Quantum Mechanics I
Quantum Mechanics I PHY 851
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This 2 page Class Notes was uploaded by Quinn Larkin on Saturday September 19, 2015. The Class Notes belongs to PHY 851 at Michigan State University taught by Staff in Fall. Since its upload, it has received 17 views. For similar materials see /class/207632/phy-851-michigan-state-university in Physics 2 at Michigan State University.
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Date Created: 09/19/15
Lecture 16 February 7 2001 Approximation Methods 11 In this lecture we present stationarystate perturbation theory which is also known RayleighSchrodinger perturbation theory In such a theory one solves for states and their energies in an expansion of powers of A where the Hamiltonian is H H0 AV 1 The potential V is known the perturbation and is assumed to be small while A is assumed to be unity and is only used to keep tally of the expansion of V ie and expansion in A is an expansion in V Before one embarks on perturbation theory one assumes that one has already solved for the states m and the energies 6 which are eigenstates of Hg The goal is to express solutions for the new Hamiltonian an expansion in terms of solutions of H0 We assume that both the eigenstates and eigenenergies of the new Hamiltonian can be written an expansion if powers of A N m A N1 Aggngl A 2 En 6 AES A215 A 3 Here the terms mm and ES denote the corrections to the eigenstates and energies of order We are also free to make an assumption about normalization of the state N wmd w which is equivalent to saying that the additional parts of the wave function have no m component MAkl U 5 The Schrodinger equation mHWMmm must be satis ed to every power of A by looking at the jth power of A this gives HNUgtVNU1gt Z WWWw 7 190 Here the sum over k goes from zero to j with the understanding that Nm m and EN n We solve for the expressions iteratively That is one rst nds Elk then nds gm then given those states move onto 16 1 To nd Eff one takes the overlap of Eq 17 with 71 and using the normalization de nitions one obtains TlilEJVU U EU 8 Lecture 16 February 7 2001 For the case where j 1 one gets the lowestorder perturbation theory answer for the energy E79 Mlm 9 The state Njgt is de ned by its overlap with the states m where m 71 liven one knows Eff one can nd the HighWU by taking the overlap of m with Eq Hymn390 mVNU1gt Z Effl7n 1fj kgt 10 k0j Solving for the jth part which is unknown mgNw E E mgvgiv01gt Z Egtm1v0kgt 11 m n k1j One can then solve for the rstorder correction to the wave function EN Z Em 1 6m 772quot mV n 12 67 Using the state N 1 one can then nd the expression for E53 ES Z min 2 ltm V ngt 13 6m 67 Several important principles can be realized by observing the form of E3 First two states energies are pushed apart in 2ndorder perturbation theory Secondly if the levels are initially close the energies are more affected In fact if they are degenerate perturbation theory breaks down and one must apply degenerate perturbation theory which is the topic of the next lecture Of special signi cance is noticing that the ground state is always lowered in 2nd order perturbation theory Example We will work the problem of the Harmonic oscillator Hamiltonian with a pertur bation V 63 14 and show that the correction to the ground state energy Em answer for this case surprisingly gives the exact DegenerateState Perturbation Theory Due to terms of the form m V m 6m 6n perturbation theory falls apart when the perturbation mixes degenerate states This can be corrected by rst separting the part of the potential that mixes the degenerate states Va from the potential that mixes the degenerate states with the other states V then diagonalizing Va and use perturbation theory for V