Question

A hydrogen atom electron is in a 2p state. If no experiment has been done to establish a z-component of angular momentum, the atom is equally likely to be found with any allowed value of $L_Z$. Show that if the probability densities for these different possible states are added (with equal weighting), the result is independent of both $\mathit{\varphi }$and$\theta$

Short answer

The probability densities for these different possible states are independent of $\varphi$and$\theta$

Step-by-step solution

 Angular momentum

Angular momentum is defined as the product of the angular velocity and its moment of inertia.

 Given data

The state of the hydrogen atom is 2p.

Show the probability densities are independent of and

We already know that the probability density does not depend on the azimuthal angleφ. In the absence of information about the z−component of angular momentum it is equally likely that the electron would have any of the allowed values. Adding the angular probability densities with equal weights, we have

$\begin{array}{rcl}{\Theta }_{1.0}\left(\theta \right)+{\Theta }_{1.+1}\left(\theta \right)+{\Theta }_{1.-1}\left(\theta \right)& =& {\left(\sqrt{\frac{3}{4\pi }\cos \theta }\right)}^2+{\left(\sqrt{\frac{3}{8\pi }\sin \theta }\right)}^2+{\left(\sqrt{\frac{3}{8\pi }\sin \theta }\right)}^2\\ & =& \frac{3}{4\pi }{\cos }^2\theta +2\frac{3}{8\pi }{\sin }^2\theta \\ & =& \frac{3}{4\pi }\\ & & \end{array}$

Where,$\theta$ = colatitude angle,$\varphi$ = azimuth angle

This has no dependence on either $\varphi \text{or}\theta$

This is clear that the above-mentioned expression is independent of $\theta$and$\varphi$

Hence, the probability densities for these different possible states are independent of $\varphi$and $\theta$