Quantum Two 1 2 Time Independent Approximation Methods 3

4 Non-Degenerate Perturbation Theory 5 We now turn to a more general and systematic method for determining the eigenvectors and eigenvalues for observables with a discrete spectrum. As in the last section we use the language of energy eigenstates and Hamiltonians even though the method itself is perfectly applicable to other observables.

Consider, therefore, the eigenvalue problem for a time-independent Hamiltonian having a discrete nondegenerate spectrum, and which can be written in the form where the eigenvalue problem for , the zero-th order or "unperturbed part" of the Hamiltonian, is assumed to have been solved, and the perturbation is presumed to be small compared to 6 We now turn to a more general and systematic method for determining the eigenvectors and eigenvalues for observables with a discrete spectrum.

As in the last section we use the language of energy eigenstates and Hamiltonians even though the method itself is perfectly applicable to other observables. Consider, therefore, the eigenvalue problem for a time-independent Hamiltonian having a discrete nondegenerate spectrum, and which can be written in the form where the eigenvalue problem for , the zero-th order or "unperturbed part" of the Hamiltonian, is assumed to have been solved, and the perturbation is presumed to be small compared to 7

We now turn to a more general and systematic method for determining the eigenvectors and eigenvalues for observables with a discrete spectrum. As in the last section we use the language of energy eigenstates and Hamiltonians even though the method itself is perfectly applicable to other observables. Consider, therefore, the eigenvalue problem for a time-independent Hamiltonian having a discrete nondegenerate spectrum, and which can be written in the form where the eigenvalue problem for , the zero-th order or "unperturbed part" of the Hamiltonian, is assumed to have been solved, and the perturbation is presumed to be small compared to

8 We now turn to a more general and systematic method for determining the eigenvectors and eigenvalues for observables with a discrete spectrum. As in the last section we use the language of energy eigenstates and Hamiltonians even though the method itself is perfectly applicable to other observables. Consider, therefore, the eigenvalue problem for a time-independent Hamiltonian having a discrete nondegenerate spectrum, and which can be written in the form where the eigenvalue problem for , the zero-th order or "unperturbed part" of

the Hamiltonian, is assumed to have been solved, and the perturbation is presumed to be small compared to 9 We now turn to a more general and systematic method for determining the eigenvectors and eigenvalues for observables with a discrete spectrum. As in the last section we use the language of energy eigenstates and Hamiltonians even though the method itself is perfectly applicable to other observables. Consider, therefore, the eigenvalue problem for a time-independent Hamiltonian having a discrete nondegenerate spectrum, and which can be written in the form

where the eigenvalue problem for , the zero-th order or "unperturbed part" of the Hamiltonian, is assumed to have been solved, and the perturbation is presumed to be small compared to 10 Our goal is to obtain expressions for the eigenstates and eigenvalues of as an expansion in powers of the small parameter . These eigenstates of the full Hamiltonian are to be expressed as linear

combinations and simple functions of the known eigenstates and eigenvalues of the unperturbed Hamiltonian . The exact and unperturbed states of the system are thus assumed to satisfy 11 Our goal is to obtain expressions for the eigenstates and eigenvalues of as an expansion in powers of the small parameter . These eigenstates of the full Hamiltonian are to be expressed as linear

combinations and simple functions of the known eigenstates and eigenvalues of the unperturbed Hamiltonian . The exact and unperturbed states of the system are thus assumed to satisfy 12 Our goal is to obtain expressions for the eigenstates and eigenvalues of as an expansion in powers of the small parameter . These eigenstates of the full Hamiltonian are to be expressed as linear

combinations and simple functions of the known eigenstates and eigenvalues of the unperturbed Hamiltonian . The exact and unperturbed states of the system are thus assumed to satisfy 13 Our goal is to obtain expressions for the eigenstates and eigenvalues of as an expansion in powers of the small parameter . These eigenstates of the full Hamiltonian are to be expressed as linear

combinations and simple functions of the known eigenstates and eigenvalues of the unperturbed Hamiltonian . The exact and unperturbed states of the system are thus assumed to satisfy 14 Our goal is to obtain expressions for the eigenstates and eigenvalues of as an expansion in powers of the small parameter . These eigenstates of the full Hamiltonian are to be expressed as linear

combinations and simple functions of the known eigenstates and eigenvalues of the unperturbed Hamiltonian . The exact and unperturbed states of the system are thus assumed to satisfy 15 Our goal is to obtain expressions for the eigenstates and eigenvalues of as an expansion in powers of the small parameter . These eigenstates of the full Hamiltonian are to be expressed as linear

combinations and simple functions of the known eigenstates and eigenvalues of the unperturbed Hamiltonian . The exact and unperturbed states of the system are thus assumed to satisfy 16 Our goal is to obtain expressions for the eigenstates and eigenvalues of as an expansion in powers of the small parameter . These eigenstates of the full Hamiltonian are to be expressed as linear

combinations and simple functions of the known eigenstates and eigenvalues of the unperturbed Hamiltonian . The exact and unperturbed states of the system are thus assumed to satisfy 17 Our goal is to obtain expressions for the eigenstates and eigenvalues of as an expansion in powers of the small parameter . These eigenstates of the full Hamiltonian are to be expressed as linear

combinations and simple functions of the known eigenstates and eigenvalues of the unperturbed Hamiltonian . The exact and unperturbed states of the system are thus assumed to satisfy 18 To proceed, we then assume that there exist expansions of the form and will refer to the terms and as the th order correction to the th eigenstate and energy eigenvalue, respectively.

The corresponding correction to the energy is also generally referred to as the th order energy shift, for obvious reasons. Note that = approaches as . But this still leaves the relative phase of the two sets of basis vectors undetermined for , since we could multiply the basis vectors of any such set by phases so long as as and still satisfy these equations. 19 To proceed, we then assume that there exist expansions of the form

and will refer to the terms and as the th order correction to the th eigenstate and energy eigenvalue, respectively. The corresponding correction to the energy is also generally referred to as the th order energy shift, for obvious reasons. Note that = approaches as . But this still leaves the relative phase of the two sets of basis vectors undetermined for , since we could multiply the basis vectors of any such set by phases so long as as and still satisfy these equations. 20

To proceed, we then assume that there exist expansions of the form and will refer to the terms and as the th order correction to the th eigenstate and energy eigenvalue, respectively. The corresponding correction to the energy is also generally referred to as the th order energy shift, for obvious reasons. Note that = approaches as . But this still leaves the relative phase of the two sets of basis vectors undetermined for , since we could multiply the basis vectors of any such set by phases so long as as and still satisfy these equations.

21 To proceed, we then assume that there exist expansions of the form and will refer to the terms and as the th order correction to the th eigenstate and energy eigenvalue, respectively. The corresponding correction to the energy is also generally referred to as the th order energy shift, for obvious reasons. Note that = approaches as . But this still leaves the relative phase of the two sets of basis vectors

undetermined for , since we could multiply the basis vectors of any such set by phases so long as as and still satisfy these equations. 22 To proceed, we then assume that there exist expansions of the form and will refer to the terms and as the th order correction to the th eigenstate and energy eigenvalue, respectively. The corresponding correction to the energy is also generally referred to as the th order energy shift, for obvious reasons.

Note from the expansion above that = approaches as . But this still leaves the relative phase of the two sets of basis vectors undetermined for , since we could multiply the basis vectors of any such set by phases so long as as and still satisfy these equations. 23 To proceed, we then assume that there exist expansions of the form and will refer to the terms and as the th order correction to the th eigenstate and energy eigenvalue, respectively.

The corresponding correction to the energy is also generally referred to as the th order energy shift, for obvious reasons. Note from the expansion above that = approaches as . But this still leaves the relative phase of the two sets of basis vectors undetermined for , since we could multiply the basis vectors of any such set by phases so long as as and still satisfy these equations. 24 For nonzero values of , therefore, we further fix the relative phase between these two sets of states and by requiring that for corresponding elements the inner

product be real and positive, so that and To determine the corrections, then, we will simply require that the exact eigenstates of satisfy the full eigenvalue equation to all orders in . 25

For nonzero values of , therefore, we further fix the relative phase between these two sets of states and by requiring that for corresponding elements the inner product be real and positive, so that and To determine the corrections, then, we will simply require that the exact eigenstates of satisfy the full eigenvalue equation

to all orders in . 26 For nonzero values of , therefore, we further fix the relative phase between these two sets of states and by requiring that for corresponding elements the inner product be real and positive, so that and

To determine the corrections, then, we will simply require that the exact eigenstates of satisfy the full eigenvalue equation to all orders in . 27 Substituting the expansions and into the eigenvalue equation (in the form ) we obtain

() ( ) () The trick is now to collect together all terms multiplied by the same power of . Note that the superscripts tell us how many powers of each term carries. So, lets work our way through a few terms 28 Substituting the expansions and into the eigenvalue equation (in the form ) we obtain

() ( ) () The trick is now to collect together all terms multiplied by the same power of . Note that the superscripts tell us how many powers of each term carries. So, lets work our way through a few terms 29 Substituting the expansions and into the eigenvalue equation (in the form ) we obtain

() ( ) () The trick is now to collect together all terms multiplied by the same power of . Note that the superscripts tell us how many powers of each term carries. So, lets work our way through a few terms 30 Substituting the expansions and into the eigenvalue equation (in the form ) we obtain

() ( ) () The trick is now to collect together all terms multiplied by the same power of . Note that the superscripts tell us how many powers of each term carries. So, lets work our way through a few terms 31 Substituting the expansions and into the eigenvalue equation (in the form ) we obtain

() ( ) () The trick is now to collect together all terms multiplied by the same power of . Note that the superscripts tell us how many powers of each term carries. So, lets work our way through a few terms 32 Substituting the expansions and into the eigenvalue equation (in the form ) we obtain

() ( ) () The trick is now to collect together all terms multiplied by the same power of . Note that the superscripts tell us how many powers of each term carries. So, lets work our way through a few terms 33 Substituting the expansions and into the eigenvalue equation (in the form ) we obtain

() ( ) () The trick is now to collect together all terms multiplied by the same power of . Note that the superscripts tell us how many powers of each term carries. So, lets work our way through a few terms 34 Substituting the expansions and into the eigenvalue equation (in the form ) we obtain

() ( ) () The trick is now to collect together all terms multiplied by the same power of . Note that the superscripts tell us how many powers of each term carries. So, lets work our way through a few terms 35 () ( ) ()

The factors which combine to yield no powers of are shown above in green. Thus, the coefficient of to the zeroeth power is seen to be 36 () ( ) () The factors which combine to yield no powers of are shown above in green. Thus, the coefficient of to the zeroeth power is seen to be

37 () ( ) () The factors which combine to yield no powers of are shown above in green. Thus, the coefficient of to the zeroeth power is seen to be 38

() ( ) () The factors which combine to yield no powers of are shown above in green. Thus, the coefficient of to the zeroeth power is seen to be 39 ()

( ) () The factors which combine to yield one power of are shown above in green and orange. Thus, the coefficient of the first power of is [ 40 ()

( ) () The factors which combine to yield one power of are shown above in green and orange. Thus, the coefficient of the first power of is 41 () ( ) ()

The factors which combine to yield are shown above in green, orange, and blue. Thus, the coefficient of is Proceeding in this way . 42 () ( ) ()

The factors which combine to yield are shown above in green, orange, and blue. Thus, the coefficient of is Proceeding in this way . 43 () ( ) ()

The factors which combine to yield are shown above in green, orange, and blue. Thus, the coefficient of is Proceeding in this way . 44 () ( ) ()

The factors which combine to yield are shown above in green, orange, and blue. Thus, the coefficient of is Proceeding in this way . 45 can be expressed as

[ ] [ [ ] [ ] 46 can be expressed as

[ ] [ [ ] [ ] 47 can be expressed as

[ ] [ [ ] [ ] 48 can be expressed as

[ ] [ [ ] [ ] 49 can be expressed as

[ ] [ [ ] [ ] 50 can be expressed as

[ ] [ [ ] [ ] 51 can be expressed as

[ ] [ [ ] [ ] 52 For this last equation to hold for small but arbitrary values of , the coefficients of each power of must vanish independently.

The reason for this is that the polynomials form a linearly independent set of functions on the real numbers, so any relation of the form can be satisfied for all only if for all . Applying this requirement to the previous equation generates an infinite hierarchy of equations, one for each power of . 53 For this last equation to hold for small but arbitrary values of , the coefficients of

each power of must vanish independently. The reason for this is that the polynomials form a linearly independent set of functions on the real numbers, so any relation of the form can be satisfied for all only if for all . Applying this requirement to the previous equation generates an infinite hierarchy of equations, one for each power of . 54

For this last equation to hold for small but arbitrary values of , the coefficients of each power of must vanish independently. The reason for this is that the polynomials form a linearly independent set of functions on the real numbers, so any relation of the form can be satisfied for all only if for all . Applying this requirement to the previous equation generates an infinite hierarchy of equations, one for each power of . 55

For this last equation to hold for small but arbitrary values of , the coefficients of each power of must vanish independently. The reason for this is that the polynomials form a linearly independent set of functions on the real numbers, so any relation of the form can be satisfied for all only if for all . Applying this requirement to the previous equation generates an infinite hierarchy of equations, one for each power of .

56 For this last equation to hold for small but arbitrary values of , the coefficients of each power of must vanish independently. The reason for this is that the polynomials form a linearly independent set of functions on the real numbers, so any relation of the form can be satisfied for all only if for all . Applying this requirement to the previous equation generates an infinite

hierarchy of equations, one for each power of . 57 The equation generated by setting the coefficient of equal to zero is referred to as the th order equation. Collecting coefficients of the first few powers of we obtain the zeroth order equation the 1st order equation the 2nd order equation the 3rd order equation

and the kth order equation 58 The equation generated by setting the coefficient of equal to zero is referred to as the th order equation. Collecting coefficients of the first few powers of we obtain the zeroth order equation the 1st order equation the 2nd order equation the 3rd order equation

and the kth order equation 59 The equation generated by setting the coefficient of equal to zero is referred to as the th order equation. Collecting coefficients of the first few powers of we obtain the zeroth order equation the 1st order equation the 2nd order equation the 3rd order equation

and the kth order equation 60 The equation generated by setting the coefficient of equal to zero is referred to as the th order equation. Collecting coefficients of the first few powers of we obtain the zeroth order equation the 1st order equation the 2nd order equation the 3rd order equation

and the kth order equation 61 The equation generated by setting the coefficient of equal to zero is referred to as the th order equation. Collecting coefficients of the first few powers of we obtain the zeroth order equation the 1st order equation the 2nd order equation the 3rd order equation

and the kth order equation 62 The equation generated by setting the coefficient of equal to zero is referred to as the th order equation. Collecting coefficients of the first few powers of we obtain the zeroth order equation the 1st order equation the 2nd order equation the 3rd order equation

and the kth order equation 63 The equation generated by setting the coefficient of equal to zero is referred to as the th order equation. Collecting coefficients of the first few powers of we obtain the zeroth order equation the 1st order equation the 2nd order equation the 3rd order equation

and the kth order equation 64 The structure of these equations allows for the general th order corrections to be obtained from those solutions of lower order, allowing for the development of a systematic expansion of the eigenstates and energy eigenvalues in powers of . In the next segment we carry out the details of this exercise to first order in the state, and to second order in the energy. 65

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