#7 Possible values of the total angular momentum J for Scandium

The electron configuration of Scandium is from

Having a single outer 3d electron, we have l=2 and s=1/2 to work with.

The total angular momentum can then take the value j=5/2 and j=3/2

The actual angular momentum values from sqrt(j(j+1))hbar are then

J = sqrt(5/2(7/2))hbar=sqrt(35)*hbar/2 and J=sqrt(3/2(5/2))hbar= sqrt(15)*hbar/2

#8 Possible total angular momentum values for a pair of 3d electrons,
e.g. , just above Scandium.

With two electrons, each with l=2 and s=1/2, we can have total spin
0 or 1

The orbital angular momentum can take the values 0,1,2,3,4

Since the wavefunction for the system describes fermions, it must be
antisymmetric.

The wavefunction is the product of the spin wavefunction and the spatial
wavefunction,

so if the spin part is symmetric, space is antisymmetric and vice versa.

If spin is 1 and therefore symmetric, then l=1 or l=3 are the possibilities
for the orbital wavefunction.

#9 Possible quantum numbers for outer electron configuration 2p13d1?

If l=1 for the p and l=2 for the d are combined, you could get L=1,2
or 3. Looks like to me that

spin can be either zero or 1, so j=0,1,2,3,4 possible.

#10. For two electrons in d subshell, what values for z-component of
angular momentum possible?

The orbital angular momentum can stretch from 0 to 4. For S=0 it can
take L=0,2,4 and for S=1, L=1,3.

So the total angular momentum can take values 0,1,2,3,4. Mj can then
take -4,-3, … 2,3,4.

#16 Energy level diagram for silicon

With two outer 3p electrons, the energy levels are expected to be very
much like those for carbon

shown in Figure 9-12, p269. My guess is a reproduction of the carbon
level diagram

with all the principal quantum numbers jogged up by one. Expect that
the ground

state is a 3p triplet state, and that of the other two 3p states, the
one with the higher total angular

momentum (j=2) will be lower. Sp3 states could be expected as in carbon.

#19 Predict splitting for Stern-Gerlach experiment with other atoms.

No splitting: No electrons outside of closed shells, so the noble gases
should exhibit no splitting.

Threefold splitting: Triplet state, spin=1. Can occur with two outer
electrons, like carbon and silicon.

Fourfold splitting would come with spin=3/2, which could come with
three electrons, like nitrogen.

#20 Magnetic splitting with a 3d 5/2 state.

The number of magnetic levels is 2j+1 = 6

#21. Splitting of a 2p state in a strong magnetic field.

The strong field magnetic splitting (Paschen-Back effect) depends upon
ml + 2ms.

You look at the orbital and spin values separately, then combine. The
values for

ml are -1,0,1 and the values of 2ms are -1 and 1. The possible combinations
are

-2,-1,0,1,2. Since the splitting is (ml+2ms)* Bohr magneton*
magnetic field in Tesla

Then the splitting is numerically equal to the quantum numbers -2,-1,0,1,2
times the

Bohr magneton.

This gives -11.8, -5.9, 0, 5.9, 11.8 times 1E-5 eV in a 1 Tesla field.