Conversion of Kinetic Energy operator from Cartesian to Spherical

Conversion of Kinetic Energy operator from Cartesian to spherical polar Co-ordinates
x = r sin θ cos φ
y = r sin θ sin φ
z = r cosθ
(
r= x +y +z
2
2
cosθ =
tan φ =
)
2 12
z
( x2 + y2 + z 2 )
12
y
x
∂r
∂x
= rx = sin θ cos φ
∂r
∂y
= ry = sin θ sin φ
∂r
∂z
= rz = cosθ
sec φ dφ =
dy
x
2
dφ
dx
=−
dφ
dz
=0
y cos 2 φ
x2
− sin θ dθ =
−
ydx
x2
⇒
dφ
dy
=
cos 2 φ
x
=
cos φ
r sin θ
φ
= − rsin
sin θ
dz
r
−
z 2 dz
r3
⇒
dθ
dz
dθ
dy
=
yz
r 3 sin θ
=
dθ
dx
=
zx
3
r sin θ
= cosθrcos φ
− zxd3 x −
r
θ
r sin θ
cos θ sin φ
1 + r cos
= − r sin
3
θ
2
r
2
yzdy
r3
1 + cos θ = − sin θ
= − r sin
r sin θ
r
θ
2
∂
∂x
φ
= sin θ cos φ ∂∂r + cosθrcosφ ∂∂θ − rsin
sin θ
∂
∂y
φ
= sin θ sin φ ∂∂r + cosθrsin φ ∂∂θ + rcos
sin θ
∂
∂z
= cosθ ∂∂r − sinr θ
∂2
∂x 2
φ
= sin θ cos φ ∂∂r + cosθrcos φ ∂∂θ − rsin
sin θ
(
= sin 2 θ cos 2 φ
+ cos
2
− sin θ cos2θ cos
∂2
∂r 2
θ cos 2 φ ∂
cos θ sin θ cos
+
∂r
r
r
2
− sin θ cos2θ cos
φ ∂
∂θ
2
r
∂2
∂φ∂r
2
φ ∂2
∂θ∂r
+ cos2θ sin
2
r
φ ∂2
∂r ∂θ
r
= sin θ sin φ
− sin θ cos2θ sin
2
+ cosθ sinrθ sin
2
r
2
+ cosr
∂2
∂r 2
φ ∂
∂θ
−
cos θ sin θ sin 2 φ ∂
∂θ
r2
+ sin θ cosrθ sin
φ ∂2
∂θ∂r
φ ∂
+ cos
2
2
φ ∂2
∂r ∂θ
2
= ( cosθ ∂∂r − sinr θ
= cos θ
sin 2 θ ∂
+ r ∂r
∂2
∂r 2
−
+
∂
∂θ
r sin θ
cos 2 θ sin 2 φ ∂ 2
∂θ 2
r2
θ sin φ cos φ ∂
∂φ
r 2 sin 2 θ
−
∂
∂r
+ cosθrsin φ
cos φ sin φ ∂
r 2 sin 2 θ ∂φ
+ cosθ 2sin φ cosφ
+ cosθ 2sin φ cosφ
r sin θ
+
−
r sin θ
∂2
∂φ∂θ
cosθ sin θ ∂ 2
∂r ∂θ
r
sin 2 θ ∂ 2
r 2 ∂θ 2
φ
− rsin
sin θ
∂
∂φ
∂2
∂θ∂φ
∂2
∂r ∂φ
)( cosθ ∂∂r − sinr θ ∂∂θ )
cos θ sin θ ∂
∂θ
r2
cos θ sin θ ∂ 2
∂θ∂r
r
− cosθ 2cosφ sin φ
− cosφrsin φ
)(sin θ sin φ
2
∂
∂θ
sin 2 φ ∂ 2
r 2 sin 2 θ ∂φ 2
+
∂
∂θ
+ cosθ 2sin θ
r
∂
∂θ
φ
+ rcos
sin θ
cos 2 φ ∂ 2
r 2 sin 2 θ ∂φ 2
r
φ ∂
r sin θ ∂θ
2
+
2
− cos
∂r
r
∂
r sin θ ∂φ
φ ∂
sin θ cos φ ∂
− sin θ cos
+
2
r
∂φ
∂r ∂φ
θ sin 2 φ ∂
+ sin φrcosφ ∂∂φ∂r + cos2θ cos
∂r
2
+
∂
∂φ
∂
∂φ
+ cosθrcosφ
∂2
∂φ∂θ
r sin θ
+ cosφ 2sin φ
∂
∂r
φ
+ sin2 φ cos
2
cos θ ∂
sin 2 θ ∂φ
− cosθ 2sin φ cosφ
2
(
2
θ cos 2 φ ∂ 2
∂θ 2
r2
+ cosθ sin2φ cos φ
φ ∂
r sin θ ∂θ
+ sin θ cosrθ cos
2
∂
∂φ
)( sinθ cosφ
∂
∂φ
+ cos
φ
= sin θ sin φ ∂∂r + cosθrsin φ ∂∂θ + rcos
sin θ
2
∂2
∂z 2
φ ∂
∂θ
2
r
+ sinr φ ∂∂r − sin φrcosφ
∂2
∂y 2
∂
∂θ
∂
∂φ
∂2
∂θ∂φ
∂
∂φ
)
)
∇2 ≡
∂2
∂x 2
+
∂2
∂y 2
+
∂2
∂z 2
=
≡
∂2
∂r 2
+ 2r ∂∂r +
1
r2
⎡ 1
⎢⎣ sin θ
≡
∂2
∂r 2
+ 2r ∂∂r +
1
r2
Λ2
∂2
∂r 2
∂
∂θ
+ 2r ∂∂r +
cos θ ∂
r 2 sin θ ∂θ
( sin θ ∂∂θ ) +
+
1 ∂2
r 2 ∂θ 2
∂2 ⎤
1
sin 2 θ ∂φ 2 ⎥
⎦
+
∂2
1
r 2 sin 2 θ ∂φ 2