2.1 The Schr ̈odinger equation 23where the operatorl^2 contains the terms that depend onθandφ,namely
l^2 =−{
1
sinθ∂
∂θ(
sinθ∂
∂θ)
+
1
sin^2 θ∂^2
∂φ^2}
, (2.3)
and^2 l^2 is the operator for the orbital angular momentum squared. Fol-
lowing the usual procedure for solving partial differential equations, we
look for a solution in the form of a product of functionsψ=R(r)Y(θ, φ).
The equation separates into radial and angular parts as follows:
1
R∂
∂r(
r^2∂R
∂r)
−
2 mer^2
^2{V(r)−E}=1
Y
l^2 Y. (2.4)Each side depends on different variables and so the equation is only
satisfied if both sides equal a constant that we callb.Thus
l^2 Y=bY. (2.5)This is an eigenvalue equation and we shall use the quantum theory of
angular momentum operators to determine the eigenfunctionsY(θ, φ).
2.1.1 Solution of the angular equation
To continue the separation of variables we substituteY=Θ(θ)Φ(φ)into
eqn 2.5 to obtain
sinθ
Θ∂
∂θ(
sinθ∂Θ
∂θ)
+bsin^2 θ=−1
Φ
∂^2 Φ
∂φ^2=const. (2.6)The equation for Φ(φ) is the same as in simple harmonic motion, so^33 A andB are arbitrary constants.
Alternatively, the solutions can be
written in terms of real functions as
A′sin(mφ)+B′cos(mφ).
Φ=Aeimφ+Be−imφ. (2.7)
The constant on the right-hand side of eqn 2.6 has the valuem^2 .Phys-
ically realistic wavefunctions have a unique value at each point and this
imposes the condition Φ(φ+2π)=Φ(φ), sommust be an integer.
The function Φ(φ) is the sum of eigenfunctions of the operator for the
z-component of orbital angular momentum
lz=−i∂
∂φ. (2.8)
The function eimφhas magnetic quantum numbermand its complex
conjugate e−imφhas magnetic quantum number−m.^4
(^4) The operator−∂ (^2) /∂φ (^2) ≡l (^2) zand con-
sequently Φ(φ) is an eigenfunction ofl^2 z
with eigenvaluem^2.
A convenient way to find the functionY(θ, φ) and its eigenvaluebin
eqn 2.5^5 is to use theladder operatorsl+=lx+ilyandl−=lx−ily.
(^5) The solution of equations involving
the angular part of∇^2 arises in many
situations with spherical symmetry, e.g.
in electrostatics, and the same mathe-
matical tools could be used here to de-
termine the properties of the spherical
harmonic functions, but angular mo-
mentum methods give more physical in-
sight for atoms.
These operators commute withl^2 , the operator for the total angular
momentum squared (becauselxandlycommute withl^2 ); therefore, the
three functionsY, l+Y andl−Y are all eigenfunctions ofl^2 with the
same eigenvalueb(if they are non-zero, as discussed below). The ladder