The solution for the hydrogen atom was reported by Schrửdinger in 1926 in the same paper where he introduced the time-independent Schrửdinger equation [149].
We recall that separation of the wave function in Eq. (7.27)
ψ(r)=ψ(r)Y,m(ϑ, ϕ) (7.139) and use ofM2|, m = ¯h2(+1)|, myields the radial Schrửdinger equation
− ¯h2 2μ 1 r
d2
dr2rψ(r)+
h¯2(+1) 2μr2 − e2
4π 0r
ψ(r)=Eψ(r), (7.140)
8Both sets of functions√
2/πsin[kr−(π/2)]/(kr)and√
2/πcos[kr−(π/2)]/(kr)are complete on the semiaxesr≥0 andk≥0 through completeness relations of the form (7.118) and (7.119).
7.8 Bound Energy Eigenstates of the Hydrogen Atom 153 where the attractive Coulomb potential between chargeseand−ehas been inserted.
This yields asymptotic equations for smallr,
− r2d2
dr2rψ(r)+(+1)rψ(r)=0, (7.141) and for larger,
− d2
dr2rψ(r)= 2μE
¯
h2 rψ(r). (7.142)
The Euler type differential equation (7.141) has basic solutions rψ(r) = Ar+1+Br−, but with ≥ 0 only the first solutionrψ(r) ∝ r+1 will yield a finite probability density|ψ(r)|2near the origin.
The normalizable solution of (7.142) forE <0 is rψ(r)∝exp
−
−2μEr/h¯
. (7.143)
We combine the asymptotic solutions with a polynomialw(r)=
ν≥0cνrν, rψ(r)=r+1w(r)exp(−κr), κ =
−2μEr/h.¯ (7.144)
Substitution in (7.140) yields the condition r d2
dr2w(r)+2(+1−κr) d drw(r)+
μe2
2π 0h¯2 −2κ(+1)
w(r)=0, (7.145) which in turn yields a recursion relation for the coefficients in the polynomialw(r),
cν+1=cν
2κ(ν++1)−2π μe2
0h¯2
(ν+1)(ν+2+2) . (7.146)
Normalizability of the solution requires termination of the polynomialw(r)with a maximal powerN≡max(ν)≥0 ofr, i.e.cN+1=0 and therefore
κ≡
√−2μE
¯
h = μe2
4π 0h¯2(N++1). (7.147) This implies energy quantization for the bound states in the form
En= − μe4 32π202h¯2
1
n2 = −αS2 2 μc21
n2 (7.148)
with the principal quantum numbern ≡N ++1. Note thatN ≥ 0 implies the relationn≥+1 between the principal and the magnetic quantum number.
We used the definition αS = e2
4π 0hc¯ =7.29735. . .×10−3= 1
137.036. . .. (7.149) of Sommerfeld’sfine structure constantin (7.148).
We will also use Eq. (7.147) in the formκ=(na)−1with the Bohr radius a ≡4π 0h¯2
μe2 = h¯
αSμc. (7.150)
The recursion relation is then cν+1=cν
2 na
ν++1−n
(ν+1)(ν+2+2), 0≤ν≤N ≡n−−1. (7.151) This defines all coefficientscν inw(r)in terms of the coefficientc0, which finally must be determined from normalization. The factor 2/nain the recursion relation will generate a power(2/na)ν incν, such thatw(r)will be a polynomial in 2r/na.
The factor(ν+1)−1will generate a factor 1/ν!incν, and the factor(ν+α)/(ν+β) withα=+1−n,β=2+2 will finally yield a polynomial of the form
w(r)=c0
1+α β
2r na+ 1
2!
α(α+1) β(β+1)
2r na
2
+1 3!
α(α+1)(α+2) β(β+1)(β+2)
2r na
3
+. . .
=c0×1F1(α;β;2r/na).
As indicated in this equation, the series for c0 = 1 defines the confluent hypergeometric function 1F1(α;β;x) ≡ M(α;β;x) (also known as Kummer’s function [1]). For −α ∈ N0 andβ ∈ Nthis function can also be expressed as an associated Laguerre polynomial. The normalized radial wave functions can then be written as
ψn,(r)= 2 n2
(n+)! (n−−1)!a3
1F1(−n++1;2+2;2r/na) (2+1)!
× 2r
na
exp − r
na
= 2 n2
(n−−1)! (n+)!a3
2r na
L2n−+1−1 2r
na
exp − r
na
. (7.152)
7.8 Bound Energy Eigenstates of the Hydrogen Atom 155 Substitution of the explicit series representation forw(r)shows that the radial wave functions are products of a polynomial in 2r/na of ordern−1 withn− terms, multiplied with the exponential function exp(−r/na),
ψn,(r)= 2 n2(−)
(n+)!(n−−1)!
a3 exp
− r na
×
n−1
k=
(−2r/na)k
(k−)!(n−k−1)!(k++1)!. (7.153) The representation (7.152) in terms of the associated Laguerre polynomials differs from older textbook representations by a factor(n+)!due to the modern convention for the normalization of Laguerre polynomials,
Ln(x)= 1
n!exp(x)dn dxn
!xnexp(−x)"
=1F1(−n;1;x), (7.154) and associated Laguerre polynomials,
Lmn(x)=(−)m dm
dxmLn+m(x)
= (−)m (n+m)!
dm dxm
exp(x) dn+m dxn+m
!xn+mexp(−x)"
= (m+n)!
n! ãm! 1F1(−n;m+1;x), (7.155) which is also used in symbolic calculation programs. The normalization follows from
∞
0
dx exp(−x)xm+1[Lmn(x)]2=(2n+m+1)(n+m)!
n! , (7.156)
but their standard orthogonality relation is ∞
0
dx exp(−x)xmLmn(x)Lmn(x)= (n+m)!
n! δn,n. (7.157)
Since they appear as eigenstates of the hydrogen Hamiltonian, the normalized bound radial wave functions must satisfy the orthogonality relation
∞
0
dr r2ψn,(r)ψn,(r)=δn,n. (7.158) This implies that the associated Laguerre polynomials must also satisfy a peculiar additional orthogonality relation which generalizes (7.156),
∞
0
dx exp
− (n+n+m+1)x (2n+m+1)(2n+m+1)
xm+1Lmn
x 2n+m+1
×Lmn
x 2n+m+1
=(2n+m+1)m+3(n+m)!
n! δn,n. (7.159) Squaresψn,2 (r)andr2ψn,2 (r)of the radial wave functions are plotted forn=1 andn=3 in Figs.7.3,7.4,7.5, and7.6.
Fig. 7.3 The functionsa3ψ1,02 (r)(upper panel) andar2ψ1,02 (r)(lower panel)
7.8 Bound Energy Eigenstates of the Hydrogen Atom 157
Fig. 7.4 The functionsa3ψ3,02 (r)(upper panel) andar2ψ3,02 (r)(lower panel). The upper panel is limited to radiir≥ato make the additional maxima visible. We havea3ψ3,02 (0)=4/27=0.148
For the meaning of the radial wave function, recall that the full three-dimensional wave function is
ψn,,m(r)=ψn,(r)Y,m(ϑ, ϕ). (7.160) This implies thatψn,2 (r)is a radial profile of the probability density|ψn,,m(r)|2to find the particle (or rather the quasiparticle which describes relative motion in the
Fig. 7.5 The functionsa3ψ3,12 (r)(upper panel) andar2ψ3,12 (r)(lower panel)
hydrogen atom) in the locationr, but note that in each particular direction(ϑ, ϕ)the radial profile is scaled by the factorY,m2 (ϑ, ϕ)to give the actual radial profile of the probability density in that direction. Furthermore, note that the probability density for finding the electron-proton pair with separation betweenrandr+dr is
π 0
dϑ 2π
0
dϕ r2sinϑ ψn,,m(r)2=r2ψn,2 (r). (7.161)
7.8 Bound Energy Eigenstates of the Hydrogen Atom 159
Fig. 7.6 The functionsa3ψ3,22 (r)(upper panel) andar2ψ3,22 (r)(lower panel)
The function ψn,2 (r) is proportional to the radial probability density in fixed directions, whiler2ψn,2 (r)samples the full spherical shell betweenrandr+dr in all directions, and therefore the latter probability density is scaled by the geometric size factorr2for thin spherical shells.
Radial expectation values rνn,=
∞
0
dr rν+2ψn,2 (r) (7.162)
have been calculated recursively using the Kramers-Pasternak relation9 ν+1
n2 rνn,−(2ν+1)arν−1n,+ν
4[(2+1)2−ν2]a2rν−2n,=0, (7.163) by starting withν = 0, which yieldsr−1n, = (n2a)−1. Using thenν =1 and ν=2 yields the results
rn,= 3n2−(+1)
2 a, r2n, =n2
2 [5n2+1−3(+1)]a2. (7.164) Equation (7.163) holds forν >−1−2, i.e. the lowest integerνthat can appear in (7.163) isν=2and the lowest exponent that can be reached through the third term isνmin= −2−2. However, Eq. (7.163) cannot be used to calculate the expectation value forν= −2,
r−2n,= 2
n3(2+1)a2. (7.165)
This can be calculated e.g. by a method which Waller introduced to calculate the expectation values (7.162), see Problem7.11.
The uncertainty in distance between the proton and the electron (r)n,=
r2n,− r2n,= a 2
n2(n2+2)−2(+1)2 (7.166) is relatively large for most states in the sense that(r/r)n,is not small, except for largenstates with large angular momentum. For example, we have(r/r)n,0 = 1+(2/n2)/3 >1/3 but(r/r)n,n−1 =1/√
2n+1. However, even for large nand, the particle could still have magnetic quantum number m = 0, whence its probability density would be uniformly spread over directions (ϑ, ϕ). This means that a hydrogen atom with sharp energy generically cannot be considered as consisting of a well localized electron near a well localized proton. This is just another illustration of the fact that simple particle pictures make no sense at the quantum level.
We also note from (7.152) or (7.153) that the bound eigenstatesψn,,m(r) = ψn,(r)Y,m(ϑ, ϕ)have a typical linear scale
na=n4π 0h¯2
μZe2 ∝nμ−1
Ze2. (7.167)
9See e.g. Drake and Swainson [44], and references there. See Problem7.10for the derivation of the Kramers-Pasternak relation.
7.8 Bound Energy Eigenstates of the Hydrogen Atom 161 Here we have generalized the definition of the Bohr radiusato the case of an elec- tron in the field of a nucleus of chargeZe. Equation (7.167) is another example of the competition between the kinetic termp2/2μdriving wave packets apart, and an attractive potential, hereV (r)= −Ze2/4π 0r, trying to collapse the wave function into a point. Metaphorically speaking, pressure from kinetic terms stabilizes the wave function. For given ratio of force constantZe2and kinetic parameterμ−1the attractive potential cannot compress the wave packet to sizes smaller thana, and therefore there is no way for the system to release any more energy. Superficially, there seems to exist a classical analog to the quantum mechanical competition between kinetic energy and attractive potentials in the Schrửdinger equation. In classical mechanics, competition between centrifugal terms and attractive potentials can yield stable bound systems. However, the classical analogy is incomplete in a crucial point. The centrifugal term for =0 is also there in Eq. (7.140) exactly as in the classical Coulomb or Kepler problems. However, what stabilizes the wave function against core collapse in the crucial lowest energy case with = 0 is the radial kinetic term, whereas in the classical case bound Coulomb or Kepler systems with vanishing angular momentum always collapse. To understand the quantum mechanical stabilization of atoms against collapse a little better, let us repeat Eq. (7.140) for=0 and nuclear chargeZe, and for low values ofr, where we can assumeψ(r)=0:
¯ h2 2μ
1 ψ(r)
d2
dr2rψ(r)= −Er− Ze2 4π 0
. (7.168)
The radial probability amplituderψ(r) must satisfyψ−1(r)d2(rψ(r))/dr2 < 0 near the origin, to bend the function around to eventually yield limr→∞rψ(r)=0, which is necessary for normalizability ofr2ψ2(r)on the semiaxisr >0. However, nearr = 0, the only term that bends the wave function in the right direction for normalizability is essentially the ratioZe2/μ−1,
1 ψ(r)
d2
dr2rψ(r) − Ze2μ
2π 0h¯2. (7.169)
If we want to concentrate as much as possible of the radial probability density r2ψ2(r)near the originr0, we have to start out with a large slope and then bend around the wave functionrψ(r)already close to r = 0 to reach (and maintain) small valuesar2ψ2(r)%1 very early. However, the only parameter that bends the wave function near the originr 0 is the ratio between attractive force constant and kinetic parameter,Ze2/μ−1. This limits the minimal spatial extension of the wave function and therefore prevents the classically inevitable core collapse in the bound Coulomb system with vanishing angular momentumM =0. In a nutshell, there is only so much squeezing of the wave function thatZe2/μ−1can do. See also Problem7.18for squeezing or stretching of a hydrogen atom near its ground state.
Fig. 7.7 The functions√arψ1,0(r)(blue),√arψ2,0(r)(orange), and√arψ2,1(r)(green)
The radial probability amplitudesrψ1,0(r),rψ2,0(r), andrψ2,1(r)are plotted in Fig.7.7.