[dakren12]

When the hydrogen atomic 2p_x is acted upon by the operator for z-component of the angular momentum L_z we find that...attached image. This problem is from the ACS Physical Chemistry Study Guide.


Since the answer is already provided, what I'm looking for is actually the LOGIC/EXPLANATION between each step of the solution process, obviously I am not understanding how one led to the next. Such as where the sqrt 2 came from, and how the h-bar comes into play and the overall conclusion that Lz is not a eigenfunction of Lz.



 how did they go from ψ2px to  (ψ211+ψ21-1 / sqrt2)  ?

[Corribus]

Many of the atomic orbital wavefunctions as you probably are aware have an imaginary component to them. This renders them less useful for understanding the electron probability distribution in real space. There is a principle that if two functions are both individually solutions to the Schrodinger equation, then a linear combination of them also is a solution. (It is a common exercise for students new to quantum mechanics to prove this; if you haven't done it, it's not a bad idea to do so in your spare time.) The mathematical formulations of what we know as the p_x or p_y orbitals are linear combinations of the actual eigenfunctions of the L_z operator. Because of the aforementioned properties of energy eigenstates, these linear combinations are still solutions to the Schrodinger equation (they have the same energy eigenvalues) but they can be plotted in real space.

ψ(211) and ψ(21-1) are both eigenfunctions of the L_z operator with imaginary components. To plot them in real space, we take linear combinations of them, which gives us the "
ψ(211) + ψ(21-1)" and "ψ(211) - ψ(21-1)" portions. The SQRT(2) factor comes from the fact that the wavefunctions still have to be normalized, so each component in the linear combination has a coefficient, c. With only two components, the coefficients have equal magnitude of 1/SQRT(2) and (depending on the combination) either opposite or similar signs. Can you see why this is so?

(Note that the other eigenfunction of L_z, ψ(210) has no imaginary component and so can be plotted "as is" in real space. This becomes the 2p_z orbital. So only 2p_x and 2p_y are linear combinations of L_z eigenfunctions. Note that the energy eigenvalue for all the functions, linear combinations and starting functions, have the same energy eigenvalue, as you would expect. Again, showing this is a very good, if a little tedious, exercise to perform on your own time.)