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INTRODUCTION

Harmonic oscillator perturbation X, X 2   and X 3 Presented by Abdulaziz Alfehaid Supervisor: Prof. Dr. M.A.ZAIDI. INTRODUCTION.

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INTRODUCTION

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  1. Harmonic oscillator perturbation X, X2  and X3Presented by AbdulazizAlfehaidSupervisor: Prof. Dr. M.A.ZAIDI

  2. INTRODUCTION Perturbation Theory is an important and powerful method in physics. It enables us to make progress when a physical system is too complicated to be analyzed exactly. The essential idea is to solve the behavior in steps. First we approximate the system by some simple Hamiltonian whose Schrödinger equation we know how to solve. Then we add the bit missed out, and use perturbation theory to calculate how our previous results (energy Eigenfunctions and eigenvalues) must be changed.

  3. UNPERTURBED HARMONIC OSCILLATOR IN ONEDIMENSION Hamiltonian of system , n=0,1,2,3…

  4. TIME INDEPENDENT PERTURBATION THEORY * *

  5. PERTURBATION and PERTURBATION in The first order correction on energy is zero =0

  6. Second order correction on energy The only non zero matrix items are for and for so Second order correction on energy is Finally

  7. PERTURBATION in

  8. PERTURBATION in We assume that the perturbation has the following form: where σ is a constant real without dimension much less than 1 So

  9. First order correction on energy =0 The W effect is to displace levels down (whatever the sign of ). It is all the more so that n is greater: The distance between two consecutive levels is:

  10. Conclusion • We applied time perturbation theory in the non degenerate case to calculate the first and second order correction on energy and first order correction on eigenstates. We treated three perturbations: linear, quadratic and anharmonic on a one-dimensional harmonic oscillator. • In the first case when the perturbation is linear, the energy levels have shifted down all of the same quantity (constant shift) but the eigenfunction, in addition to are linear combination with and • The second case that is to say when the perturbation is quadratic. The level of energies shift to the top but more so as n is large and the eigenfunction, in addition to are linear combination with and • In third case when the perturbation is anharmonic. The level of energies shift in decreasing order of energies (whatever the sign of ). It is all the more so that n is greater. The distance between two consecutive levels is constant and the eigenfunction, in addition to are linear combination with , and

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