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The uncertainty principle is a set of mathematical inequalities that establish a basic limit on the precision with which the values for certain pairs of physical properties of a particle, such as position and momentum, can be anticipated from initial conditions.

$\mathit{螖}\mathit{x}\mathit{螖}\mathit{p}\mathbf{鈮�}\frac{\mathbf{h}}{\mathbf{4}\mathbf{蟺}}$

Calculate the commutator of the two operators, A and B in the following way,

$$\begin{gathered} \left[\mathrm{A},\mathrm{B}\right]=\left[{\mathrm{X}}^2,{\mathrm{L}}_{\mathrm{z}}\right] \\ =\left[{\mathrm{x}}^2,{\mathrm{yp}}_{\mathrm{x}}-{\mathrm{xp}}_{\mathrm{y}}\right] \\ =\left[{\mathrm{x}}^2,{\mathrm{yp}}_{\mathrm{x}}\right]-\left[{\mathrm{x}}^2,{\mathrm{xp}}_{\mathrm{y}}\right] \\ =\left[{\mathrm{x}}^2,\mathrm{y}\right]{\mathrm{p}}_{\mathrm{x}}+\mathrm{y}\left[{\mathrm{x}}^2,{\mathrm{p}}_{\mathrm{x}}\right]-\left[{\mathrm{x}}^2,\mathrm{x}\right]{\mathrm{p}}_{\mathrm{y}}-\mathrm{x}\left[{\mathrm{x}}^2,{\mathrm{p}}_{\mathrm{y}}\right] \end{gathered}$$

Evaluate the above expression further.

$$\begin{gathered} \left[\mathrm{A},\mathrm{B}\right]=0+\mathrm{y}\left\{\mathrm{x}\left[\mathrm{x},{\mathrm{p}}_{\mathrm{x}}\right]+\left[\mathrm{x},{\mathrm{p}}_{\mathrm{x}}\right]\mathrm{x}\right\}-0-\mathrm{x}\left\{\mathrm{x}\left[\mathrm{x},{\mathrm{p}}_{\mathrm{y}}\right]+\left[\mathrm{x},{\mathrm{p}}_{\mathrm{y}}\right]\mathrm{x}\right\} \\ =\mathrm{y}\left\{\mathrm{x}\left(-\mathrm{i}\sout{\mathrm{h}}\right)+\left(-\mathrm{i}\sout{\mathrm{h}}\right)\mathrm{x}\right\}-\mathrm{x}\left\{0+0\right\} \\ =-2\mathrm{i}\sout{\mathrm{h}}\mathrm{yx} \end{gathered}$$

It is known that .

Apply it in expression.

Thus, the uncertainty principle for ${\mathrm{蟽}}_{\mathrm{A}}{\mathrm{蟽}}_{\mathrm{B}}鈮�\mathrm{魔}|鉄�\mathrm{xy}鉄�|$.

Now get the expectation values of both $L_z$ and $L_z^2$, in some arbitrary hydrogenic normalized state$|{蠄}_{n}lm鉄�$ .

Evaluate the variation in $B,{蟽}_B$,

Similarly, evaluate the variation in ${\mathrm{B}}^2,{\mathrm{蟽}}_{\mathrm{B}}$

role="math" localid="1658137986506"

Find the value of${蟽}_B$.

Thus,${蟽}_B$ in the hydrogen state ${蠄}_{nim}$ is ${\mathrm{蟽}}_{\mathrm{B}}=0$.

Since the variation in$\mathrm{B},{\mathrm{蟽}}_{\mathrm{B}},$ is zero and the right hand side of the uncertainty principle, $鈩�|鉄�xy鉄�$, is always positive , then the right hand side must also be zero. Hence,it is concluded that $鉄�\mathrm{xy}鉄�$ is also 0.

Thus, the conclusion about $鉄�\mathrm{xy}鉄�$ in the given state is 0.