University of Connecticut
DigitalCommons@UConn
Chemistry Education Materials
Department of Chemistry
2-6-2007
The Laplacian in Spherical Polar Coordinates
Carl W. David
University of Connecticut, [email protected]
Recommended Citation
David, Carl W., "The Laplacian in Spherical Polar Coordinates" (2007). Chemistry Education Materials. Paper 34.
http://digitalcommons.uconn.edu/chem_educ/34
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The Laplacian in Spherical Polar Coördinates
C. W. David
Department of Chemistry
University of Connecticut
Storrs, Connecticut 06269-3060
(Dated: February 6, 2007)
I.
and
SYNOPSIS
In treating the Hydrogen Atom’s electron quantum mechanically, we normally convert the Hamiltonian from its
Cartesian to its Spherical Polar form, since the problem is
variable separable in the latter’s coördinate system. This
reading treats the brute-force method of effecting the
transformation of the kinetic energy operator, normally
called the Laplacian, from one to the other coördinate
systems.
z = r cos θ
and their inverses
r=
θ = cos−1
II.
PRELIMINARY DEFINITIONS
z
z
= cos−1 p
2
r
x + y2 + z2
φ = tan−1
x = r sin θ cos φ
y = r sin θ sin φ
∂
=
∂y
∂
=
∂z
x2 + y 2 + z 2
and
We start with the primitive definitions
∂
=
∂x
p
y
x
and attempt to write (using the chain rule)
∂r
∂x
∂r
∂y
∂r
∂z
y,z
∂
∂r
∂
∂r
∂
∂r
+
θ,φ
∂θ
∂x
∂θ
∂y
∂θ
∂z
y,z
∂
∂θ
∂
∂θ
∂
∂θ
+
r,φ
∂φ
∂x
∂φ
∂y
∂φ
∂z
∂
∂φ
∂
∂φ
x,y
∂
∂φ
y,z
r,θ
and
x,z
+
θ,φ
x,z
+
r,φ
x,z
r,θ
and
III.
x,y
+
θ,φ
PRELIMINARY PARTIAL DERIVATIVES
The needed (above) partial derivatives are:
∂r
= sin θ cos φ
∂x y,z
∂r
∂y
Typeset by REVTEX
x,y
+
r,φ
∂r
∂z
r,θ
= cos θ
(3.3)
x,y
(3.1)
= sin θ sin φ
x,z
(3.2)
and we have as a starting point for doing the θ terms,
d cos θ = − sin θdθ =
1
dz
1
dz
dr
+z·d
=
−z·
r
r
r
r2
dz
z 1
=
− 2 (xdx + ydy + zdz)
r
r r
or
so that, for example (when dy = dz = 0) we have
− sin θdθ = −
(3.4)
z x
dx
r2 r
1
cos2 φ
dφ =
dy
y
− 2 dx
x
x
which is
r2 1 − cos2 ϑ
r cos θ
− sin θdθ = − 2 sin θ cos φdx =
dz
r
r3
so, after multiplying across by cos2 φ leads to (at constant
x)
so that
∂θ
∂x
∂θ
∂y
cos θ cos φ
r
=
y,z
(3.5)
x,z
cos θ sin φ
=
r
(3.6)
but, for the z-equation, we have
− sin θdθ =
=
x,z
cos φ
r sin θ
(3.8)
and (at constant y)
∂φ
∂y
∂φ
∂x
dz
z 1
− 2 zdz
r
r r
=−
y,z
∂φ
∂z
sin φ
r sin θ
(3.9)
=0
(3.10)
x,y
which is
− sin θdθ =
1 z2
− 3
r
r
2
IV.
r2 − z 2
dz
r3
dz =
THE FIRST PARTIAL DERIVATIVE
TERMS
Given these results (above) we write
− sin θdθ =
1 z
− 3
r
r
2
dz =
2
r sin θ
dz
r3
∂
∂
= cos θ
−
∂z
∂r
sin θ
r
∂
∂θ
(4.1)
so one has
∂θ
∂z
=−
x,y
sin θ
r
and
(3.7)
Next, we have (as an example)
tan φ =
sin φ
y
= tan−1
cos φ
x
and taking the partial derivatives on both sides, we obtain
1
sin φ
d(sin φ) +
d(cos φ)
cos ϑ
− cos2 φ
so
sin2 φ
1+
cos2 φ
dφ =
dy
y
− 2 dx
x
x
∂
∂
= (sin θ sin φ)
+
∂y
∂r
cos θ sin φ
r
∂
+
∂θ
cos φ
r sin θ
∂
∂φ
(4.2)
and
∂
∂
cos θ cos φ ∂
sin φ
∂
= (sin θ cos φ) +
+ −
∂x
∂r
r
∂θ
r sin θ ∂φ
(4.3)
V.
GATHERING TERMS TO FORM THE
LAPLACIAN
From Equation 4.1 we form
∂
∂ cos θ ∂r
−
∂2
=
cos
θ
2
∂z
∂r
sin θ
r
∂
∂θ
−
sin θ
r
∂
∂ cos θ ∂r
−
∂θ
sin θ
r
∂
∂θ
(5.1)
while from Equation 4.2 we obtain
h
i
φ
∂
∂
∂
∂ sin θ sin φ ∂r
+ cos θrsin φ ∂θ
+ rcos
∂r
sin θ ∂φ
2
∂
= (sin θ sin φ)
∂y 2
∂r
h
i
cos
θ sin φ
cos φ
∂
∂
∂
r
∂θ + r sin θ ∂φ
cos θ sin φ ∂ sin θ sin φ ∂r +
+
r
∂θ
h
i
∂ sin θ sin φ ∂ + cos θ sin φ ∂ + cos φ ∂
∂r
r
∂θ
r sin θ ∂φ
cos φ
+
r sin θ
∂φ
(5.2)
and from Equation 4.3 we obtain
h
i
φ
∂
∂
∂
+ cos θrcos φ ∂θ
− rsin
∂ sin θ cos φ ∂r
sin θ ∂φ
2
∂
= (sin θ cos φ)
∂x2
h
∂r i
∂ sin θ cos φ ∂ + cos θ cos φ ∂ − sin φ ∂
∂r
r
∂θ
r sin θ ∂φ
cos θ cos φ
+
r
∂θ
h
i
∂ sin θ cos φ ∂ + cos θ cos φ ∂ − sin φ ∂
∂r
r
∂θ
r sin θ ∂φ
sin φ
−
r sin θ
∂φ
(5.3)
Expanding, we have
∂2
cos θ sin θ ∂
sin θ cos θ ∂ 2
∂2
= cos2 θ 2 +
−
2
2
∂z
∂r
r
∂θ
r
∂r∂θ
2 2
sin θ
∂
∂
sin θ cos θ ∂
sin θ
∂
−
− sin θ
− cos θ
−
+
r
∂r
∂θ
r2
∂θ
r
∂θ2
(5.4)
while for the y-equation we have
∂2
∂2
= sin2 θ sin2 φ 2
2
∂y
∂r
2 cos θ sin φ
∂
cos θ sin φ ∂
+ sin θ sin φ +
+
r2
∂θ
r
∂r∂θ
2 cos φ
∂
cos φ
∂
+ sin θ sin φ − 2
+
r sin θ ∂φ
r sin θ ∂r∂φ
cos θ sin φ
∂
∂2
+
cos θ sin φ
+ sin θ sin φ
r
∂r
∂r∂θ
2 cos θ sin φ
sin θ sin φ ∂
cos θ sin φ ∂
+
−
+
r
r
∂θ
r
∂θ2
2 cos θ sin φ
cos φ cos θ ∂
cos φ
∂
+
−
+
r
∂φ
r sin θ ∂φ∂θ
r sin2 θ
cos φ
∂
∂2
+
sin θ cos φ
+ sin θ sin φ
r sin θ
∂r
∂r∂φ
2 cos φ
cos θ cos φ ∂
cos θ sin φ
∂
+
+
+
r sin θ
r
∂θ
r
∂θ∂φ
2 cos φ
sin φ cos φ ∂
cos φ
∂
+
−
+
r sin θ
r sin θ
∂φ
r sin θ ∂φ2
(5.5)
(5.6)
(5.7)
(5.8)
(5.9)
(5.10)
(5.11)
(5.12)
(5.13)
and finally
∂2
∂2
=
(sin
θ
cos
φ)
sin
θ
cos
φ
2
∂x2
2∂r
cos θ cos φ ∂
cos θ cos φ
∂
+ (sin θ cos φ) −
+
r2
∂θ
r
∂θ∂r
2 sin φ
∂
sin φ
∂
− (sin θ cos φ) −
+
r2 sin θ ∂φ
r sin θ ∂φ∂r
cos θ cos φ
∂
∂2
+
cos θ cos φ
+ sin θ cos φ
r
∂r
∂r∂θ
2 cos θ cos φ
sin θ cos φ ∂
cos θ cos φ ∂
+
−
+
r
r
∂θ
r
∂θ2
2 cos θ cos φ
sin φ
∂
sin φ
∂
+
−
+
r
r sin θ ∂φ∂θ
r sin2 θ ∂φ
sin φ
∂
∂2
−
sin θ sin φ
+ sin θ cos φ
r sin θ
∂r
∂r∂φ
2 sin φ
cos θ sin φ ∂
cos θ cos φ
∂
−
+
−
r sin θ
r
∂θ
r
∂θ∂φ
2 cos φ
∂
sin φ
∂
sin φ
−
−
−
r sin θ
r sin θ ∂φ
r sin θ ∂φ2
Now, one by one, we expand completely each of these
three terms. We have
∂2
∂2
2
=
cos
θ
∂z 2
∂r2
cos θ sin θ ∂
+
r2
∂θ
sin θ cos θ ∂ 2
−
r
∂r∂θ
2 sin θ ∂
+
r
∂r
2
sin θ cos θ
∂
−
r
∂r∂θ
sin θ cos θ ∂
+
r2
∂θ
2 2
sin θ ∂
+
r2
∂θ2
(5.22)
(5.23)
(5.24)
(5.25)
(5.26)
(5.27)
(5.28)
and, for the y-equation:
∂2
∂2
= sin2 θ sin2 φ 2
2
∂y
∂r
2 sin θ cos θ sin φ ∂
(5.6) → +
r2
∂θ
2 cos θ sin θ sin φ
∂2
+
r
∂r∂θ
sin φ cos φ ∂
(5.7) → −
r2
∂φ
2
cos φ sin φ
∂
+
r
∂r∂φ
(5.29)
(5.30)
(5.31)
(5.32)
(5.33)
(5.14)
(5.15)
(5.16)
(5.17)
(5.18)
(5.19)
(5.20)
(5.21)
cos2 θ sin2 φ ∂
(5.8) → +
r
∂r
2 cos θ sin θ sin φ
∂2
+
r
∂r∂θ
sin θ cos θ sin2 φ ∂
−
r2
∂θ
2
2 cos θ sin φ ∂ 2
(5.9) → +
r2
∂θ2
2
cos θ cos φ sin φ ∂
−
∂φ
r sin2 θ
2
∂
cos θ cos φ sin φ
+
r2 sin θ
∂φ∂θ
2 cos φ ∂
(5.10) → +
r
∂r
2
cos φ sin φ
∂
+
r
∂r∂φ
2
cos φ cos θ ∂
+
r2 sin θ
∂θ
2
cos θ cos φ sin φ
∂
(5.12) → +
r2 sin θ
∂θ∂φ
2
cos φ sin φ ∂
(5.13) → −
∂φ
r sin2 θ
2
cos2 φ
∂
+
r2 sin2 θ ∂φ2
(5.34)
(5.35)
(5.36)
(5.37)
(5.38)
(5.39)
(5.40)
(5.41)
(5.42)
(5.43)
(5.44)
(5.45)
and finally, for the x-equation, we have
2
2
∂
∂
= sin2 θ cos2 φ 2
2
∂x
∂r
sin θ cos θ cos2 φ ∂
(5.14) → −
r2
∂θ
2
2
sin θ cos θ cos φ
∂
(5.14) → +
r
∂θ∂r
cos φ sin φ ∂
r2
∂φ
2
sin φ cos φ
∂
−
r
∂φ∂r
2
cos θ cos2 φ ∂
(5.15) → +
r
∂r
2
2
sin θ cos θ cos φ
∂
+
r
∂r∂θ
sin θ cos θ cos2 φ ∂
(5.15) → −
r2
∂θ
2
2
2
cos θ cos φ ∂
+
r2
∂θ2
cos θ cos φ
cos φ sin φ ∂
(5.16) → +
r
r sin θ
∂φ
2
∂
sin φ cos φ cos θ
−
2
r sin θ
∂φ∂θ
∂
∂θ
+
(5.46)
(5.47)
(5.48)
(5.49)
(5.50)
2 sin φ ∂
(5.17) → −
r
∂r
2
sin φ cos φ
∂
−
r
∂r∂φ
cos θ sin2 φ ∂
(5.19) → +
r2 sin θ
∂θ
2
cos θ sin φ cos φ
∂
−
r2 sin θ
∂θ∂φ
sin φ cos φ ∂
(5.20) → +
∂φ
r sin2 θ
2
2
∂
sin φ
+
2
2
r sin θ ∂φ2
(5.57)
(5.58)
(5.59)
(5.60)
(5.61)
(5.62)
(5.51)
(5.52)
Gathering terms as coefficients of partial derivatives,
we obtain (from Equations 5.22, 5.29 and 5.46)
(5.53)
(5.54)
∂2
∂2
2
2
2
2
2
cos
θ
+
sin
θ
sin
φ
+
sin
θ
cos
φ
→
∂r2
∂r2
(5.55)
(5.56)
and (from Equations 5.23, 5.26, 5.30, 5.36, 5.42, 5.47,
5.53, and 5.59)
cos θ sin θ sin θ cos θ sin θ cos θ sin2 φ sin θ cos θ sin2 φ cos2 φ cos θ
+
−
−
+
r2
r2
r2
r2
r2 sin θ
2
2
sin θ cos θ cos φ sin θ cos θ cos φ cos θ sin2 φ
−
−
+
r2
r2
r2 sin θ
cos θ ∂
→ 2
r sin θ ∂θ
while we obtain from Equations 5.28, 5.37, and 5.54:
∂ 2 sin2 θ cos2 θ sin2 φ cos2 θ cos2 φ
1 ∂2
+
+
→
∂θ2
r2
r2
r2
r2 ∂θ2
From Equations 5.25, 5.34, 5.40, 5.51, 5.57,
∂
sin2 θ cos2 θ sin2 φ cos2 φ cos2 θ cos2 φ sin2 φ
2 ∂
→
+
+
+
+
−
∂r
r
r
r
r
r
r ∂r
From Equations 5.32, 5.38, 5.44, 5.49, 5.55 and 5.61 we obtain
∂
sin φ cos φ cos2 θ cos φ sin φ cos2 θ cos φ sin φ cos φ sin φ
cos θ cos φ
−
+
−
−
+
∂φ
r2
r2
r
r sin2 θ
r sin2 θ
2
cos θ cos φ sin φ
sin φ cos φ
→0
+
+
r2 sin θ
r sin2 θ
From Equations 5.45 and 5.62 we obtain
2
∂2
cos2 φ
sin2 φ
1
∂
→
+
∂φ2 r2 sin2 θ r2 sin2 θ
r2 sin2 θ ∂φ2
(5.63)
(5.64)
(5.65)
(5.66)
(5.67)
The mixed derivatives yield, first, from Equations 5.33, 5.41, 5.50, and 5.58 leading to
cos φ sin φ cos φ sin φ sin φ cos φ sin φ cos φ
∂2
+
−
−
→0
∂r∂φ
r
r
r
r
From Equations 5.24, 5.27, 5.35, 5.31 5.52, 5.48
sin θ cos θ sin θ cos θ cos θ sin θ sin2 φ
∂2
−
−
+
∂r∂θ
r
r
r
2
2
2
sin θ cos θ cos φ cos θ sin θ sin φ sin θ cos θ cos φ
+
+
+
→0
r
r
r
(5.68)
(5.69)
From Equations 5.39 5.43 5.56 5.60
∂2
cos θ cos φ sin φ cos φ sin φ
+ 2
∂φ∂θ
r2 sin θ
r sin θ
sin φ cos φ cos θ
cos θ sin φ cos φ
−
−
→0
r2 sin θ
r2 sin θ
Gathering together the non-vanishing terms, we obtain
VI.
∂2
2 ∂
1 ∂2
cos θ ∂
1
∂2
+
+ 2 2+ 2
+ 2 2
2
∂r
r ∂r r ∂θ
r sin θ ∂θ r sin θ ∂φ2
MAPLE EQUIVALENT
A.
which is one of the two “classic” forms for ∇2 . The other
is
!
∂
∂
∂ sin θ ∂θ
1 ∂ r2 ∂r
1
∂2
+ 2 2
sin θ
+
r2
∂r
∂θ
∂φ2
r sin θ
ty2:=expand(sin(theta)*sin(phi)*diff(ty,r)+((cos(theta)*sin(phi))/r)
*diff(ty,theta)+(cos(phi)/(r*sin(theta)))*diff(ty,phi));
tz := cos(theta)*diff(f,r)
-(sin(theta)/r)*diff(f,theta);
tz2 := expand(cos(theta)*diff(tz,r)-(sin(theta)/r)*diff(tz,theta));
del := tx2+ty2+tz2:
del := algsubs( cos(theta)^2=1-sin(theta)^2, del ):
del := expand(algsubs( cos(phi)^2=1-sin(phi)^2, del ));
Here is another version of the same thing:
Example 1
Here is a set of Maple instructions adjusted from the
2-dimensional code [1] for our 3-dimensional case, which
will get you the same result:
restart;
f:=g(r,theta,phi);
tx :=
sin(theta)*cos(phi)*diff(f,r)+((cos(theta)*cos(phi))/r)*diff(f,theta)
-(sin(phi)/(r*sin(theta)))*diff(f,phi);
tx2:=expand(
sin(theta)*cos(phi)*diff(tx,r)+((cos(theta)*cos(phi))/r)*diff(tx,theta)
-(sin(phi)/(r*sin(theta)))*diff(tx,phi));
ty :=
sin(theta)*sin(phi)*diff(f,r)+((cos(theta)*sin(phi))/r)*diff(f,theta)
+(cos(phi)/(r*sin(theta)))*diff(f,phi);
B.
(5.70)
Example 2
>
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#CARTESIAN TO SPHERICAL POLAR
restart;
with(plots):
Warning, the name changecoords has been redefined
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uu:=u(sqrt(x^2+y^2+z^2),arccos(z/sqrt(x^2+y^2+z^2)),arctan(y,x));
p
z
uu := u( x2 + y 2 + z 2 , arccos( p
), arctan(y, x))
2
x + y2 + z2
ux:=diff(uu,x):
uy:=diff(uu,y):
uz:=diff(uu,z):
uxx:=diff(ux,x):
uyy:=diff(uy,y):
uzz:=diff(uz,z):
Lapu:=simplify(uxx+uyy+uzz):
assume(r,positive);
Lapu:=simplify(subs(x=r*sin(theta)*cos(phi),
y=r*sin(theta)*sin(phi),
z = r*cos(theta),
arctan(sin(theta)*sin(phi),sin(theta)*cos(phi))=phi,
arccos(cos(theta))=theta,
Lapu),trig):
Lapu := subs(arctan(sin(theta)*sin(phi),sin(theta)*cos(phi))=phi,
arccos(cos(theta))=theta,
Lapu):
Lapu := algsubs(-1+cos(theta)^2=-sin(theta)^2,Lapu):
Lapu:=expand(Lapu);
D2 (u)(r ˜, θ, φ) sin(θ)2 cos(θ) D2, 2 (u)(r ˜, θ, φ) D3, 3 (u)(r ˜, θ, φ)
+
+
r ˜2 (sin(θ)2 )(3/2)
r ˜2
r ˜2 sin(θ)2
2 D1 (u)(r ˜, θ, φ)
+
+ D1, 1 (u)(r ˜, θ, φ)
r˜
Lapu :=
It takes some getting used to Maple notation to see that
this is the expected result.
VII.
are better ways to carry out the transformation from
Cartesian to Spherical Polar (and indeed any orthogonal) coördinate system.
COMMENTS
The reader should be aware that the brute force methods used here are primitive in the extreme, and that there
[1] Mathias Kawski, http://math.la.asu.edu/∼kawski/MAPLE/MAPLE.html