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Angular Momentum Operator in Spherical Coordinates 📂Quantum Mechanics

Angular Momentum Operator in Spherical Coordinates

Formula

The angular momentum operator is expressed in spherical coordinates as follows.

Lx=iℏ(sin⁡ϕ∂∂θ+cos⁡ϕcot⁡θ∂∂ϕ)Ly=−iℏ(cos⁡ϕ∂∂θ−sin⁡ϕcot⁡θ∂∂ϕ)Lz=−iℏ∂∂ϕ \begin{align*} L_{x} &= \i\hbar \left(\sin\phi\dfrac{\partial }{\partial \theta} + \cos\phi \cot\theta \dfrac{\partial }{\partial \phi}\right) \\ L_{y} &= -\i\hbar \left( \cos\phi \dfrac{\partial }{\partial \theta} - \sin\phi \cot\theta \dfrac{\partial }{\partial \phi}\right) \\ L_{z} &= -\i\hbar \dfrac{\partial }{\partial \phi} \end{align*}

Derivation

The definition of the angular momentum operator is as follows.

L=r×P=−iℏ(r×∇) L = \mathbf{r} \times P = - \i\hbar (\mathbf{r} \times \nabla)

At this time, r=(X,Y,Z)\mathbf{r} = (X, Y, Z) is the position operator, PP is the momentum operator, ∇\nabla is the del operator. The del operator in spherical coordinates is as follows.

∇=r^∂∂r+θ^1r∂∂θ+ϕ^1rsin⁡θ∂∂ϕ \nabla = \widehat{\mathbf{r}}\dfrac{\partial}{\partial r} + \widehat{\boldsymbol{\theta}}\dfrac{1}{r}\dfrac{\partial}{\partial \theta} + \widehat{\boldsymbol{\phi}}\dfrac{1}{r\sin\theta}\dfrac{\partial}{\partial \phi}

Since r=rr^\mathbf{r} = r \widehat{\mathbf{r}}, LL is as follows.

L=−iℏ(rr^×∇)=−iℏ(r^×r^r∂∂r+r^×θ^∂∂θ+r^×ϕ^1sin⁡θ∂∂ϕ)=−iℏ(ϕ^∂∂θ−θ^1sin⁡θ∂∂ϕ) \begin{align*} L &= -\i\hbar (r \widehat{\mathbf{r}} \times \nabla) \\ &= -\i\hbar \left( \widehat{\mathbf{r}} \times \widehat{\mathbf{r}} r \dfrac{\partial}{\partial r} + \widehat{\mathbf{r}} \times \widehat{\boldsymbol{\theta}}\dfrac{\partial}{\partial \theta} + \widehat{\mathbf{r}} \times \widehat{\boldsymbol{\phi}} \dfrac{1}{\sin\theta}\dfrac{\partial}{\partial \phi} \right) \\ &= -\i\hbar \left( \widehat{\boldsymbol{\phi}} \dfrac{\partial}{\partial \theta} - \widehat{\boldsymbol{\theta}} \dfrac{1}{\sin\theta}\dfrac{\partial}{\partial \phi} \right) \\ \end{align*}

At this time, the two unit vectors can be represented as follows in Cartesian coordinates.

θ^=cos⁡ϕcos⁡θx^+sin⁡ϕcos⁡θy^−sin⁡θz^ϕ^=−sin⁡ϕx^+cos⁡ϕy^ \begin{align*} \widehat{\boldsymbol{\theta}} &= \cos\phi \cos\theta \widehat{\mathbf{x}} + \sin\phi \cos\theta \widehat{\mathbf{y}} - \sin\theta\widehat{\mathbf{z}} \\ \widehat{\boldsymbol{\phi}} &= -\sin\phi \widehat{\mathbf{x}} + \cos\phi \widehat{\mathbf{y}} \end{align*}

Therefore, LL is as follows.

L=−iℏ[(−sin⁡ϕx^+cos⁡ϕy^)∂∂θ−(cos⁡ϕcos⁡θx^+sin⁡ϕcos⁡θy^−sin⁡θz^)1sin⁡θ∂∂ϕ]=−iℏ[x^(−sin⁡ϕ∂∂θ−cos⁡ϕcot⁡θ∂∂ϕ)+y^(cos⁡ϕ∂∂θ−sin⁡ϕcot⁡θ∂∂ϕ)+z^(∂∂ϕ)] \begin{align*} L &= -\i\hbar \left[ \left( -\sin\phi \widehat{\mathbf{x}} + \cos\phi \widehat{\mathbf{y}} \right) \dfrac{\partial}{\partial \theta} - \left( \cos\phi \cos\theta \widehat{\mathbf{x}} + \sin\phi \cos\theta \widehat{\mathbf{y}} - \sin\theta\widehat{\mathbf{z}} \right) \dfrac{1}{\sin\theta}\dfrac{\partial}{\partial \phi} \right] \\ &= -\i\hbar \left[ \widehat{\mathbf{x}}\left(-\sin\phi\dfrac{\partial }{\partial \theta} - \cos\phi \cot\theta \dfrac{\partial }{\partial \phi}\right) + \widehat{\mathbf{y}}\left( \cos\phi \dfrac{\partial }{\partial \theta} - \sin\phi \cot\theta \dfrac{\partial }{\partial \phi}\right) + \widehat{\mathbf{z}}\left( \dfrac{\partial }{\partial \phi} \right) \right]\\ \end{align*}

Therefore, each component is as follows.

Lx=iℏ(sin⁡ϕ∂∂θ+cos⁡ϕcot⁡θ∂∂ϕ)Ly=−iℏ(cos⁡ϕ∂∂θ−sin⁡ϕcot⁡θ∂∂ϕ)Lz=−iℏ∂∂ϕ \begin{align*} L_{x} &= \i\hbar \left(\sin\phi\dfrac{\partial }{\partial \theta} + \cos\phi \cot\theta \dfrac{\partial }{\partial \phi}\right) \\ L_{y} &= -\i\hbar \left( \cos\phi \dfrac{\partial }{\partial \theta} - \sin\phi \cot\theta \dfrac{\partial }{\partial \phi}\right) \\ L_{z} &= -\i\hbar \dfrac{\partial }{\partial \phi} \end{align*}

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