Magnetic Dipole due to an electron's orbital motion

[xtrubambinoxpr]

Homework Statement

Select all of the following which are possible combinations of Lz and θ for hydrogen atoms in a d state, where Lz is the z component of the angular momentum L, and θ is the angle between the +zaxis and the magnetic dipole moment µℓ due to the electron's orbital motion.

-2hbar, 35
hbar, 114
2hbar, 90
2hbar, 35
2hbar, 145

Homework Equations

cosΘ= Lz/L = m(hbar)/√(l(l+1))
mu = -e/2m * L

The Attempt at a Solution

I was able to figure out that cos^-1(Lz/L) will give me all the possible combinations of the d state and the angles associated with each, but cannot figure out the angle between +z and the magnetic dipole moment. Originally I selected 2hbar, 35 as the answer but I was incorrect. I did this because it was the only answer that gave me an angle in the positive z axis and looked like the right choice through elimination. Does anyone have any ideas?

 

[anathema]

it is important to approach problems with a clear understanding of the relevant equations and concepts. In this case, the key equation is μ = -e/2m * L, where μ is the magnetic dipole moment, e is the charge of the electron, and m is the mass of the electron. This equation shows that the magnetic dipole moment is directly proportional to the angular momentum L.

Additionally, the value of Lz is related to the magnetic quantum number m, which can take on values from -l to l, where l is the orbital quantum number. For a d state, l = 2, so m can take on values of -2, -1, 0, 1, or 2.

Using these relationships, we can determine the possible combinations of Lz and θ for hydrogen atoms in a d state. Here are the steps to follow:

1. Determine the possible values of Lz based on the allowed values of m. For a d state, m can take on values of -2, -1, 0, 1, or 2. This corresponds to Lz = -2hbar, -hbar, 0, hbar, or 2hbar, respectively.

2. Use the equation cosΘ = Lz/L = m(hbar)/√(l(l+1)) to determine the angle θ for each value of Lz. For a d state, l = 2, so the equation becomes cosΘ = Lz/√(2(2+1)) = Lz/√6. Plugging in the values of Lz from step 1, we get θ = cos^-1(-2/√6), cos^-1(-1/√6), cos^-1(0), cos^-1(1/√6), and cos^-1(2/√6) for Lz = -2hbar, -hbar, 0, hbar, and 2hbar, respectively.

3. Look for the given combinations of Lz and θ in the list of possible values calculated in step 2. The correct answers are 2hbar, 90 and -2hbar, 35. This is because the magnetic dipole moment μ is always perpendicular to the z axis, so the angle θ must be 90 degrees for any non-zero