Veff(r)=V (r)+ ¯h2(+1)
2μr2 , (7.105)
with a “centrifugal barrier” term M2/(2μr2) just like in classical mechanics.
The reason for this term is essentially the same consistency requirement as in classical mechanics. Classically, two particles with non-vanishing relative angular momentumM can never be in the same location, and the centrifugal barrier term simply reflects this property. Quantum mechanically, non-vanishing relative angular momentumMimplies that the particular valueψ(r =0)of the radial wave function must be suppressed, and it must be more strongly suppressed for largerM2.
Equation (7.104) is usually solved by the Sommerfeld method. In the first step one studies the asymptotic equations for smallrand for larger, and keeps only the normalizable solutions or those solutions which approximate Fourier monomials in the asymptotic regions. In the next step one makes anansatzfor the full solution by multiplying the asymptotic solutions with a polynomial. Before we apply this method to the hydrogen atom, we will do something that one might find odd at first sight. The simplest case of a radially symmetric potential isV =0, i.e. free motion.
It is of interest both for scattering theory and for ionization or decay of rotationally symmetric systems to discuss free motion with defined angular momentum, when the wave function for a free particle has the form (7.102).
7.7 Free Spherical Waves: The Free Particle with Sharp Mz, M2
The radial Schrửdinger equation for a free particle with fixed angular momentum Mz,M2and energyE= ¯h2k2/2μis
− ¯h2 2μ 1 r
d2
dr2rψ(r)+ ¯h2(+1)
2μr2 ψ(r)=Eψ(r), (7.106) or
d2
dr2 −(+1) r2 +k2
rψ(r)=0. (7.107)
The regular solution for=0 is ψk,0(r)=
2 π
sin(kr)
kr , (7.108)
where the normalization was determined from the condition
∞
0
dr r2ψk,0(r)ψk,0(r)= 1
kkδ(k−k). (7.109) For the study of solutions ψk,(r) for higher , we observe that solutions of Eq. (7.107) forkr % √
(+1)areψ(r) ∝ r orψ(r) ∝ r−−1. We will first study solutions which are regular forr =0, i.e. forkr %√
(+1)our solutions must approximater. Therefore we substituteψk,(r)=rfk,(r)into Eq. (7.107),
d2 dr2 +2
r(+1)d dr +k2
fk,(r)= 1
r d2 dr2r+2
r d dr +k2
fk,(r)=0.
It is useful to write this as
ArBr +2Ar +k2
fk,(r)=0 (7.110)
with operators
Ar = 1 r
d
dr, Br = d
drr. (7.111)
These operators satisfy the commutation relation
[Ar, Br] =2Ar, (7.112)
and this implies Ar
%ArBr +k2&
=Ar
%BrAr +2Ar+k2&
=%
ArBr +2Ar +k2&
Ar, Ar%
ArBr +k2&
=%
ArBr+2Ar +k2&
Ar. (7.113)
This yields fk,(r)∝ 1
r d
drfk,−1(r)∝ 1
r d dr
fk,0(r)= 2
π 1
r d dr
sin(kr) kr ,
(7.114) ψk,(r)∝
2 πr
1 r
d dr
sin(kr)
kr =(−)k 2
πj(kr). (7.115) Here we used the definition of the spherical Bessel functions,
j(x)=(−x) 1
x d dx
sinx
x =
π 2xJ+1
2
(x). (7.116)
The asymptotic expansion ofj(x)for large argument is (see Problem7.4for the proof)
7.7 Free Spherical Waves: The Free Particle with SharpMz,M2 149
j(x)
x1≈ 1 xsin
x−π
2
. (7.117)
However, the dominant contribution to the completeness relations on the conjugate semiaxesk≥0 andr≥0,
∞
0
dr r2ψk,+ (r)ψk,(r)= 1
k2δ(k−k), (7.118) ∞
0
dk k2ψk,+ (r)ψk,(r)= 1
r2δ(r−r), (7.119) in the singular cases k = k or r = r come from the region kr 1, and therefore the asymptotic limit ofψk,(r)must also correspond to functions which satisfy the completeness relations. This implies that the properly normalized radial eigenfunctions are
ψk,(r)= 2
πij(kr)= 2
π r
ik
1 r
d dr
sin(kr)
kr , (7.120)
and the free spherical waves with sharp angular momentaM2,Mzare r|k, , m =
2
πij(kr)Y,m(ϑ, ϕ)= i
√krJ+1 2
(kr)Y,m(ϑ, ϕ). (7.121) The expansion for small argumentkr,
ψk,(r) 2
π
(ikr)
(2+1)!!, (7.122)
is in agreement with our initial finding that regular solutions of equation (7.107) should behave likeψk,(r)∝rnear the origin, and follows from
x
−1 x
d dx
sinx x =x
−1 x
d dx
∞ n=0
(−)n x2n (2n+1)!
= ∞
n=
(−)n+2nã(2n−2)ã. . .ã(2n−2+2)
× x2n−
(2n+1)! = x
(2+1)!!+O(x+2). (7.123) Our conventions for the phase and the normalization of the radial wave function are motivated by the expansion of plane waves in terms of spherical harmonics. If
we define
k|k, , m = 1
kkδ(k−k)Y,m(k),ˆ (7.124) we automatically get the expansion of plane waves in terms of spherical harmonics,
r|k = 1
√2π3
exp(ikãr)= 2
π
∞
=0
m=−
ij(kr)Y,m(r)Yˆ ,m+ (k)ˆ
= 1
√kr
∞
=0
m=−
iJ+1 2
(kr)Y,m(rˆ)Y,m+ (ˆk). (7.125)
This expansion is also particularly useful for exp(ikz). We havePm(1)=δm,0and therefore from (7.90)Y,m(ez)=Y,m(ϑ =0)=√
(2+1)/4π δm,0. This yields exp(ikz)= ∞
=0
(2+1)ij(kr)P(cosϑ). (7.126)
If our discussion above does not refer to motion of a single particle with massμ, but to relative motion of two non-interacting particles at locations
x1=R+ m2
m1+m2r, x2=R− m1
m1+m2r (7.127)
we can write a full two-particle wave function with sharp angular momentum quantum numbers for the relative motion as
R,r|K, k, , m = i
2π2exp(iKãR)j(kr)Y,m(r),ˆ (7.128) or we could also require sharp angular momentum quantum numbersL, Mfor the center or mass motion,6
R,r|K, L, M, k, , m = 2
πiL+jL(KR)j(kr)YL,M(R)Yˆ ,m(r).ˆ (7.129)
6. . . or we could use total angular momentum, i.e. quantum numbersK, k, j∈ {|L−|, . . . , L+ }, mj=M+m, L, .
7.7 Free Spherical Waves: The Free Particle with SharpMz,M2 151
Asymptotically Free Angular Momentum Eigenstates
We will denote potentialsU (x)which satisfy the condition limr→∞r2U (x) = 0 as localized potentials. Particles which are scattered by a localized potential satisfy Eq. (7.107) asymptotically at large distance. Although the regular solutions (7.120) describe the radial factors of a complete system of wave functions (7.121), it would be very cumbersome (through the use of singular integral representations) to rely only on those functions for the description of the asymptotic wave functions of the scattered particles. Instead, one also uses the solutions of Eq. (7.107) which are singular forr→0 and result from the second solution of (7.107) for=0,
φk,0(r)= − 2
π
cos(kr)
kr . (7.130)
Theansatzφk,(r) =rgk,(r)and the same reasoning which led to (7.120) then leads to singular solutions of Eq. (7.107) for other values of,
φk,(r)= 2
πin(kr)= − 2
π r
ik
1 r
d dr
cos(kr)
kr , (7.131)
with the spherical Neumann functions7(or spherical Bessel functions of the second kind)
n(x)= −(−x) 1
x d dx
cosx
x . (7.132)
The spherical Neumann functions have asymptotic behavior for large argument n(x)
x1≈ −1 x cos
x−π
2
, (7.133)
see Problem7.4. The asymptotic behavior forx ≡kr%1, n(x)
|x|%1 −(2−1)!! ãx−−1, (7.134) can also be physically relevant for localized potentials if k is sufficiently small, viz.if the wavelength of the scattered particles is large compared to the range of the scattering potential. Equation (7.134) follows from
7The spherical Neumann functionsn(x)are denoted byy(x)in [1]. They can also be considered as spherical Bessel functionsjn(x)with negative index,n(x)=(−)+1j−−1(x). The notation n(x)is common in the quantum mechanics literature, see e.g. [113,153].
x
−1 x
d dx
cosx x
|x|%1
x
−1 x
d dx
1
x, (7.135)
and shows that the solutions (7.131) realize the solutions of Eq. (7.107) which behave likeψ(r) ∝ r−−1for r → 0. However, using both ψk,(r)andφk,(r) for large distances from a localized scattering potential corresponds to the use of an overcomplete set of states, because for larger we now have two sets of wave functions which provide complete sets8on the semiaxesk ≥ 0,r ≥ 0. Therefore, if we also need to use the spherical Neumann functions, we should use unitary combinations of solutions of Eq. (7.107), e.g.
k,(r)=cosδk,ãψk,(r)−sinδk,ãφk,(r). (7.136) This corresponds for large argumentkr to
k,(r) 2
π i kr sin
kr−π 2 +δk,
, (7.137)
and the full wave functions for given angular momentum quantum numbers,mare r|k, , m =k,(r)Y,m(ϑ, ϕ). (7.138) The parameters δk, are known as scattering phase shifts. They describe how much of the singular solutions (7.131) has been mixed into the free spherical waves due to scattering at a localized potential. We will encounter them again in Chap.11.