1. Quantum Two 1

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3. Time Independent Approximation Methods 3

4. Non-Degenerate Perturbation Theory III 4

5. Time Independent Approximation Methods Non-Degenerate Perturbation Theory III The Second Order Energy Correction 5

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7. This second order energy correction first appears in the 2 nd order equation 7

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12. It is convenient to re-write this in the even longer form in which, based upon our earlier results and arguments, the quantities in green reduce to zero, the quantities in blue reduce to unity, and the quantities in orange survive the reduction unchanged. Thus, upon re-arrangement, the very long equation above reduces to the following simpler expression for the second order energy: 12

13. It is convenient to re-write this in the even longer form Based upon our earlier results and arguments, the quantities in green reduce to zero, the quantities in blue reduce to unity, and the quantities in orange survive the reduction unchanged. Thus, upon re-arrangement, the very long equation above reduces to the following simpler expression for the second order energy: 13

14. It is convenient to re-write this in the even longer form Based upon our earlier results and arguments, the quantities in green reduce to zero, the quantities in blue reduce to unity, and the quantities in orange survive the reduction unchanged. Thus, upon re-arrangement, the very long equation above reduces to the following simpler expression for the second order energy: 14

15. It is convenient to re-write this in the even longer form Based upon our earlier results and arguments, the quantities in green reduce to zero, the quantities in blue reduce to unity, and the quantity in orange survives the reduction unchanged. Thus, upon re-arrangement, the very long equation above reduces to the following simpler expression for the second order energy: 15

16. It is convenient to re-write this in the even longer form Based upon our earlier results and arguments, the quantities in green reduce to zero, the quantities in blue reduce to unity, and the quantity in orange survives the reduction unchanged. Thus, upon re-arrangement, the very long equation above reduces to the following simpler expression for the second order energy: 16

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26. Comments: 1. In problems involving weak perturbations it usually suffices to determine corrections and energy shifts to lowest non-vanishing order in the perturbation, and so it is unusual that one needs to go beyond second order for "simple" problems in perturbation theory. Exceptions to this general observation arise quite often when dealing with many-body problems, where diagrammatic methods have been developed that take the ideas of perturbation theory to an extremely high level, and where it is not uncommon to find examples where effects of the perturbation are calculated to all orders. 26

27. Comments: 1. In problems involving weak perturbations it usually suffices to determine corrections and energy shifts to lowest non-vanishing order in the perturbation, and so it is unusual that one needs to go beyond second order for "simple" problems in perturbation theory. Exceptions to this general observation arise quite often when dealing with many-body problems, where diagrammatic methods have been developed that take the ideas of perturbation theory to an extremely high level, and where it is not uncommon to find examples where effects of the perturbation are calculated to all orders. 27

28. Comments: 2.Even in simple perturbation theory problems, in many cases it is not possible to analytically perform the sum appearing in the 2 nd order energy correction and so it is useful to develop simple means for estimating the magnitude of the second order energy shift. For example, a general upper bound for the second order shift can be obtained for any nondegenerate level by observing that 28

29. Comments: 2.Even in simple perturbation theory problems, in many cases it is not possible to analytically perform the sum appearing in the 2 nd order energy correction and so it is useful to develop simple means for estimating the magnitude of the second order energy shift. For example, a general upper bound for the second order shift can be obtained for any nondegenerate level by observing that 29

30. Comments: 2.Even in simple perturbation theory problems, in many cases it is not possible to analytically perform the sum appearing in the 2 nd order energy correction and so it is useful to develop simple means for estimating the magnitude of the second order energy shift. For example, a general upper bound for the second order shift can be obtained for any nondegenerate level by observing that 30

31. This upper bound also applies to the magnitude of the 2 nd order energy shift: Moreover, each term in the sum on the right can itself be bounded. If we denote by the energy spacing, then and so 31

32. This upper bound also applies to the magnitude of the 2 nd order energy shift: Moreover, each term in the sum on the right can itself be bounded. If we denote by the energy spacing, then and so 32

33. This upper bound also applies to the magnitude of the 2 nd order energy shift: Moreover, each term in the sum on the right can itself be bounded. If we denote by the energy spacing, then and so 33

34. This upper bound also applies to the magnitude of the 2 nd order energy shift: Moreover, each term in the sum on the right can itself be bounded. If we denote by the energy spacing, then and so 34

35. This upper bound also applies to the magnitude of the 2 nd order energy shift: Moreover, each term in the sum on the right can itself be bounded. If we denote by the energy spacing, then and so 35

36. Using a simple trick we can perform the infinite sum by "removing" the restriction on the summation index. We write where and 36

37. Using a simple trick we can perform the infinite sum by "removing" the restriction on the summation index. We write where and 37

38. Using a simple trick we can perform the infinite sum by "removing" the restriction on the summation index. We write where and 38

39. Using a simple trick we can perform the infinite sum by "removing" the restriction on the summation index. We write where and 39

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44. In this segment, we computed the second order contribution to the energy, obtaining the general result and computed a general upper bound on the magnitude of the second order energy shift. Other bounds, e.g., for the ground state energy are also straightforward to derive. In the next segment, to make some of these formal results more concrete, we apply the formulae of non-degenerate perturbation theory to a specific example. 44

45. In this segment, we computed the second order contribution to the energy, obtaining the general result and computed a general upper bound on the magnitude of the second order energy shift. unds, e.g., for the ground state energy are also straightforward to derive. In the next segment, to make some of these formal results more concrete, we apply the formulae of non-degenerate perturbation theory to a specific example. 45

46. In this segment, we computed the second order contribution to the energy, obtaining the general result and computed a general upper bound on the magnitude of the second order energy shift. Other bounds, e.g., for the ground state energy are also straightforward to derive. In the next segment, to make some of these formal results more concrete, we apply the formulae of non-degenerate perturbation theory to a specific example 46

47. In this segment, we computed the second order contribution to the energy, obtaining the general result and computed a general upper bound on the magnitude of the second order energy shift. Other bounds, e.g., for the ground state energy are also straightforward to derive. In the next segment, to make some of these formal results more concrete, we apply the formulae of non-degenerate perturbation theory to a specific example. 47

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