The_Cambridge_Handbook_of_Physics_Formulas

96 Quantum physics
Hydrogenlike atoms – Schrödinger solution a
Schrödinger equation
− ¯h2
2µ ∇2 Ψ nlm − Ze2
4πɛ 0 r Ψ nlm = E n Ψ nlm with µ = m em nuc
m e +m nuc
(4.79)
Eigenfunctions
Ψ nlm (r,θ,φ)=
with
a = m e
µ
[ (n−l −1)!
2n(n+l)!
a 0
Z ,
] 1/2 ( ) 3/2 2
x l e −x/2 L 2l+1
n−l−1
an
(x)Y l m (θ,φ) (4.80)
2r
x= , and L2l+1
an
∑
n−l−1 (x)= n−l−1
k=0
(l +n)!(−x) k
(2l +1+k)!(n−l −1−k)!k!
Total energy E n = − µe4 Z 2
8ɛ 2 0 h2 n 2 (4.81)
E n
ɛ 0
total energy
permittivity of free space
Radial
expectation
values
〈r〉 = a 2 [3n2 −l(l +1)] (4.82)
〈r 2 〉 = a2 n 2
2 [5n2 +1−3l(l +1)] (4.83)
〈1/r〉 = 1
an 2 (4.84)
〈1/r 2 2
〉 =
(2l +1)n 3 a 2 (4.85)
h Planck constant
m e massofelectron
¯h h/2π
µ reduced mass (≃ m e )
m nuc mass of nucleus
Ψ nlm eigenfunctions
Ze charge of nucleus
−e electronic charge
selection rules b ∆l = ±1 (4.90)
n =1,2,3,... (4.86)
l =0,1,2,...,(n−1) (4.87)
Allowed
quantum m =0,±1,±2,...,±l (4.88)
numbers and ∆n ≠ 0 (4.89)
∆m =0 or ±1 (4.91)
L q p associated Laguerre
polynomials c
a classical orbit radius, n =1
r electron–nucleus separation
Yl
m spherical harmonics
a 0 Bohr radius = ɛ 0h 2
πm ee 2
Ψ 100 = a−3/2
π 1/2 e−r/a Ψ 200 =
(2− a−3/2 r )
e −r/2a
4(2π) 1/2 a
Ψ 210 =
Ψ 300 =
a−3/2 r
4(2π) 1/2 a e−r/2a cosθ
Ψ 21±1 = ∓ a−3/2
8π 1/2 r
a e−r/2a sinθe ±iφ
(
a−3/2
27−18 r )
81(3π) 1/2 a +2r2 a 2 e −r/3a Ψ 310 = 21/2 a −3/2
Ψ 31±1 = ∓ a−3/2
81π 1/2 (6− r a
) r
a e−r/3a cosθ
(6− r 81π 1/2 a
) r
a e−r/3a sinθe ±iφ Ψ 320 = a−3/2 r 2
81(6π) 1/2 a 2 e−r/3a (3cos 2 θ −1)
Ψ 32±1 = ∓ a−3/2
81π 1/2 r 2
a 2 e−r/3a sinθcosθe ±iφ Ψ 32±2 = a−3/2
162π 1/2 r 2
a 2 e−r/3a sin 2 θe ±2iφ
a For a single bound electron in a perfect nuclear Coulomb potential (nonrelativistic and spin-free).
b For dipole transitions between orbitals.
c The sign and indexing definitions for this function vary. This form is appropriate to Equation (4.80).