# Angular momentum operator in spherical coordinates

The angular momentum operator in quantum mechanics is a crucial observable that represents the rotational motion of a particle. In spherical coordinates (r,θ,ϕ), the angular momentum operator is expressed as:

x = rsinθcos ϕ

y = rsinθsinϕ

z = rcosθ ....i

Then x² + y² = r² sin²θ ....ii

z² = r² cos²θ ......iii

Adding these two equations, x² + y² + z² = r² ......iv

Dividing ii by iii

tan θ = sq root x² + y² / z² .......v

From i,

y/x = tan ϕ ....vi

Now differentiating iv with respect to x, y and z, we get

2x = 2r ∂r /∂x

∂r /∂x = x/ r = sinθcos ϕ .....vii

2y = 2r ∂r /∂y

∂r /∂y = y/ r = sinθsin ϕ ....viii

∂r /∂z = cos θ......ix

Now differentiating eq v with respect to x, y , z we get

∂𝜃/∂z = - 1/r sin 𝜃 ...xii

Again differentiating equation vi with respect to x, y and z

sec²𝜙 ∂𝜙/∂x = - y/x²

sec²𝜙 ∂𝜙/∂y = 1/x

∂𝜙/∂y = cos²𝜙 /r sin 𝜃 cos𝜙 = cos𝜙 / r sin 𝜃 .....xiv

sec²𝜙 ∂𝜙/∂z = 0

∂𝜙/∂z = 0 .....xv

To find the relation between them

∂/∂x -> ∂/∂r

∂/∂x -> ∂/∂𝜃

∂/∂x -> ∂/∂𝜙

From equation vii, x and xiii we get

∂/∂x = sin 𝜃 cos𝜙 ∂/∂r ....xvi

∂/∂x = (cos 𝜃 cos𝜙 /r )∂/∂𝜃 ....xvii

∂/∂x = - sin 𝜙 / rsin𝜃 ∂/∂𝜙 ....xviii

Similarly from equation viii, xi and xiv we have

∂/∂y = sin 𝜃 sin 𝜙 ∂/∂r ....xix

∂/∂y = cos 𝜃 sin 𝜙 /r ∂/∂𝜃 ....xx

∂/∂y = cos𝜙 /r sin 𝜃 ∂/∂𝜙 ....xxi

∂/∂y = sin 𝜃 sin 𝜙 ∂/∂r + cos 𝜃 sin 𝜙 /r ∂/∂𝜃+ cos𝜙 /r sin 𝜃 ∂/∂𝜙

Similarly, ∂/∂z = cos 𝜃 ∂/∂r - sin 𝜃 /r ∂/∂𝜃 ....xxii

and ∂/∂x = sin 𝜃 cos𝜙 ∂/∂r + cos 𝜃 cos𝜙 /r ∂/∂𝜃 - sin 𝜙 / rsin𝜃 ∂/∂𝜙 ....xxiii

The components of the orbital angular momentum L can be expressed in terms of cartesian coordinate system as

now we should have following conversions

x-> r, 𝜃, 𝜙

y-> r, 𝜃, 𝜙

z-> r, 𝜃, 𝜙

and

∂/∂x -> ∂/∂r, ∂/∂𝜃, ∂/∂𝜙

∂/∂y -> ∂/∂r, ∂/∂𝜃, ∂/∂𝜙

∂/∂z -> ∂/∂r, ∂/∂𝜃, ∂/∂𝜙

Now consider an arbitrary ket vector l 𝜓 >,

Lx l 𝜓 > = ℏ / i ( y ∂/∂z l 𝜓 > - z∂/∂y l 𝜓 > )

Substituting the values of y, ∂/∂z, z and ∂/∂y we get

Again for y component of the orbital angular momentum

Substituting the value of z, ∂/∂x, x and ∂/∂z from above equations, we get

for y component of the orbital angular momentum

Substituting the value of x, ∂/∂y, y and ∂/∂x from above equations, we get

Hence, equation xxiv, xxv and xxvi are the expressions for angular momentum components in spherical polar coordinate.

The significance of the angular momentum operator lies in its role in describing the rotational properties of quantum systems. It is a conserved quantity in systems with rotational symmetry, meaning that the total angular momentum of a closed quantum system remains constant over time. This principle is known as the conservation of angular momentum.

The operator L² corresponds to the magnitude of the angular momentum, and its eigenvalues represent the possible values of the squared angular momentum observable. The eigenstates of L² are simultaneous eigenstates of L, meaning that they have well-defined values of both the magnitude of the angular momentum and one of its components.

This note is a part of the Physics Repository.