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11.2: Non-Commuting Operators

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    56868
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    In the previous section, we saw that if a particle can be in a definite state for two observables, then the two operators associated with those observables will commute. The converse is therefore also true; if two operators do not commute, then it is not possible for a quantum state to have a definite value of the corresponding two observables at the same time.

    We’ve already seen examples of this. A particle can’t have a definite \(x\) spin and a definite \(y\) spin at the same time. If our theory is to be useful, then we would hope that \(\hat{S}_{x}\) and \(\hat{S}_{y}\) would not commute when they operate on a general normalized state \(|\psi\rangle\). Let’s try it first in one order:

    \[\begin{aligned}
    \hat{S}_{x} \hat{S}_{y}|\psi\rangle &=\frac{\hbar^{2}}{4}\left[\begin{array}{ll}
    0 & 1 \\
    1 & 0
    \end{array}\right]\left[\begin{array}{cc}
    0 & -i \\
    i & 0
    \end{array}\right]\left[\begin{array}{l}
    \psi_{1} \\
    \psi_{2}
    \end{array}\right] \\
    &=\frac{\hbar^{2}}{4}\left[\begin{array}{ll}
    0 & 1 \\
    1 & 0
    \end{array}\right]\left[\begin{array}{l}
    -i \psi_{2} \\
    i \psi_{1}
    \end{array}\right] \\
    &=\frac{\hbar^{2}}{4}\left[\begin{array}{c}
    i \psi_{1} \\
    -i \psi_{2}
    \end{array}\right] \\
    &=i \frac{\hbar^{2}}{4}\left[\begin{array}{c}
    \psi_{1} \\
    -\psi_{2}
    \end{array}\right]
    \end{aligned}\tag{11.4}\]

    Now let’s try it in the other order:

    \[\begin{aligned}
    \hat{S}_{y} \hat{S}_{x}|\psi\rangle &=\frac{\hbar^{2}}{4}\left[\begin{array}{cc}
    0 & -i \\
    i & 0
    \end{array}\right]\left[\begin{array}{ll}
    0 & 1 \\
    1 & 0
    \end{array}\right]\left[\begin{array}{l}
    \psi_{1} \\
    \psi_{2}
    \end{array}\right] \\
    &=\frac{\hbar^{2}}{4}\left[\begin{array}{cc}
    0 & -i \\
    i & 0
    \end{array}\right]\left[\begin{array}{l}
    \psi_{2} \\
    \psi_{1}
    \end{array}\right] \\
    &=\frac{\hbar^{2}}{4}\left[\begin{array}{c}
    -i \psi_{1} \\
    i \psi_{2}
    \end{array}\right] \\
    &=-i \frac{\hbar^{2}}{4}\left[\begin{array}{c}
    \psi_{1} \\
    -\psi_{2}
    \end{array}\right]
    \end{aligned}\tag{11.5}\]

    Clearly the two are not the same; one is the negative of the other. Therefore, \(\hat{S}_{x}\) and \(\hat{S}_{y}\) do not commute when operating on a general state \(\psi\), as expected.

    It is interesting to note the effect of \(\hat{S}_{z}\) on this same general state:

    \[\begin{aligned}
    \hat{S}_{z}|\psi\rangle &=\frac{\hbar}{2}\left[\begin{array}{cc}
    1 & 0 \\
    0 & -1
    \end{array}\right]\left[\begin{array}{l}
    \psi_{1} \\
    \psi_{2}
    \end{array}\right] \\
    &=\frac{\hbar}{2}\left[\begin{array}{c}
    \psi_{1} \\
    -\psi_{2}
    \end{array}\right]
    \end{aligned}\tag{11.6}\]

    Notice that except for the constant out front, the vector produced by \(\hat{S}_{z}\) on this state is the same as the vector produced by \(\hat{S}_{x} \hat{S}_{y}\) and \(\hat{S}_{y} \hat{S}_{x}\). In fact, we can put the two together:

    \[\begin{gathered}
    \left(\hat{S}_{x} \hat{S}_{y}-\hat{S}_{y} \hat{S}_{x}\right)|\psi\rangle=i \frac{\hbar^{2}}{2}|\psi\rangle \\
    {\left[\hat{S}_{x}, \hat{S}_{y}\right]|\psi\rangle=i \hbar \hat{S}_{z}|\psi\rangle}
    \end{gathered}\tag{11.7}\]

    The term in brackets, \(\left[\hat{S}_{x}, \hat{S}_{y}\right]\) is called the commutator of \(\hat{S}_{x}\) and \(\hat{S}_{y}\). It’s defined by the term in parentheses above it: \(\left(\hat{S}_{x} \hat{S}_{y}-\hat{S}_{y} \hat{S}_{x}\right)\). It works out for the commutators of all three spin angular momentum operators that:

    \[\left[\hat{S}_{x}, \hat{S}_{y}\right]=i \hbar \hat{S}_{z}\tag{11.8}\]

    \[\begin{aligned}
    &{\left[\hat{S}_{y}, \hat{S}_{z}\right]=i \hbar \hat{S}_{x}} \\
    &{\left[\hat{S}_{z}, \hat{S}_{x}\right]=i \hbar \hat{S}_{y}}
    \end{aligned}\tag{11.9}\]


    This page titled 11.2: Non-Commuting Operators is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Pieter Kok via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.