PHYSICS 244 HOMEWORK SOLUTION WEEK 9

9.32 It is known that a certain hydrogen atomhas n=5 and m=2.

a) How many different states are consistent with this information?

Since m ranges from -l to +l, and l< n, l can be 2, 3, or 4. Therefore there are three different states with n=5 and m=2.

b) What is the answer for arbitrary n and m?

l must be greater than or equal to m as well as less than n. This implies that there are n-|m| possible states with identical n and m.

9.34 The hydrogenic radial wave functions R(r) are relatively simple for the case l=n-1:.

a)Write down the radial Schrodinger equation for this case.

b) Verify that R(r) satifies this equation if and only if . Set .

9.42 Consider the radial probability density P(r) for the ground state of hydrogen .

Find the most probable radius for this state by finding the maximum P(r).

10.2 There exist subatomic particles with spin magnitudes different from that of the electron. However, in all cases they obey the same rules: The magnitude of S is , where s is a fixed integer of half-integer; and the possible values of where m has the values s, s-1,....,-s.

a) For a particle with s=3/2, how many different values of Sz are there and what are they?

There are four values of ms = -3/2, -1/2, +1/2, +3/2.

b) Draw a vector-model diagram showing the orientations of S.

c) What is the minimum possible angle between S and the z axis?

10.5 Make a table showing the values of the four quantum-numbers n, l, m, ms and the energies for each of the 10 lowest-lying quantum states (not energy levels) of the hydrogen atom.

 n 1 2 2 2 2 l 0 0 1 1 1 m 0 0 -1 0 +1 ms +-1/2 +-1/2 +-1/2 +-1/2 +-1/2

10.10 The energy of a magnetic moment in a magnetic field B pointing along the z axis is . For an electron in orbit around a proton . If B = 10T and if the electron is in a p state with Lz = hbar, what is the magnetic energy due to the orbital angular momentum (Joules and eV)?