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Time Independent Approximation Methods 3

Non-Degenerate Perturbation Theory IV 4

Time Independent Approximation Methods Non-Degenerate Perturbation Theory IV Example: The Harmonic Oscillator in a Uniform Field 5

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In terms of these operators so where Assuming that the applied field is small, we now wish to apply our perturbation theoretic formulae. We start with energy, and compute the first order energy shift 13

In terms of these operators so where Assuming that the applied field is small, we now wish to apply our perturbation theoretic formulae. We start with energy, and compute the first order energy shift 14

In terms of these operators so where Assuming that the applied field is small, we now wish to apply our perturbation theoretic formulae. We start with energy, and compute the first order energy shift 15

In terms of these operators so where Assuming that the applied field is small, we now wish to apply our perturbation theoretic formulae to this system. 16

In terms of these operators so where Assuming that the applied field is small, we now wish to apply our perturbation theoretic formulae to this system. We start with the energy 17

In terms of these operators so where Assuming that the applied field is small, we now wish to apply our perturbation theoretic formulae to this system. We start with the energy, and compute the first order energy shift 18

We find in a straightforward calculation that the 1st order shift vanishes due to the orthogonality of the unperturbed states. Physically this vanishing of the first order energy shift occurs for the unperturbed states because they have equal weight on each side of the origin, and so their net position, and the net change in energy due to the linear applied potential vanishes. 19

We find in a straightforward calculation that the 1st order shift vanishes due to the orthogonality of the unperturbed states. Physically this vanishing of the first order energy shift occurs for the unperturbed states because they have equal weight on each side of the origin, and so their net position, and the net change in energy due to the linear applied potential vanishes. 20

We find in a straightforward calculation that the 1st order shift vanishes due to the orthogonality of the unperturbed states. Physically this vanishing of the first order energy shift occurs for the unperturbed states because they have equal weight on each side of the origin, and so their net position, and the net change in energy due to the linear applied potential vanishes. 21

We find in a straightforward calculation that the 1st order shift vanishes due to the orthogonality of the unperturbed states. Physically this vanishing of the first order energy shift occurs for the unperturbed states because they have equal weight on each side of the origin, and so their net position, and the net change in energy due to the linear applied potential vanishes. 22

We find in a straightforward calculation that the 1st order shift vanishes due to the orthogonality of the unperturbed states. Physically this vanishing of the first order energy shift occurs for the unperturbed states because they have equal weight on each side of the origin, and so their net position, and the net change in energy due to the linear applied potential vanishes. 23

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We now consider the second order energy shift and note that but, as is easily verified so 38

We now consider the second order energy shift and note that but, as is easily verified so 39

We now consider the second order energy shift and note that but, as is easily verified so 40

We now consider the second order energy shift and note that but, as is easily verified so 41

We now consider the second order energy shift and note that but, as is easily verified so 42

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This completes our discussion and exploration of non-degenerate time- independent perturbation theory. As we have seen, the formulae derived for the corrections to the energies and eigenstates diverge whenever the perturbation connects to or more states having the same unperturbed energy. We need a means, therefore of treating these divergences that occur when the original unperturbed energy levels are degenerate. In the next segment, therefore, we begin a discussion of what is generally referred to as degenerate perturbation theory. 67

This completes our discussion and exploration of non-degenerate time- independent perturbation theory. As we have seen, the formulae derived for the corrections to the energies and eigenstates diverge whenever the perturbation connects two or more states having the same unperturbed energy. We need a means, therefore of treating these divergences that occur when the original unperturbed energy levels are degenerate. In the next segment, therefore, we begin a discussion of what is generally referred to as degenerate perturbation theory. 68

This completes our discussion and exploration of non-degenerate time- independent perturbation theory. As we have seen, the formulae derived for the corrections to the energies and eigenstates diverge whenever the perturbation connects two or more states having the same unperturbed energy. We need a means, therefore of treating these divergences that occur when the original unperturbed energy levels are degenerate. In the next segment, therefore, we begin a discussion of what is generally referred to as degenerate perturbation theory. 69

This completes our discussion and exploration of non-degenerate time- independent perturbation theory. As we have seen, the formulae derived for the corrections to the energies and eigenstates diverge whenever the perturbation connects two or more states having the same unperturbed energy. We need a means, therefore of treating these divergences that occur when the original unperturbed energy levels are degenerate. In the next segment, therefore, we begin a discussion of what is generally referred to as degenerate perturbation theory. 70

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