ABSTRACT
The operator is contained in the expression (31.1) of the central field Hamiltonian, and this can be shown using Eq.(31.6), where the component operators are replaced by their forms (29.16) written in Cartesian coordinates, and this gives
⎟⎠ ⎞⎜⎝
⎛ ∂ ∂−∂
∂⎟⎠ ⎞⎜⎝
⎛ ∂ ∂−∂
∂−⎟⎟⎠ ⎞
⎜⎜⎝ ⎛
∂ ∂−∂
∂⎟⎟⎠ ⎞
⎜⎜⎝ ⎛
∂ ∂−∂
∂−= z
x x
z z
x x
z y
z z
y y
z z
yL 222ˆ ==
⎟⎟⎠ ⎞
⎜⎜⎝ ⎛
∂ ∂−∂
∂⎟⎟⎠ ⎞
⎜⎜⎝ ⎛
∂ ∂−∂
∂− x
y y
x x
y y
x2= It is a straightforward matter to show that this relation may also be written as
⎥⎥⎦ ⎤
⎢⎢⎣ ⎡
⎟⎟⎠ ⎞
⎜⎜⎝ ⎛
∂ ∂+∂
∂+⎟⎟⎠ ⎞
⎜⎜⎝ ⎛
∂ ∂+∂
∂+⎟⎟⎠ ⎞
⎜⎜⎝ ⎛
∂ ∂+∂
∂−= 2 2
yx z
xz y
zy xL =
⎥⎦ ⎤⎢⎣
⎡ ∂ ∂+∂
∂+∂ ∂+∂∂
∂+∂∂ ∂+∂∂
∂+ z
z y
y x
x xz
zx zy
yz yx
xy 222222 222
2= We now use the expansion
( )2 2ˆ ˆr p x y z x y zx y z x y z⎛ ⎞⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂⋅ = − + + + +⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠⎝ ⎠G G =
⎥⎦ ⎤⎢⎣
⎡ ∂ ∂+∂
∂+∂ ∂+∂∂
∂+∂∂ ∂+∂∂
∂+∂ ∂+∂
∂+∂ ∂−=
z z
y y
x x
xz zx
zy yz
yx xy
z z
y y
x x
so that the addition of the last two relations yields
( ) ( ) 2 2 222 2 2 2 2 22 2 2ˆ ˆLˆ r p x y z x y zx y zx y z⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂+ ⋅ = − + + + + + + +⎜ ⎟ ⎜ ⎟∂ ∂ ∂∂ ∂ ∂ ⎝ ⎠⎝ ⎠G G = =
( )pripr ˆˆˆ 22 GG=G ⋅+= (31.10) It follows that the momentum can be expressed as
( ) ( ) ( ) rrr
r r
L r
priprL r
p ∂ ∂−⎟⎠
⎞⎜⎝ ⎛
∂ ∂−=⎥⎦
⎤⎢⎣ ⎡ ⋅−⋅+=
2 ˆ1ˆˆˆˆˆ1ˆ ==GG=GGG (31.11) where the radial part results from the substitution of Eq. (I.20), Appendix I, which gives
1 1ˆ sinr
p i i e e e r r rθ ϕθ θ ϕ
⎛ ⎞∂ ∂ ∂= − ∇ = − + +⎜ ⎟∂ ∂ ∂⎝ ⎠ G G G G= =
and hence
ˆ ˆr p i r r
∂⋅ = − ∂ G G =
Therefore the Hamiltonian (31.1) becomes
( )rUL rrrr
r rm
H + ⎥⎥⎦ ⎤
⎢⎢⎣ ⎡ −∂
∂+⎟⎠ ⎞⎜⎝
⎛ ∂ ∂−= 222
2 ˆ111
2 ˆ
= =
and reduces to the equivalent form
( )rUL rr
r rrm
H +⎥⎦ ⎤⎢⎣
⎡ −⎟⎠ ⎞⎜⎝
⎛ ∂ ∂
2 ˆ11
2 ˆ
= = (31.12)
used for a particle bound by a central force field. 31.2. EIGENVALUE EQUATIONS IN A CENTRAL FIELD Spherically symmetric systems, where the potential energy ( )rU is independent of the direction of ,rG are usually treated using the spherical coordinates ( )ϕθ ,,r related to the Cartesian coordinates as referred to in Appendix I, Eq.(I.19). As the angle θ is defined by the inclination to the z-axis, it is mathematically convenient to take the direction of the component of
( zyx ,, ) L G
that is compatible with (in the sense introduced in Chapter 34) to be the z-axis. Therefore we consider the eigenvalue equations of the operators and given by
ˆ ˆ, zH L 2Lˆ
( ) ( )ˆ , , , ,H r rθ ϕ ε θ ϕΨ = Ψ
( ) ( )ϕθϕθ ,,,,ˆ rLrL zz Ψ=Ψ (31.13)
( ) ( )ϕθϕθ ,,,,ˆ 22 rLrL Ψ=Ψ
The appropriate form of the Hamiltonian Hˆ is conveniently written using the standard expression (I.23) of the Laplace operator in spherical coordinates as 2∇
( )rU rrr
r rrm
H +⎥⎦ ⎤⎢⎣
⎡ ∂ ∂+⎟⎠
⎞⎜⎝ ⎛
∂ ∂
∂ ∂+⎟⎠
⎞⎜⎝ ⎛
∂ ∂
∂ ∂−= 2
sin 1sin
sin 11
2 ˆ
ϕθθθθθ = (31.14)
Using the transformation (I.19) from Cartesian to spherical coordinates and the chain rule for differentiation, we find
sin sin sin cosx y z r r x y x y z x y y
θ ϕ θ ϕϕ ϕ ϕ ϕ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂= + + = − + = −∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂x
so that the operator takes the form zLˆ
ϕ∂ ∂=⎟⎟⎠
⎞ ⎜⎜⎝ ⎛
∂ ∂−∂
∂= ix
y y
x i
Lz ==ˆ (31.15)
Furthermore, from Eqs.(31.12) and (31.14) it follows that is in fact expressed by that part of Hamiltonian which operates on the angular variables
2Lˆ ,θ ϕ only, that is,
⎥⎦ ⎤⎢⎣
⎡ ∂ ∂+⎟⎠
⎞⎜⎝ ⎛
∂ ∂
∂ ∂−= 2
sin 1sin
sin 1ˆ
ϕθθθθθ=L (31.16) The energy eigenfunctions in central field, which are simultaneously eigenfunctions of and are subjected, in addition to the usual bound state requirement (30.21), to the single-valueness condition
zLˆ 2ˆ ,L
( ) ( )πϕθϕθ 2,,,, +Ψ=Ψ rr since the points ϕ and πϕ 2+ are in fact the same physical point. The solution to the time-independent Schrödinger equation (31.13) in central field can be found by a process of separation of variables, yielding three ordinary differential equations in the variables ,r θ and ,ϕ respectively, to which a physical significance will be assigned. It is common practice to separate the variables in two stages, first substituting the trial solution ( ) ( ) ( ), , ,r R r Yθ ϕΨ = θ ϕ (31.17) into the energy eigenvalue equation. If the Hamiltonian is expanded in spherical coordinates, as given by Eq.(31.14), Eq.(31.13) becomes
( )2 22 2 2 21 1 1 sin2 2 sin sin 21R Y Yr U r r
m R r r m Y ε θθ θ θ θ ϕ
⎡ ⎤⎡ ⎤∂ ∂ ∂ ∂ ∂⎛ ⎞ ⎛ ⎞⎡ ⎤+ − = − +⎢ ⎥⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎣ ⎦∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦ = =
(31.18) One side is independent of θ and ,ϕ and the other side is independent of r, so they must be separately equal to some constant, which is conveniently chosen to be equal to
In this way we obtain a radial equation, given by .2/2 mL
( ) ( ) ( )rRrR mr LrU
dr dr
dr d
rm ε=⎥⎦
⎤⎢⎣ ⎡ ++⎟⎠
⎞⎜⎝ ⎛− 2
2 1
2 = (31.19)
where was replaced by for obvious reasons, and an angular equation, having the so-called spherical harmonics
r∂∂ / drd / ( ),Y θ ϕ as solutions, which reads
2 2 ˆor
sin 1sin
sin 1 ==⎥⎦
⎤⎢⎣ ⎡
∂ ∂+⎟⎠
⎞⎜⎝ ⎛
∂ ∂
∂ ∂− ϕθθθθθ= (31.20)
The form of the spherical harmonics is obtained as a result of the second stage of the separation process, where we assume that ( ) ( ) ( )ϕθϕθ Φ= PY ,
so that Eq.(31.20) becomes
2 2 1sinsinsin ϕθθθθ
θ ∂
Φ∂ Φ−=⎥⎦
⎤⎢⎣ ⎡ +⎟⎠
⎞⎜⎝ ⎛
∂ ∂
∂ ∂ ===
LP P
(31.21)
Since one side is independent of ϕ while the other is independent of ,θ they must each be equal to a constant, which we call We thus obtain the differential equations .2zL
d L dϕ
Φ 2− = Φ= (31.22) where again ϕ∂∂ / has been replaced by ϕdd / , and
0 sin
sin sin
1 22
⎥⎥⎦ ⎤
⎢⎢⎣ ⎡
⎟⎟⎠ ⎞
⎜⎜⎝ ⎛ −+⎟⎠
⎞⎜⎝ ⎛ PLL
d dP
d d z
θθθθθ === (31.23) where θ∂∂ / has been replaced by θdd / . Therefore the energy equation (31.13) completely separates, in central field, into a radial equation (31.19) and two angular equations (31.22) and (31.23). The radial equation, giving the energy levels ε of the system, takes the form
( ) ( ) ( ) ( )rRrR mr LrUrR
dr d
rdr d
m ε=⎥⎦
⎤⎢⎣ ⎡ ++⎟⎟⎠
⎞ ⎜⎜⎝ ⎛ +− 2
2 2
2 = (31.24)
which can be simplified by making the substitution ( ) ( )rrRr =ξ (31.25)
and using the identity
2 1r dd d r dr r rdr dr
ξ ξ⎛ ⎞+ =⎜ ⎟⎝ ⎠ r
Thus Eq.(31.24) reduces to
( ) ( ) ( ) ( )rr mr LrU
dr rd
m εξξξ =⎥⎦
⎤⎢⎣ ⎡ ++− 2
22 = (31.26)
which is formally identical to the time-independent Schrödinger equation in one dimension (30.17), provided the central force potential ( )rU is replaced by an effective potential energy of the form
( ) ( ) 2 2
2mr LrUrUeff +=
The repulsive potential, associated with the angular momentum, modifies the central potential as illustrated before, see Figure 3.3, for ( ) rCrU /−= . Further progress with the solution of the radial equation depends on the particular form of the potential function
. We shall consider the significant example of one-electron atoms in the next chapter.
