Quantum Two 1 2 Time Independent Approximation Methods 3 Time Independent Approximation Methods Degenerate Perturbation Theory
4 The expressions that we derived above for the first order correction to the state and the second order correction to the energy are clearly inappropriate to situations in which degenerate or nearly-degenerate eigenstates are connected by the perturbation: the corresponding corrections all diverge as the spacing between the energy levels goes to zero. This divergence is an indication of the strength with which the perturbation tries to mix together states that are very-nearly degenerate (or exactly so).
5 The expressions that we derived above for the first order correction to the state and the second order correction to the energy are clearly inappropriate to situations in which degenerate or nearly-degenerate eigenstates are connected by the perturbation: the corresponding corrections all diverge as the spacing between the energy levels goes to zero. This divergence is an indication of the strength with which the perturbation tries to mix together states that are very-nearly degenerate (or exactly so). 6
So, when things blow up on you its a good time to re-think the situation. Clearly we need to treat differently those states that are known in advance to be very close in energy. In this segment, therefore, we discuss the general approach taken to deal with this issue. The result will not be formulae of the sort we have already derived, but a general strategy or prescription for dealing with the situation. We assume, as before, that the Hamiltonian of interest can be separated into two parts and we again seek the exact eigenstates and eigenvalues of , expressed, in terms of eigenstates and eigenvalues of , the latter of which are now assumed to be at
least partially degenerate. 7 So, when things blow up on you its a good time to re-think the situation. Clearly we need to treat differently those states that are known in advance to be very close in energy. In this segment, therefore, we discuss the general approach taken to deal with this issue. The result will not be formulae of the sort we have already derived, but a general strategy or prescription for dealing with the situation. We assume, as before, that the Hamiltonian of interest can be separated into two parts
and we again seek the exact eigenstates and eigenvalues of , expressed, in terms of eigenstates and eigenvalues of , the latter of which are now assumed to be at least partially degenerate. 8 So, when things blow up on you its a good time to re-think the situation. Clearly we need to treat differently those states that are known in advance to be very close in energy. In this segment, therefore, we discuss the general approach taken to deal with this issue. The result will not be formulae of the sort we have already derived,
but a general strategy or prescription for dealing with the situation. We assume, as before, that the Hamiltonian of interest can be separated into two parts and we again seek the exact eigenstates and eigenvalues of , expressed, in terms of eigenstates and eigenvalues of , the latter of which are now assumed to be at least partially degenerate. 9 So, when things blow up on you its a good time to re-think the situation. Clearly we need to treat differently those states that are known in advance to be
very close in energy. In this segment, therefore, we discuss the general approach taken to deal with this issue. The result will not be formulae of the sort we have already derived, but a general strategy or prescription for dealing with the situation. We assume, as before, that the Hamiltonian of interest can be separated into two parts and we again seek the exact eigenstates and eigenvalues of , expressed, in terms of eigenstates and eigenvalues of , the latter of which are now assumed to be at least partially degenerate. 10
So, when things blow up on you its a good time to re-think the situation. Clearly we need to treat differently those states that are known in advance to be very close in energy. In this segment, therefore, we discuss the general approach taken to deal with this issue. The result will not be formulae of the sort we have already derived, but a general strategy or prescription for dealing with the situation. We assume, as before, that the Hamiltonian of interest can be separated into two parts and we again seek the exact eigenstates and eigenvalues of , expressed, in terms of eigenstates and eigenvalues of , the latter of which are now assumed to be at
least partially degenerate. 11 So, when things blow up on you its a good time to re-think the situation. Clearly we need to treat differently those states that are known in advance to be very close in energy. In this segment, therefore, we discuss the general approach taken to deal with this issue. The result will not be formulae of the sort we have already derived, but a general strategy or prescription for dealing with the situation. We assume, as before, that the Hamiltonian of interest can be separated into two parts
and we again seek the exact eigenstates and eigenvalues of , expressed, in terms of eigenstates and eigenvalues of , the latter of which are now assumed to be at least partially degenerate. 12 We also assume, as before, that we have already solved the eigenvalue problem for , and that we have therefore obtained for that operator its eigenvalues, their degeneracies, and an ONB of eigenstates where we now obviously need an index to distinguish the different - fold
degenerate orthonormal basis states of having the same unperturbed energy . The reason for "cluttering" our notation with the symbol , rather than simply writing these states as will become obvious shortly. 13 We also assume, as before, that we have already solved the eigenvalue problem for , and that we have therefore obtained for that operator its eigenvalues, their degeneracies, and an ONB of eigenstates where we now obviously need an index to distinguish the different - fold degenerate orthonormal basis states of having the same unperturbed energy .
The reason for "cluttering" our notation with the symbol , rather than simply writing these states as will become obvious shortly. 14 We also assume, as before, that we have already solved the eigenvalue problem for , and that we have therefore obtained for that operator its eigenvalues, their degeneracies, and an ONB of eigenstates where we now obviously need an index to distinguish the different - fold degenerate orthonormal basis states of having the same unperturbed energy . The reason for "cluttering" our notation with the symbol , rather than simply
writing these states as will become obvious shortly. 15 We also assume, as before, that we have already solved the eigenvalue problem for , and that we have therefore obtained for that operator its eigenvalues, their degeneracies, and an ONB of eigenstates where we now obviously need an index to distinguish the different - fold degenerate orthonormal basis states of having the same unperturbed energy . The reason for "cluttering" our notation with the symbol , rather than simply writing these states as will become obvious shortly.
16 We also assume, as before, that we have already solved the eigenvalue problem for , and that we have therefore obtained for that operator its eigenvalues, their degeneracies, and an ONB of eigenstates where we now obviously need an index to distinguish the different - fold degenerate orthonormal basis states of having the same unperturbed energy . The reason for "cluttering" our notation with the symbol , rather than simply writing these states as will become obvious shortly.
17 We also assume, as before, that we have already solved the eigenvalue problem for , and that we have therefore obtained for that operator its eigenvalues, their degeneracies, and an ONB of eigenstates where we now obviously need an index to distinguish the different - fold degenerate orthonormal basis states of having the same unperturbed energy . The reason for "cluttering" our notation with the symbol , rather than simply writing these states as will become obvious shortly. 18
We also assume, as before, that we have already solved the eigenvalue problem for , and that we have therefore obtained for that operator its eigenvalues, their degeneracies, and an ONB of eigenstates where we now obviously need an index to distinguish the different - fold degenerate orthonormal basis states of having the same unperturbed energy . Recall that the basis states with fixed , associated with same eigenvalue span an dimensional eigenspace of , all the states of which are also eigenstates associated with that particular -fold degenerate energy.
19 It will be critical in what follows to point out that, when the degeneracy , the choice of the basis set for a given degenerate subpace is not unique; We can always form linear combinations within of any such basis vectors to generate a new orthonormal basis which could be used as readily as any other for expanding the exact eigenstates of . A complete ONB of unperturbed eigenstates for the entire space is then formed by collecting together the particular ONB's that we have chosen for each subspace.
20 It will be critical in what follows to point out that, when the degeneracy , the choice of the basis set for a given degenerate subpace is not unique; We can always form linear combinations within of any such basis vectors to generate a new orthonormal basis which could be used as readily as any other for expanding the exact eigenstates of . A complete ONB of unperturbed eigenstates for the entire space is then formed by collecting together the particular ONB's that we have chosen for each subspace.
21 It will be critical in what follows to point out that, when the degeneracy , the choice of the basis set for a given degenerate subpace is not unique; We can always form linear combinations within of any such basis vectors to generate a new orthonormal basis which could be used as readily as any other for expanding the exact eigenstates of . A complete ONB of unperturbed eigenstates for the entire space is then formed by collecting together the particular ONB's that we have chosen for each subspace.
22 Clearly, though, all such basis sets of this sort will satisfy the same set of relations Moreover, in every such complete basis of its own eigenstates the matrix representing will be the same, i.e., it will be diagonal, and the diagonal elements, once properly ordered, will just contain the eigenvalues , each one repeated along the diagonal a number of times equal to that eigenvalue's degeneracy . 23 Clearly, though, all such basis sets of this sort will satisfy the same set of relations
Moreover, in every such complete basis of its own eigenstates the matrix representing will be the same, i.e., it will be diagonal, and the diagonal elements, once properly ordered, will just contain the eigenvalues , each one repeated along the diagonal a number of times equal to that eigenvalue's degeneracy . 24 Clearly, though, all such basis sets of this sort will satisfy the same set of relations Moreover, in every such complete basis of its own eigenstates the matrix representing will be the same, i.e., it will be diagonal, and the diagonal
elements, once properly ordered, will just contain the eigenvalues , each one repeated along the diagonal a number of times equal to that eigenvalue's degeneracy . 25 Clearly, though, all such basis sets of this sort will satisfy the same set of relations Moreover, in every such complete basis of its own eigenstates the matrix representing will be the same, i.e., it will be diagonal, and the diagonal elements, once properly ordered, will just contain the eigenvalues , each one repeated along the diagonal a number of times equal to that eigenvalue's degeneracy .
26 Clearly, though, all such basis sets of this sort will satisfy the same set of relations Moreover, in every such complete basis of its own eigenstates the matrix representing will be the same, i.e., it will be diagonal, and the diagonal elements, once properly ordered, will just contain the eigenvalues , each one repeated along the diagonal a number of times equal to that eigenvalue's degeneracy . 27 Clearly, though, all such basis sets of this sort will satisfy the same set of relations
Moreover, in every such complete basis of its own eigenstates the matrix representing will be the same, i.e., it will be diagonal, and the diagonal elements, once properly ordered, will just contain the eigenvalues , each one repeated along the diagonal a number of times equal to that eigenvalue's degeneracy . 28 It is useful to express this in block matrix form in which the elements are themselves matrices and in particular in which (i) the off-diagonal submatrix [ 0 ] in the -th block is the null matrix, all the
elements of which are zero, and (ii) the -th diagonal block is an diagonal submatrix [1], in which [1] is the identity matrix, i.e., 29 It is useful to express this in block matrix form in which the elements are themselves matrices and in particular in which (i) the off-diagonal submatrix [ 0 ] in the -th block is the null matrix, all the elements of which are zero, and (ii) the -th diagonal block is an diagonal submatrix [1], in which [1] is the identity matrix, i.e.,
30 It is useful to express this in block matrix form in which the elements are themselves matrices and in particular in which (i) the off-diagonal submatrix [ 0 ] in the -th block is the null matrix, all the elements of which are zero, and (ii) the -th diagonal block is an diagonal submatrix [1], in which [1] is the identity matrix, i.e., 31 It is useful to express this in block matrix form
in which the elements are themselves matrices and in particular in which (i) the off-diagonal submatrix [ 0 ] in the -th block is the null matrix, all the elements of which are zero, and (ii) the -th diagonal block is an diagonal submatrix [1], in which [1] is the identity matrix, i.e., 32 Although the matrix representing in any such basis of it's eigenstates is the same, the matrix representing any other operator generally depends explicitly on which particular basis of eigenstates of that one choses to work in. For example the perturbation in a particular basis of eigenstates of will have a
particular set of complex numbers associated with its matrix elements, and will be represented by a matrix that we can again express in block matrix form as 33 Although the matrix representing in any such basis of it's eigenstates is the same, the matrix representing any other operator generally depends explicitly on which particular basis of eigenstates of that one choses to work in. For example the perturbation in a particular basis of eigenstates of will have a particular set of complex numbers associated with its matrix elements, and will be represented by a matrix that we can again express in block matrix form as 34
Although the matrix representing in any such basis of it's eigenstates is the same, the matrix representing any other operator generally depends explicitly on which particular basis of eigenstates of that one choses to work in. For example the perturbation in a particular basis of eigenstates of will have a particular set of complex numbers associated with its matrix elements, and will be represented by a matrix that we can again express in block matrix form as 35 in which the -th block now contains an matrix whose elements connect the basis states in the subspace with energy to those in the subspace with energy , i.e., 36
in which the -th block now contains an matrix whose elements connect the basis states in the subspace with energy to those in the subspace with energy , i.e., 37 In a different basis of eigenstates of there will be a different set of complex numbers that will then appear in a different matrix in which the -th block now contains a different matrix than before, i.e., with elements
that connect a different set of basis states in to those in . 38 In a different basis of eigenstates of there will be a different set of complex numbers that will then appear in a different matrix in which the -th block now contains a different matrix than before, i.e., with elements that connect a different set of basis states in to those in . 39
In a different basis of eigenstates of there will be a different set of complex numbers that will then appear in a different matrix in which the -th block now contains a different matrix than before, i.e., with elements that connect this different set of basis states in to those in . 40 With these rather lengthy preliminary observations in hand, we now return to the problem of the divergences that appear in our expansions for the states and eigenvalues, which in the notation of the present discussion we write in a
particular unperturbed basis in the form and 41 With these rather lengthy preliminary observations in hand, we now return to the problem of the divergences that appear in our expansions for the states and eigenvalues, which in the notation of the present discussion we write in a particular unperturbed basis in the form and
42 With these rather lengthy preliminary observations in hand, we now return to the problem of the divergences that appear in our expansions for the states and eigenvalues, which in the notation of the present discussion we write in a particular unperturbed basis in the form and 43
As we have remarked, these formulae will diverge, and thus fail, if in this particular basis the perturbation actually connects states within each degenerate eigenspace of , i.e., if there actually exists any non-zero matrix elements for i.e., connecting different basis states of the same energy. We also remark that if connects any such basis state to itself, i.e., if is not zero than that is, at least in principle, a good thing, since it then gives us a non-vanishing first order energy shift which can then break (or lift) the degeneracy in first order.
44 As we have remarked, these formulae will diverge, and thus fail, if in this particular basis the perturbation actually connects states within each degenerate eigenspace of , i.e., if there actually exists any non-zero matrix elements for i.e., connecting different basis states of the same energy. We also remark that if connects any such basis state to itself, i.e., if is not zero than that is, at least in principle, a good thing, since it then gives us a non-vanishing first order energy shift which can then break (or lift) the
degeneracy in first order. 45 As we have remarked, these formulae will diverge, and thus fail, if in this particular basis the perturbation actually connects states within each degenerate eigenspace of , i.e., if there actually exists any non-zero matrix elements for i.e., connecting different basis states of the same energy. We also remark that if connects any such basis state to itself, i.e., if
is not zero than that is, at least in principle, a good thing, since it then gives us a non-vanishing first order energy shift which can then break (or lift) the degeneracy in first order. 46 But even if it does, and we incorporate that first order energy shift in a redefined version of , we will still face the possibility of previously degenerate states that now differ in energy by an amount of order , which will be of the same order as the matrix element
of the perturbation that connects them, and which will therefore tend to negate the validity of the perturbation expansion. So we really do have to think a little harder. 47 But even if it does, and we incorporate that first order energy shift in a redefined version of , we will still face the possibility of previously degenerate states that now differ in energy by an amount of order , which will be of the same order as the matrix element of the perturbation that connects them, and which will therefore tend to negate
the validity of the perturbation expansion. So we really do have to think a little harder. 48 But even if it does, and we incorporate that first order energy shift in a redefined version of , we will still face the possibility of previously degenerate states that now differ in energy by an amount of order , which will be of the same order as the matrix element of the perturbation that connects them, and which will therefore tend to negate the validity of the perturbation expansion.
So we really do have to think a little harder. 49 So we now note that our perturbation theoretic formulae could, in fact, be applied (at least to the level that we have developed them), if it just so happened (i.e., fortuitously) that all the matrix elements of between different basis states of the same unperturbed energy were identically equal to zero. Indeed, if that actually happened, the numerators in the problematic terms in our first and second order equations would have just disappeared before they even had a chance of being divided by those offending (vanishing) energy denominators. Of course, one presumes that when we originally solved the eigenvalue problem
for we obtained some set of eigenstates , which we just sort of inherit when we later consider applying a perturbation to our quantum system. So there is no a priori reason to believe that such a fortuitous vanishing of the matrix elements actually occurs. 50 So we now note that our perturbation theoretic formulae could, in fact, be applied (at least to the level that we have developed them), if it just so happened (i.e., fortuitously) that all the matrix elements of between different basis states of the same unperturbed energy were identically equal to zero.
Indeed, if that actually happened, the numerators in the problematic terms in our first and second order equations would have just disappeared before they even had a chance of being divided by those offending (vanishing) energy denominators. Of course, one presumes that when we originally solved the eigenvalue problem for we obtained some set of eigenstates , which we just sort of inherit when we later consider applying a perturbation to our quantum system. So there is no a priori reason to believe that such a fortuitous vanishing of the matrix elements actually occurs. 51
So we now note that our perturbation theoretic formulae could, in fact, be applied (at least to the level that we have developed them), if it just so happened (i.e., fortuitously) that all the matrix elements of between different basis states of the same unperturbed energy were identically equal to zero. Indeed, if that actually happened, the numerators in the problematic terms in our first and second order equations would have just disappeared before they even had a chance of being divided by those offending (vanishing) energy denominators. Of course, one presumes that when we originally solved the eigenvalue problem for we obtained some set of eigenstates , which we just sort of inherit when we later consider applying a perturbation to our quantum system.
So there is no a priori reason to believe that such a fortuitous vanishing of the matrix elements actually occurs. 52 So we now note that our perturbation theoretic formulae could, in fact, be applied (at least to the level that we have developed them), if it just so happened (i.e., fortuitously) that all the matrix elements of between different basis states of the same unperturbed energy were identically equal to zero. Indeed, if that actually happened, the numerators in the problematic terms in our first and second order equations would have just disappeared before they even
had a chance of being divided by those offending (vanishing) energy denominators. Of course, one presumes that when we originally solved the eigenvalue problem for we obtained some set of eigenstates , which we just sort of inherit when we later consider applying a perturbation to our quantum system. So there is no a priori reason to believe that such a fortuitous vanishing of the matrix elements actually occurs. 53 So instead of just hoping that this happens, we have to take charge of our lives
and actually make it happen. Indeed, we argue, that a potential strategy for dealing with perturbations applied to degenerate energy levels actually arises from the two key insights that (i) we are free to choose whatever ONB we want in each degenerate subspace, and (ii) the offensive divergences entirely disappear if we can somehow choose a new basis set in each degenerate subspace whose elements are not connected by the perturbing operator , i.e., an ONB for which Can we do that? 54
So instead of just hoping that this happens, we have to take charge of our lives and actually make it happen. Indeed, we argue, that a potential strategy for dealing with perturbations applied to degenerate energy levels actually arises from the two key insights that (i) we are free to choose whatever ONB we want in each degenerate subspace, and (ii) the offensive divergences entirely disappear if we can somehow choose a new basis set in each degenerate subspace whose elements are not connected by the perturbing operator , i.e., an ONB for which Can we do that?
55 So instead of just hoping that this happens, we have to take charge of our lives and actually make it happen. Indeed, we argue, that a potential strategy for dealing with perturbations applied to degenerate energy levels actually arises from the two key insights that (i) we are free to choose whatever ONB we want in each degenerate subspace, and (ii) the offensive divergences entirely disappear if we can somehow choose a new basis set in each degenerate subspace whose elements are not connected by the perturbing operator , i.e., an ONB for which
Can we do that? 56 So instead of just hoping that this happens, we have to take charge of our lives and actually make it happen. Indeed, we argue, that a potential strategy for dealing with perturbations applied to degenerate energy levels actually arises from the two key insights that (i) we are free to choose whatever ONB we want in each degenerate subspace, and (ii) the offensive divergences entirely disappear if we can somehow choose a
new basis set in each degenerate subspace whose elements are not connected by the perturbing operator , i.e., an ONB for which Can we do that? 57 So instead of just hoping that this happens, we have to take charge of our lives and actually make it happen. Indeed, we argue, that a potential strategy for dealing with perturbations applied to degenerate energy levels actually arises from the two key insights that (i) we are free to choose whatever ONB we want in each degenerate subspace,
and (ii) the offensive divergences entirely disappear if we can somehow choose a new basis set in each degenerate subspace whose elements are not connected by the perturbing operator , i.e., an ONB for which Can we do that? 58 So instead of just hoping that this happens, we have to take charge of our lives and actually make it happen. Indeed, we argue, that a potential strategy for dealing with perturbations applied
to degenerate energy levels actually arises from the two key insights that (i) we are free to choose whatever basis set we want in each degenerate subspace, and (ii) the offensive divergences entirely disappear if we can somehow choose a new basis set in each degenerate subspace whose elements are not connected by the perturbing operator , i.e., an ONB for which Can we do that? 59 And this becomes, perhaps, more clear when we realize that this last condition is
equivalent to the statement that we simply want to find a basis in which the submatrix is diagonal. We just want its off-diagonal elements to vanish! So the problem of implementing the proposed strategy reduces to the following: Given an Hermitian submatrix [] representing the perturbation within some degenerate eigenspace , in a particular representation of unperturbed eigenstates , find a new ONB of unperturbed eigenstates for that subspace for which the matrix representing the perturbation in this new set of basis states is diagonal. But we have a verb for this process. The verb is "diagonalize". The new basis we are looking for are just the eigenstates of
60 And this becomes, perhaps, more clear when we realize that this last condition is equivalent to the statement that we simply want to find a basis in which the submatrix diagonal. We just want its off-diagonal elements to vanish! So the problem of implementing the proposed strategy reduces to the following: Given an Hermitian submatrix [] representing the perturbation within some degenerate eigenspace , in a particular representation of unperturbed eigenstates , find a new ONB of unperturbed eigenstates for that subspace for which the matrix
representing the perturbation in this new set of basis states is diagonal. But we have a verb for this process. The verb is "diagonalize". The new basis we are looking for are just the eigenstates of 61 And this becomes, perhaps, more clear when we realize that this last condition is equivalent to the statement that we simply want to find a basis in which the submatrix diagonal. We just want its off-diagonal elements to vanish! So the problem of implementing the proposed strategy reduces to the following:
Given an Hermitian submatrix [] representing the perturbation within some degenerate eigenspace , in a particular representation of unperturbed eigenstates , find a new ONB of unperturbed eigenstates for that subspace for which the matrix representing the perturbation in this new set of basis states is diagonal. But we have a verb for this process. The verb is "diagonalize". The new basis we are looking for are just the eigenstates of 62 And this becomes, perhaps, more clear when we realize that this last condition is
equivalent to the statement that we simply want to find a basis in which the submatrix diagonal. We just want its off-diagonal elements to vanish! So the problem of implementing the proposed strategy reduces to the following: Given an Hermitian submatrix [] representing the perturbation within some degenerate eigenspace , in a particular representation of unperturbed eigenstates , find a new ONB of unperturbed eigenstates for that subspace for which the matrix representing the perturbation in this new set of basis states is diagonal. But we have a verb for this process. The verb is "diagonalize". The new basis we are looking for are just the eigenstates of
63 And this becomes, perhaps, more clear when we realize that this last condition is equivalent to the statement that we simply want to find a basis in which the submatrix diagonal. We just want its off-diagonal elements to vanish! So the problem of implementing the proposed strategy reduces to the following: Given an Hermitian submatrix [] representing the perturbation within some degenerate eigenspace , in a particular representation of unperturbed eigenstates , find a new ONB of unperturbed eigenstates for that subspace for which the matrix
representing the perturbation in this new set of basis states is diagonal. But we have a verb for this process. The verb is "diagonalize". The new basis we are looking for are just the eigenstates of 64 And this becomes, perhaps, more clear when we realize that this last condition is equivalent to the statement that we simply want to find a basis in which the submatrix diagonal. We just want its off-diagonal elements to vanish! So the problem of implementing the proposed strategy reduces to the following:
Given an Hermitian submatrix [] representing the perturbation within some degenerate eigenspace , in a particular representation of unperturbed eigenstates , find a new ONB of unperturbed eigenstates for that subspace for which the matrix representing the perturbation in this new set of basis states is diagonal. But we have a verb for this process. The verb is "diagonalize". The new basis we are looking for are just the eigenstates of 65 And this becomes, perhaps, more clear when we realize that this last condition is
equivalent to the statement that we simply want to find a basis in which the submatrix diagonal. We just want its off-diagonal elements to vanish! So the problem of implementing the proposed strategy reduces to the following: Given an Hermitian submatrix [] representing the perturbation within some degenerate eigenspace , in a particular representation of unperturbed eigenstates , find a new ONB of unperturbed eigenstates for that subspace for which the matrix representing the perturbation in this new set of basis states is diagonal. But we have a verb for this process. The verb is "diagonalize". The new basis we are looking for are just the eigenstates of
66 So, what we have just described is equivalent to the prescription: Diagonalize the submatrix [] representing the perturbation within each degenerate eigenspace . How do we do that? 1. Find the roots of the characteristic equation det. 2. For each root substitute into the eigenvalue equation and solve to find the normalized column vectors
67 So, what we have just described is equivalent to the prescription: Diagonalize the submatrix [] representing the perturbation within each degenerate eigenspace . How do we do that? 1. Find the roots of the characteristic equation det. 2. For each root substitute into the eigenvalue equation and solve to find the normalized column vectors 68
So, what we have just described is equivalent to the prescription: Diagonalize the submatrix [] representing the perturbation within each degenerate eigenspace . How do we do that? 1. Find the roots of the characteristic equation det. 2. For each root substitute into the eigenvalue equation and solve to find the normalized column vectors 69
So, what we have just described is equivalent to the prescription: Diagonalize the submatrix [] representing the perturbation within each degenerate eigenspace . How do we do that? 1. Find the roots of the characteristic equation det. 2. For each root substitute into the eigenvalue equation and solve to find the normalized column vectors 70 So, what we have just described is equivalent to the prescription:
Diagonalize the submatrix [] representing the perturbation within each degenerate eigenspace . How do we do that? 1. Find the roots of the characteristic equation det. 2. For each root substitute into the eigenvalue equation and solve to find the normalized column vectors 71 These column vectors represent the eigenstates of this matrix in the original representation , and therefore define precisely the new basis of states
that we are looking for in the subspace . In this new representation, we then have a new matrix diagonal that is strictly with diagonal elements (the eigenvalues of []), that are then simply the mean values of taken with respect to this new set of unperturbed eigenstates of , and can be identified as giving the true first order energy shifts for the problem. 72
These column vectors represent the eigenstates of this matrix in the original representation , and therefore define precisely the new basis of states that we are looking for in the subspace . In this new representation, we then have a new matrix diagonal that is strictly with diagonal elements (the eigenvalues of []), that are then simply the mean values of taken with respect to this new set of unperturbed eigenstates of , and can be identified as giving the true first order energy shifts for the problem.
73 These column vectors represent the eigenstates of this matrix in the original representation , and therefore define precisely the new basis of states that we are looking for in the subspace . In this new representation, we then have a new matrix diagonal that is strictly with diagonal elements (the eigenvalues of []), that are then simply the mean
values of taken with respect to this new set of unperturbed eigenstates of , and can be identified as giving the true first order energy shifts for the problem. 74 These column vectors represent the eigenstates of this matrix in the original representation , and therefore define precisely the new basis of states that we are looking for in the subspace . In this new representation, we then have a new matrix diagonal that is strictly
with diagonal elements (the eigenvalues of []), that are then simply the mean values of taken with respect to this new set of unperturbed eigenstates of , and can be identified as giving the true first order energy shifts for the problem. 75 OK, so in conclusion: Q: What is degenerate perturbation theory? A: It is the prescription: Diagonalize the submatrix [] representing the perturbation within each degenerate eigenspace to obtain: 1. The correct first order energy shifts for the problem (from the eigenvalues of [] thus obtained), and
2. A new basis of unperturbed states that are not connected by the perturbation, and to which the the first order corrections to the states, and the 2nd order corrections to the energies can, if needed, now be applied without any of the bothersome divergences appearing. Problem solved! 76 OK, so in conclusion: Q: What is degenerate perturbation theory? A: It is the prescription: Diagonalize the submatrix [] representing the perturbation within each degenerate eigenspace to obtain:
1. The correct first order energy shifts for the problem (from the eigenvalues of [] thus obtained), and 2. A new basis of unperturbed states that are not connected by the perturbation, and to which the the first order corrections to the states, and the 2nd order corrections to the energies can, if needed, now be applied without any of the bothersome divergences appearing. Problem solved! 77 OK, so in conclusion: Q: What is degenerate perturbation theory?
A: It is the prescription: Diagonalize the submatrix [] representing the perturbation within each degenerate eigenspace to obtain: 1. The correct first order energy shifts for the problem (from the eigenvalues of [] thus obtained), and 2. A new basis of unperturbed states that are not connected by the perturbation, and to which the the first order corrections to the states, and the 2nd order corrections to the energies can, if needed, now be applied without any of the bothersome divergences appearing. Problem solved! 78
OK, so in conclusion: Q: What is degenerate perturbation theory? A: It is the prescription: Diagonalize the submatrix [] representing the perturbation within each degenerate eigenspace to obtain: 1. The correct first order energy shifts for the problem (from the eigenvalues of [] thus obtained), and 2. A new basis of unperturbed states that are not connected by the perturbation, and to which the the first order corrections to the states, and the 2nd order corrections to the energies can, if needed, now be applied without any of the bothersome divergences appearing. Problem solved!
79 OK, so in conclusion: Q: What is degenerate perturbation theory? A: It is the prescription: Diagonalize the submatrix [] representing the perturbation within each degenerate eigenspace to obtain: 1. The correct first order energy shifts for the problem (from the eigenvalues of [] thus obtained), and 2. A new basis of unperturbed states that are not connected by the perturbation, and to which the first order corrections to the states, and the 2nd order corrections to the energies can, if needed, now be applied without any of the bothersome divergences appearing!
Problem solved! 80 OK, so in conclusion: Q: What is degenerate perturbation theory? A: It is the prescription: Diagonalize the submatrix [] representing the perturbation within each degenerate eigenspace to obtain: 1. The correct first order energy shifts for the problem (from the eigenvalues of [] thus obtained), and 2. A new basis of unperturbed states that are not connected by the perturbation, and to which the first order corrections to the states, and the
2nd order corrections to the energies can, if needed, now be applied without any of the bothersome divergences appearing! Problem solved! 81 82