- #1

keyzan

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**TL;DR Summary:**Find the possible outcomes of ]##L^2## and ##L_{z}## and their respective probabilities of an electron of an idrogen athom with function:

##\psi(r) = ze^{-\alpha r}##

Hi guys, I have a problem with this exercise.

The electron of a hydrogen atom is found with direct spin along the z axis in a state with an orbital wave function:

##\psi(r) = ze^{-\alpha r}##

with alpha greater than zero. The first exercise asks: find the possible outcomes of ]##L^2## and ##L_{z}## and their respective probabilities.

**My solution:**

First of all I have to write the function as the product of a radius function multiplied by a spherical harmonic. So I have:

##\psi(r) = r \cos{\theta} \text{ } e^{-\alpha r} = \sqrt{\frac{8 \pi}{3}} \Upsilon^{0}_{1} \text{ } \frac{1}{2\sqrt{\alpha^{3}}} \varphi_{2,1}##

Here I deduced that the radial part has quantum numbers ##n=2## and ##l=1##, because the coefficient is a power of ##1## and starts from ##r^{1}## and therefore ##n=2## and ##l=1##, but in reality this would be if we had a stationary state, but it is not since we have an ##\alpha## in the argument of the exponential. Can I still write this function with these quantum numbers or is it an error and should I consider a linear combination of radial functions? Any kind of help will be appreciated