Eigen values for a state and spherical harmonics

[samee]

Homework Statement

The complete wavefunction for a particular state an atom, is Si(r,theta,phi)=Ne^(-Zr/a_0)(Z/a_0)^3/2sqrt(1/4pi). What is the eigenvalue Lz for this state?

Homework Equations

see above

The Attempt at a Solution

This is the last one that I'm having trouble with. I have no idea how to start it. Just some pointers on how to begin would be awesome...

 

[vela]

The hydrogenic wavefunctions are of the form $\psi(r,\phi,\theta)=R_{nl}Y_l^m$. You've been given the wavefunction
$$\psi = N r^2 e^{-\frac{Zr}{3a_0}}\sin^2 \theta e^{2i\phi}$$.

Identify the pieces and see if you can identify what n, l, and m are for this state.

 

[samee]

Okay, so since R_n,l=2(Z/a₀)^3/2 e^-Zr/a₀, it means that the first part of the wavefunction is R and the second part is Y?

I know that Y_0,0=Sqrt(1/4pi)
which fits the equation except for the constant out front.

This means that l=0. The first R for which l=0 is R_1,0=2* e^(-Zr/a₀)(Z/a₀)^3/2

So, if N=1/162sqrt(pi) *(Z/a₀)⁷, then this wavefunction is R_1,0Y_0,0

This means that it is in the state |1,0,0>

 

[vela]

Good work. Now learn how the quantum numbers n, l, and m relate to Lz. (You should also know how they relate to other observables, like the energy and total angular momentum, though you don't need to know that for this particular problem.)

 

[paranormal]

As a scientist, my response to this content would be to first clarify the terminology being used. In this context, the term "eigenvalue" refers to a physical quantity that can be measured or calculated for a specific state of an atom. In quantum mechanics, eigenvalues represent the possible outcomes of a measurement for a given observable.

In this case, the observable being discussed is the angular momentum in the z-direction, denoted as Lz. The state of the atom is described by the complete wavefunction, which is a mathematical representation of the atom's quantum state. The wavefunction contains all the information about the atom's position, energy, and other physical properties.

To determine the eigenvalue Lz for this state, we need to use the spherical harmonics, which are mathematical functions that describe the spatial distribution of the wavefunction. The spherical harmonics are labeled by two quantum numbers, l and m, which correspond to the orbital angular momentum and its projection along the z-axis, respectively.

In this case, the wavefunction given in the problem is a special case known as the 1s orbital, which corresponds to l=0 and m=0. The eigenvalue for Lz in this case is therefore 0. This means that the atom has no angular momentum along the z-axis and its wavefunction is spherically symmetric.

In summary, the eigenvalue Lz for this state is 0, which indicates that the atom has no angular momentum along the z-axis. This result is consistent with the fact that the wavefunction given in the problem represents the 1s orbital, which has a spherical symmetry and no preferred direction in space.