii the delta function well7 Vz 72lba6 Shankar Exercise 523 d Work through Shankar Exercise 5267 dealing with the graphical procedure for nding bound states of the general rectangular well e Do Shankar Ex 5227 proving that every attractive potential in one dimension has at least one bound state This need not be the case in higher dimensions f Understand the qualitative effects of perturbing the potential in various ways7 eg7 making V V1 for z gt 17 where 0 lt V1 ltlt Vb or making V vary linearly from 7V6 at z 7a to 7V1 at z a 4 Double Barriers Vz 0zlia0bilxl with Vb gt 0 and b gt a gt 0 Note This problem is somewhat more complicated than those above It is included as a challenge for those students who are familiar with the more standard problems a Construct the transfer matrix element for this potential by combining the transfer matrices for two rectangular barriers ot t e transmission coe c1ent vs 0 or a m 0 an 1 a b Pl h f T E V f 18712 V d b 12 ii b 3a iii b 10a It will suf ce to calculate TE numerically

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