Commutation Relations

Multidimensional problems entail the possibility of having rotation as a part of solution. Just like in classical mechanics where we can calculate the angular momentum using vector cross product, we have a very similar form of equation. However, just like any observable in quantum mechanics, this angular momentum is expressed by a Hermitian operator. Similar to classical mechanics we write the angular momentum operator Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf L\!} as:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{L}=\mathbf{r}\times\mathbf{p}}

Working in the spatial representation, we have Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{r}} as our radial vector, while Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{p}} is the momentum operator.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{p}=-i\hbar\nabla}

Using the cross product in Cartesian coordinate system, we get a component of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bold L\!} in each direction:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L_x=yp_z-zp_y=\frac{\hbar}{i}\left(y\frac{\partial}{\partial z}-z\frac{\partial}{\partial y}\right)\!}

Similarly, using cyclic permutation on the coordinates x, y, z, we get the other two components of the angular momentum operator. All of these can be written in a more compact form using the Levi-Civita symbol as (the Einstein summation convention of summing over repeated indices is understood here)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L_{\mu}=\epsilon_{\mu\nu\lambda}r_\nu p_\lambda\!}

with

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon_{\mu\nu\lambda} = \begin{cases} +1, & \mbox{if } (\mu,\nu,\lambda) \mbox{ is } (1,2,3), (3,1,2) \mbox{ or } (2,3,1), \\ -1, & \mbox{if } (\mu,\nu,\lambda) \mbox{ is } (3,2,1), (1,3,2) \mbox{ or } (2,1,3), \\ 0, & \mbox{otherwise: }\mu=\nu \mbox{ or } \nu=\lambda \mbox{ or } \lambda=\mu. \end{cases} }

Or we simply say that the even permutation gives 1, odd permutation gives -1, otherwise, we get 0.

We can immediately verify the following commutation relations:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [L_\mu,r_\nu]=i\hbar\epsilon_{\mu\nu\lambda}r_\lambda}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [L_\mu,p_\nu]=i\hbar\epsilon_{\mu\nu\lambda}p_\lambda}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [L_\mu,L_\nu]=i\hbar\epsilon_{\mu\nu\lambda}L_\lambda} (this relation tells us :Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{L}\times\mathbf{L}=i\hbar\mathbf{L}} )

and

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [\hat{\mathbf{n}}\cdot\mathbf{L},\mathbf{r}]=i\hbar(\mathbf{r}\times\hat{\mathbf{n}})}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [\hat{\mathbf{n}}\cdot\mathbf{L},\mathbf{p}]=i\hbar(\mathbf{p}\times\hat{\mathbf{n}})}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [\hat{\mathbf{n}}\cdot\mathbf{L},\mathbf{L}]=i\hbar(\mathbf{L}\times\hat{\mathbf{n}})}

For example,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \left[L_\mu,r_\nu\right] &= [\epsilon_{\mu\lambda\rho}r_\lambda p_\rho,r_\nu] = \epsilon_{\mu\lambda\rho}[r_\lambda p_\rho,r_\nu] = \epsilon_{\mu\lambda\rho}r_\lambda[ p_\rho,r_\nu] \\ &= \epsilon_{\mu\lambda\rho}r_\lambda\frac{\hbar}{i}\delta_{\rho\nu} = \epsilon_{\mu\lambda\nu}r_\lambda\frac{\hbar}{i} \\ &= i\hbar\epsilon_{\mu\nu\lambda}r_\lambda \end{align} }

Also, note that for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L^2=L_x^2+L_y^2+L_z^2=L_{\mu} L_{\mu}} ,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \left[L_{\mu},L^2\right] &= \left[L_{\mu},L_{\nu} L_{\nu}\right] \\ &= L_{\nu}\left[L_{\mu},L_{\nu}\right]+\left[L_{\mu},L_{\nu}\right]L_{\nu} \\ &= L_{\nu} i\hbar \epsilon_{\mu\nu\lambda} L_{\lambda} + i\hbar \epsilon_{\mu\nu\lambda} L_{\lambda} L_{\nu} \\ &= i\hbar \epsilon_{\mu\nu\lambda} L_{\nu} L_{\lambda} - i\hbar \epsilon_{\mu\lambda\nu} L_{\lambda} L_{\nu} \\ &= i\hbar \epsilon_{\mu\nu\lambda} L_{\nu} L_{\lambda} - i\hbar \epsilon_{\mu\nu\lambda} L_{\nu} L_{\lambda} \\ &= 0. \end{align} }