# 6.3: The Three Components of Angular Momentum Cannot be Measured Simultaneously with Arbitrary Precision

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Consider a particle described by the Cartesian coordinates \((x, y, z)\equiv \vec{r}\) and their conjugate momenta \((p_x, p_y, p_z)\equiv \vec{p}\). The classical definition of the *orbital angular momentum* of such a particle about the origin is (i.e., via the cross product):

\[ \vec{L} = \vec{r} \times \vec{p}\]

which can be separated into projections into each of the primary axes :

\[ \begin{align} L_x &= y\, p_z - z\, p_y, \label{6.3.1a} \\[4pt] L_y &= z\, p_x - x\, p_z \label{6.3.1b} \\[4pt] L_z &= x\,p_y - y \,p_x \label{6.3.1c} \end{align}\]

Extending this discussion to the quantum mechanics, we can us assume that the operators \((\hat{L}_x, \hat{L}_y, \hat{L}_z)\equiv \vec{L}\) which represent the components of orbital angular momentum in quantum mechanics can be defined in an analogous manner to the corresponding components of classical angular momentum. In other words, we are going to assume that the above equations specify the angular momentum operators in terms of the position and linear momentum operators.

In Cartesian coordinates, the three operators for the orbital angular momentum components can be written as

\[\hat{L}_x = -{\rm i}\,\hbar\left(y\,\dfrac{\partial}{\partial z} - z\,\dfrac{\partial} {\partial y}\right) \label{6.3.2a}\]

\[ \hat{L}_y = -{\rm i}\,\hbar\left(z\,\dfrac{\partial}{\partial x} - x\,\dfrac{\partial} {\partial z}\right) \label{6.3.2b}\]

\[ \hat{L}_z = -{\rm i}\,\hbar\left(x\,\dfrac{\partial}{\partial y} - y\,\dfrac{\partial} {\partial x}\right) \label{6.3.2c}\]

These can be transforming to operators in standard spherical polar coordinates,

\[ \begin{align} x &= r \,\sin\theta\, \cos\varphi \label{6.3.3a} \\[4pt] y &= r\, \sin\theta\, \sin\varphi \label{6.3.3b} \\[4pt] z &=r \cos \theta \label{ 6.3.3c} \end{align}\]

we obtain

\[ \begin{align*} \hat{L}_x &= {\rm i}\,\hbar\,\left(\sin\varphi\, \dfrac{\partial}{\partial \theta} + \cot\theta \cos\varphi\,\dfrac{\partial}{\partial \varphi}\right) \label{6.3.4a} \\[4pt] \hat{L}_y &= -{\rm i} \,\hbar\,\left(\cos\varphi\, \dfrac{\partial}{\partial\theta} -\cot\theta \sin\varphi \,\dfrac{\partial}{\partial \varphi}\right) \label{6.3.4b} \\[4pt] \hat{L}_z &= -{\rm i}\,\hbar\,\dfrac{\partial}{\partial\varphi} \label{6.3.4c} \end{align*} \]

We can introduce a new operator \(\hat{L^2}\):

\[ \begin{align} \hat{L^2} &= \hat{L}_x^{\,2}+\hat{L}_y^{\,2}+\hat{L}_z^{\,2} \label{6.3.5} \\[4pt] &= - \hbar^2\left( \dfrac{1}{\sin\theta}\dfrac{\partial}{\partial \theta} \sin \theta \dfrac{\partial}{\partial \theta} + \dfrac{1}{\sin^2\theta}\dfrac{\partial^2} {\partial\varphi^2}\right) \label{6.3.6} \end{align} \]

The eigenvalue problem for \(\hat{L^2}\) takes the form

\[\hat{L^2} | \psi \rangle = \lambda \,\hbar^2 | \psi \rangle \label{6.3.6a}\]

where \(\psi(r, \theta, \varphi)\) is the wavefunction, and \(\lambda\) is a number. Let us write

\[ \psi(r, \theta, \varphi) = R(r) \,Y(\theta, \varphi) \label{6.3.6b}\]

By definition,

\[ \color{red}L^2 \,Y_{l}^{m_l} = l\,(l+1)\,\hbar^2\,Y_{l}^{m_l} \label{6.3.9}\]

where \(l\) is an integer. This is an important conclusion that argues the angular momentum is quantized with the square of the magnitude of the angular momentum only capable of assume one of the discrete set of values (Equation \(\ref{6.3.9}\)). From this, the **amplitude** of angular momentum can be expressed

\[ \color{red} |\vec{L}| =\sqrt{L^2} = \sqrt{l(l+1)} \hbar \label{6.3.10}\]

The properties of spherical harmonics that the z-component of the angular momentum (\(L_z\)) is also quantized and can only assume a one of a discrete set of values

\[ L_z \,Y_{l}^{m_l} = m\,\hbar\,Y_l^{m_l} \label{6.3.11}\]

where \(m_l\) is an integer lying in the range \(-l\leq m_l \leq l\).

## Simultaneous Measurements

Note that observables associated with \(\hat{L}_x\), \(\hat{L}_y\), and \(\hat{L}_z\) can, in principle, be measured. However, to determine if they can be measured *simultaneously* with infinite precision, the corresponding operators must **commute**. Remember that the fundamental commutation relations satisfied by the position and linear momentum operators are:

\[ \begin{align*} [\hat{x}_i, \hat{x}_j] &=0 \label{6.3.12} \\[4pt] [\hat{p}_i, \hat{p}_j] &=0 \label{6.3.13} \\[4pt] [\hat{x}_i, \hat{p}_j] &= {\rm i}\,\hbar \,\delta_{ij} \label{6.3.14} \end{align*}\]

where \(i\) and \(j\) stand for either \(x\), \(y\), or \(z\). Consider the commutator of the operators \(\hat{L}_x\) and \(\hat{L}_z\) :

\[ \begin{align*} [\hat{L}_x, \hat{L}_y] & = [(y\,p_z-z\,p_y), (z\,p_x-x \,p_z)] \\[4pt] &= y\,[p_z, z]\,p_x + x\,p_y\,[z, p_z] \label{6.3.15} \\[4pt] &= {\rm i}\,\hbar\,(-y \,p_x+ x\,p_y) \\[4pt] &= {\rm i}\,\hbar\, \hat{L}_z \label{6.3.16} \end{align*}\]

The **cyclic permutations** of the above result yield the fundamental commutation relations satisfied by the components of an orbital angular momentum:

\[[\hat{L}_x, \hat{L}_y] = {\rm i}\,\hbar\, \hat{L}_z \label{6.3.17a}\]

\[[\hat{L}_y, \hat{L}_z] = {\rm i}\,\hbar\, \hat{L}_x \label{6.3.17b}\]

\[[\hat{L}_z, \hat{L}_x] = {\rm i}\,\hbar\, \hat{L}_y \label{6.3.17c}\]

The three commutation relations (Equations \(\ref{6.3.17a}\) - \(\ref{6.3.17c}\)) are the foundation for the whole theory of angular momentum in quantum mechanics. Whenever we encounter three operators having these commutation relations, we know that the dynamical variables that they represent have identical properties to those of the components of an angular momentum (which we are about to derive). In fact, we shall assume that any three operators that satisfy the commutation relations (Equations \(\ref{6.3.17a}\) - \(\ref{6.3.17c}\)) represent the components of some sort of angular momentum.

In fact, we shall assume that any three operators that satisfy the commutation relations (Equations \(\ref{6.3.17a}\) - \(\ref{6.3.17c}\)) represent the components of some sort of angular momentum.

The fact that \(L_z\) is known with certainty, but \(L_x\) and \(L_y\) are **unknown**; therefore every classical vector with the appropriate length and *z*-component can drawn, forming a cone (Figure \(\PageIndex{1}\)). The expected value of the angular momentum for a given ensemble of systems in the quantum state characterized by \(l\) and \(m_l\) could be somewhere on this cone while it cannot be defined for a single system (since the components of \(L\) do not commute with each other).

## Summary

In the quantum world, angular momentum is quantized. The square of the magnitude of the angular momentum (determined by the eigenvalues of the \(\hat{L^2}\) operator) can only assume one of the discrete set of values

\[l(l + 1)\hbar^2 \nonumber\]

with \(l = 0, 1, 2, ... \nonumber\)

The z-component of the angular momentum (i.e., projection of \(L\) onto the \(z\)-axis) is also quantized with

\[L_z= m \hbar \nonumber\]

with

\[m_l = -l, 0-1, ..., 0, ... +l +1, l \nonumber\]

for a given value of \(l\). Hence, \(l\) and \(m_l\) are the *angular momentum quantum number* and the *magnetic quantum number,* respectively.

## Contributors

Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)