We consider a perturbation of a solvable time-independent Hamiltonian H0 by a time-independent term V, and for bookkeeping purposes we extract a coupling constantfrom the perturbation,
HDH0CV!HDH0CV:
After the relevant expressions for shifts of states and energy levels have been calculated to the desired order in, we usually subsume again inV, such that e.g.h.0/jVj .0/i ! h.0/jVj .0/i.
© Springer International Publishing Switzerland 2016
R. Dick,Advanced Quantum Mechanics, Graduate Texts in Physics, DOI 10.1007/978-3-319-25675-7_9
171
172 9 Stationary Perturbations in Quantum Mechanics We know the unperturbed energy levels and eigenstates of the solvable Hamilto- nianH0,
H0j j.0/i DE.0/j j j.0/i:
In the present section we assume that the energy levels E.0/j are not degenerate, and we want to calculate in particular approximations for the energy level Ei
which arises from the unperturbed energy level Ei.0/ due to the presence of the perturbationV. We will see below that consistency of the formalism requires that the differencesjE.0/i E.0/j jforj ¤ imust have a positive minimal value, i.e. the unperturbed energy levelE.0/i for which we want to calculate corrections has to be discrete1.
Orthogonality of eigenstates for different energy eigenvalues implies h i.0/j j.0/i Dıij:
In the most common form of time-independent perturbation theory we try to find an approximate solution to the equation
Hj ii DEij ii
in terms of power series expansions in the coupling constant, j ii DX
n0
nj i.n/i; h i.0/j i.n1/i D0; EiDX
n0
nEi.n/: (9.1) Depending on the properties ofV, these series may converge for small values ofjj, or they may only hold as asymptotic expansions forjj ! 0. The book by Kato [21] provides results and resources on convergence and applicability properties of the perturbation series. Here we will focus on the commonly used first and second order expressions for wave functions and energy levels.
We can require
h i.0/j i.n/i Dın;0 (9.2) because the recursion equation (9.3) below, which is derived without the assump- tion (9.2), does not determine these particular coefficients. One way to understand this is to observe that we can decomposej i.n1/iinto terms parallel and orthogonal toj i.0/i,
j i.n1/i D j i.0/ih i.0/j i.n1/i C j i.n1/i j i.0/ih i.0/j i.n1/i:
1This condition is not affected by a possible degeneracy ofEi.0/, as will be shown in Section9.2.
Inclusion of the parallel partj i.0/ih i.0/j i.n1/iin the zeroth order term, followed by a rescaling by
1C h i.0/j i.n1/i1
D1 h i.0/j i.n1/i CO.2n/
to restore a coefficient 1 in the zeroth order term, affects only terms of ordernC1or higher in the perturbation series. This implies that if we have solved the Schrửdinger equation to ordern1with the constraint
h i.0/j i.m/i Dım;0; 0mn1;
then ensuring that constraint also to ordernpreserves the constraint for the lower order terms. Therefore we can fulfill the constraint (9.2) to any desired order in which we wish to calculate the perturbation series.
Substitution of the perturbative expansions into the Schrửdinger equation Hj ii DEij iiyields
X
n0
nH0j i.n/i CX
n0
nC1Vj i.n/i D X
m;n0
mCnE.im/j i.n/i
DX
n0
Xn mD0
nE.im/j i.nm/i:
This equation is automatically fulfilled at zeroth order. Isolation of terms of order nC1forn0yields
H0j i.nC1/i CVj i.n/i D
nC1
X
mD0
E.im/j i.nmC1/i;
and projection of this equation ontoj j.0/iyields Ej.0/h j.0/j i.nC1/i C h j.0/jVj i.n/i D
Xn mD0
E.im/h j.0/j i.nmC1/i
CE.inC1/ıij: (9.3) We can first calculate the first order corrections for energy levels and wave functions from this equation, and then solve it recursively to any desired order.
First order corrections to the energy levels and eigenstates
The first order corrections are found from equation (9.3) fornD0. Substitution of jDiimplies for the first order shifts of the energy levels the result
E.1/i D h i.0/jVj i.0/i; (9.4)
174 9 Stationary Perturbations in Quantum Mechanics
andj¤iyields withE.0/i ¤E.0/j the first order shifts of the energy eigenstates h j.0/j i.1/i D h j.0/jVj i.0/i
E.0/i Ej.0/ : (9.5)
Recursive solution of equation (9.3) for n 1
We first observe thatjDiin equation (9.3) implies with the condition (9.2) E.inC1/D h i.0/jVj i.n/i
Xn mD1
Ei.m/h i.0/j i.nmC1/i D h i.0/jVj i.n/i; (9.6) andi¤jyields
Ei.0/E.0/j
h j.0/j i.nC1/i D h j.0/jVj i.n/i
Xn mD1
E.im/h j.0/j i.nmC1/i: (9.7) The right hand side of both equations depends only on lower order shifts of energy levels and eigenstates. Therefore these equations can be used for the recursive solution of equation (9.3) to arbitrary order.
Second order corrections to the energy levels and eigenstates
Substitution ofnD1into equation (9.6) yields with (9.5) and XZ
k
j k.0/ih k.0/j D1 the second order shift
E.2/i DXZ
k¤i
h i.0/jVj k.0/ih k.0/jVj i.0/i E.0/i Ek.0/
DXZ
k¤i
jh i.0/jVj k.0/ij2
Ei.0/E.0/k : (9.8) States in the continuous part of the spectrum ofH0will also contribute to the shifts in energy levels and eigenstates. It is only required that the energy levelE.0/i , for which we want to calculate the corrections, is discrete and does not overlap with any continuous energy levels.
Note that equation (9.8) implies that the second order correction to the ground state energy is always negative.
For the eigenstates, equation (9.7) yields with the first order results (9.4,9.5) the equation (recalli¤jin (9.7))
h j.0/j i.2/i DXZ
k¤i
h j.0/jVj k.0/ih k.0/jVj i.0/i Ei.0/E.0/j
Ei.0/E.0/k h i.0/jVj i.0/ih j.0/jVj i.0/i
E.0/i E.0/j 2 : (9.9) Now we can explain why it is important that our original unperturbed energy levelE.0/i is discrete. To ensure that then-th order corrections to the energy levels and eigenstates in equations (9.1) are really of ordern(or smaller than all previous terms), the matrix elementsjh j.0/jVj k.0/ijof the perturbation operator should be at most of the same order of magnitude as the energy differencesjE.0/i E.0/j jbetween the unperturbed levelE.0/i and the other unperturbed energy levels in the system.
This implies in particular that the minimal absolute energy difference betweenE.0/i and the other unperturbed energy levels must not vanish, i.e.E.0/i must be a discrete energy level.
Equations (9.4) and (9.8) (and their counterparts (9.16) and (9.24) in degenerate perturbation theory below) used to be the most frequently employed equations of time-independent perturbation theory, because historically many experiments were concerned with spectroscopic determinations of energy levels. However, measurements e.g. of local electron densities or observations of wave functions (e.g.
in scanning tunneling microscopes or through X-ray scattering using synchrotrons) are very common nowadays, and therefore the corrections to the states are also directly relevant for the interpretation of experimental data.
Summary of non-degenerate perturbation theory in second order
If we includewithV, the states and energy levels in second order are j ii D j i.0/i C j i.1/i C j i.2/i D j i.0/i
CXZ
j¤i
j j.0/ih j.0/jVj i.0/i E.0/i Ej.0/
CXZ
j;k¤i
j j.0/ih j.0/jVj k.0/ih k.0/jVj i.0/i E.0/i Ej.0/
E.0/i Ek.0/
XZ
j¤ij j.0/ih j.0/jVj i.0/ih i.0/jVj i.0/i
E.0/i Ej.0/2 (9.10)
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and
EiDEi.0/C h i.0/jVj i.0/i CXZ
j¤i
jh i.0/jVj j.0/ij2
E.0/i E.0/j : (9.11) The second order statesj iiare not normalized any more,
h ij ji DıijCO.2/ıij: Normalization is preserved in first order due to
h i.0/j j.1/i C h i.1/j j.0/i D0;
but in second order we have
h i.0/j j.2/i C h i.1/j j.1/i C h i.2/j j.0/i DXZ
k¤i
jh i.0/jVj k.0/ij2 Ei.0/Ek.0/2 ıij:
However, we can add to the leading term j i.0/i in j ii a term of the form j i.0/iO.2/and still preserve the master equation (9.3) to second order. We can therefore rescale (9.10) by a factorŒ1CO.2/1=2to a normalized second order state
j ii D j i.0/i 1
2j i.0/iXZ
j¤i
jh i.0/jVj j.0/ij2 E.0/i Ej.0/2 CXZ
j¤i
j j.0/ih j.0/jVj i.0/i E.0/i Ej.0/
CXZ
j;k¤i
j j.0/ih j.0/jVj k.0/ih k.0/jVj i.0/i E.0/i Ej.0/
E.0/i E.0/k XZ
j¤i
j j.0/ih j.0/jVj i.0/ih i.0/jVj i.0/i
Ei.0/E.0/j 2 : (9.12)
Now the second order shift is not orthogonal toj i.0/iany more, but we still have a solution of equation (9.3) to second order.