ABSTRACT
If we set the first-order eigenvalue correction to the unperturbed state is obtained as the expectation value of the perturbation operator V in this state
,ki = 0kΨ ˆ
( )1 0 ˆkk Vε 0k= Ψ Ψ (36.11) whereas for the coefficients can be obtained from Eq.(36.10) as ,ki ≠ kia
00 ˆ
V a εε −
ΨΨ= (36.12) and the coefficients must be obtained in a different manner. Since the eigenfunction (36.5) is determined up to a phase factor, we may choose the coefficients to be real. The normalization of the eigenfunctions (36.5) to first order in λ gives
kkkkkk Ψ+ΨΨ+Ψ=ΨΨ λλ 00001 kkkkkkkk aa ΨΨ+ΨΨ+= λλ 1Re21 =+= kkaλ so that the result is Substituting Eqs.(36.8), (36.11) and (36.12), Eq.(36.5) gives the first-order results for the system (36.2) under consideration, that is, for
.0=kka ,1=λ as
ˆ i
V Ψ− ΨΨ+Ψ=Ψ ∑
≠ εε , 000 ˆ kkkk V ΨΨ+= εε (36.13)
If certain eigenstates are degenerate, that is, if there are n independent eigenfunctions ,
0 αkΨ 1,2, , ,nα = … which all correspond to the same eigenvalue the
formula (36.12) does not hold and one should proceed as follows. Assuming that the degenerate states are orthonormal and allowing for degeneracy in our expansion (36.5) we obtain
kk A λα α
If we substitute this expression into the eigenvalue equation (36.4), instead of the firstorder equation (36.9), we obtain a set of n relations
words, are states orthogonal to for all α and On multiplication by ( and integration one obtains, using orthonormality, that
(36.15)
which is a set of homogenous equations for the coefficients .kA α If we set
00 ˆ αγγα kk VV ΨΨ= (36.16)
we may rewrite Eq.(36.15) as
= which is a matrix eigenvalue equation, analogous to Eq.(33.9), of the form
⎟⎟ ⎟⎟ ⎟
⎠
⎞
⎜⎜ ⎜⎜ ⎜
⎝
⎛ =
⎟⎟ ⎟⎟ ⎟
⎠
⎞
⎜⎜ ⎜⎜ ⎜
⎝
⎛
⎟⎟ ⎟⎟ ⎟
⎠
⎞
⎜⎜ ⎜⎜ ⎜
⎝
⎛
A
A A
A
A A
VVV
VVV VVV
…… …
……… … …
ε
The condition for nonidentically vanishing is given by the characteristic equation αkA
ˆ ˆdet 0
V V V
V V VV I
V V V
ε εε
ε
− −− = =
−
… …
… … … …
(36.19)
with n roots for which are denoted by ( ) ,1kε ( )1 , 1, 2, ,k nαε α = … . If the roots are all distinct, it is said that the degeneracy has been completely removed due to the perturbation otherwise it is said to be only partially removed. As the eigenvalues give the first-order corrections to the energy, the formerly n-fold degenerate state splits, under the influence of perturbation, into n close-lying states with energies (36.5) which can be written as
Inserting each eigenvalue ( )1αε k into the matrix equation (36.18) yields a set of n coefficients that is restricted by the normalization condition αkA
1 1
If all the eigenstates are degenerate, we no longer have states outside to make the expansion possible (36.8), so that Eq.(36.14) reduces to the first term and therefore the eigenfunction
kΨ can be written as a linear combination of the degenerate states for each energy (36.20). Thus, the first-order results (36.13) of the perturbation theory for nondegenerate states are replaced by Eqs.(36.14) and (36.20) in the case of degenerate states, which are of special interest in experimental physics. The second-order corrections can be obtained using a similar approach to the first-order case. We start by inserting the first-order corrections (36.13) into the secondorder equation (36.6), and this gives
If we multiply Eq.(36.22) by and integrate, one obtains for ( )∗Ψ 0i ki = the second-order eigenvalue corrections as
ε ∑ ≠ −
ΨΨ =
εε (36.23) where use has been made of Eqs.(36.8) and (36.12). It follows that the second-order correction to ground state energy is always negative. Inserting Eq.(36.23) into Eq.(36.5) provides, for
1=λ , the energy
ΨΨ +ΨΨ+=
V V
000 ˆ
ˆ εεεε (36.24)
The corresponding approximation to kΨ may be obtained making use of Eq.(36.22) for
However, it is common practice to avoid terms beyond the first-order correction to the wave function and beyond the second-order correction in energy
.ki ≠ 36.2. THE HELIUM ATOM The helium atom, consisting of a nucleus of charge eZe 2= and two orbiting electrons, is the simplest multielectron atom. If the nucleus is placed at the origin and the spin-interaction terms are ignored, the Hamiltonian operator may be written as
( ) VHH rr
e rr
e m
H e
ˆˆˆ112 2
ˆ 21
++=−+⎟⎟⎠ ⎞
⎜⎜⎝ ⎛ +−∇+∇−= GG= (36.25)
where denotes the mutual electrostatic interaction of the two electrons, which will be treated as a perturbation. The Hamiltonians are single-particle operators for two individual electrons that satisfy the eigenvalue equations
Vˆ
1 ˆ andH 2Hˆ
e m
H ϕεϕϕ =⎟⎟⎠ ⎞
⎜⎜⎝ ⎛ −∇−=
2 2 2
ˆ = (36.26) where The eigenstates have the form (34.1) previously derived for the spinning electron of one-electron atoms, which is
.2,1=i siiii mmlnϕ
( ) ( ) ( ),
i i i si i i i in l m m i n l l m r R r Yϕ θ ϕ χ=G (36.27)
and correspond to degenerate energy levels (32.9) or (28.18) given by
Z n n
ε = − = − ) (36.28)
Thus, if the mutual electrostatic interaction V is ignored, the idealized helium atom can be regarded as a system of two identical fermions, of energy
ˆ
1 154,4 eVn nE n nα ε ε ⎛ ⎞= + = − +⎜ ⎟⎝ ⎠ ) (36.29)
with the corresponding eigenfunction represented by the Slater determinant (35.28), which reads
( ) ( ) ( ) ( )1 1
1 2
r r
r rα ϕ ϕ ϕ ϕΦ =
2 G G G G (36.30)
The ground state configuration of this system contains both electrons in the lowest energy level, that is ,121 == nn ,021 == ll ,021 == mm ,211 =sm .212 −=sm The ground state energy (36.29) is eV8.108−=gE and
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )2210011100
2 1
−− ++=Φ χψχψ
χψχψ rr rr
g GG GG
( ) ( )100 1 100 2 Sr rψ ψ= χG G (36.31)
The spatial part is symmetrical whereas the spin function which is antisymmetrical under spin label interchange, as given by
,Sχ
χχχ S ) (36.32) defines a singlet spin state, of total spin ,0=S such that .112 =+S As both the spatial and spin functions are unique, the ground state is nondegenerate. The first excited state contains one electron in the ground-state level
having the spatial wave function ,1=n ,0=l
0=m 100ψ and another in the energy level defined by with or ,2=n ,0=l 0=m ,1=l 0=m , 1± and corresponding spatial wave functions The configuration is fourfold degenerate because of both exchange and spin
degeneracy, with total energy .2lmψ
68 eV,eE = − as provided by Eq.(36.29). Two of the corresponding Slater determinants (36.30) are products of a spatial function and a spin function, both normalized and of definite symmetry, written as
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )222112
2 1
++ ++=Φ χψχψ
χψχψ rr rr
++= χχψ a (36.33)
and
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )222112
2 1
−− −−=Φ χψχψ
χψχψ rr rr
−−= χχψ a (36.34)
where ( ) ( ) ( ) ( )100 1 2 2 2 1 100 2 / 2.a lm lmr r r rψ ψ ψ ψ ψ⎡= −⎣ G G G G ⎤⎦ For the other two determinants
2 1
χψχψ rr rr
GG ,
2 1
χψχψ rr rr
GG (36.35)
a linear combination will be needed, as shown below, in order to obtain eigenfunctions of definite symmetry. The influence of the mutual electrostatic interaction V is taken into account by using the perturbation theory. For the nondegenerate ground state, due to the spinindependence of the first-order energy correction (see Problem 36.2) reads
ˆ
,Vˆ
5ˆ V V 4g g
e eK V r r r r d d r r a
ψ ψ ψ ψ∗ ∗= Φ Φ = =−∫ G G G GG G (36.36) The positive energy shift represents the repulsive electrostatic interaction between
the two charge densities 10K
( ) 21 100 1e rρ ψ= G and ( ) 22 100 2 .e rρ ψ= G The magnitude of is 34 eV, and this yields a value which is
closer to the experimental ground-state energy of
10 10 108.8 34 74.8 eV,gE E K= + = − + = − 7910 −=E eV.
