23.1 Separability in Three Dimensions
261
If the potential can be written as
V (r) = V 1 (x) + V 2 (y) + V 3 (z)
(23.4a)
the quantum system becomes separable and with a product ansatz for the wave
function
= X(x) · Y (y) · Z(z)
(23.4b)
we obtain three one-dimensional differential equations
d 2
dx 2 − U x + x
= 0
(23.4c)
d 2
dy 2 − U y + y
= 0
(23.4d)
d 2
dz 2 − U z + − x − y
= 0 ,
(23.4e)
with = 2m/ ¯
hE, E the energy, and U(r) = 2m/ ¯
hV (r) and x , , y separation
constants. For example, the 3d-harmonic oscillator becomes separable in cartesian
coordinates and spherical coordinates.
(2) Spherical Coordinates
x = r sin θ cos φ
0 ≤ r < ∞
(23.5a)
y = r sin θ sin φ
0 ≤ θ < π
(23.5b)
z = r cos θ
− π ≤ φ < π.
(23.5c)
In spherical coordinates the metric tensor g ij is
diag(g ij ) = (1, r
2 , r
2 sin
2 θ)
(23.6a)
and the Laplace–Beltrami operator
Δ rθφ =
1
r 2
∂
∂r
r
2 ∂
∂r
+
1
r 2 sinθ
∂
∂θ
sin θ
∂
∂θ
+
∂ 2
∂φ 2
.
(23.6b)
A quantum system becomes separable, if the potential fulfills
V (r, θ, φ) = V 1 (r) +
V 2 (θ )
r 2 +
V 3 (φ)
r 2 sin
2 θ
(23.7a)