Physical Chemistry Third Edition

17.7 The Intrinsic Angular Momentum of the Electron. “Spin” 759

commuting observables can put a system into a known state. The four observables,

E,L^2 ,Lz, andSz, form a complete set of commuting observables for the hydrogen

atom. Assume that we can measure the energy without any experimental error. The

outcome must be an eigenvalue ofĤ, and let us assume that the outcome isE 3 , equal

to (− 13 .6eV)/ 32 −1.511 eV. After the measurement, the wave function must be a

linear combination of all of the energy eigenfunctions in the 3s,3p, and 3dsubshells,

all of which correspond to the correct energy eigenvalue. There are 18 such wave

functions.

Assume that we now measureL^2 and that the outcome is 6h ̄^2 , which means that after

the measurement the wave function is a linear combination of all energy eigenfunctions

withn3 andl2 (all of the states in the 3dsubshell). There are 10 spin orbitals in the

3 dsubshell. Now assume that we measureLzand obtain the value 2h ̄, corresponding

tom2. This means that the wave function is now a linear combination ofψ 322 αand

ψ 322 β. Now assume that we measureSzand obtain the resulth/ ̄ 2. This means that we

have put the hydrogen atom into the state corresponding toψ 322 α. Since there were

four quantum numbers to be determined, we needed four commuting operators to put

the hydrogen atom into a known state. We do not have any information about the state

before the measurements, except that it could be represented by a linear combination

in which the coefficient of theψ 322 αstate is nonzero.

PROBLEMS

Section 17.7: The Intrinsic Angular Momentum
of the Electron. “Spin”

17.40Calculate the expectation value of the square of the speed
of the electron in a hydrogen-like atom withZ26 (an
Fe^25 +ion) in the 1sstate, and from this calculate the
root-mean-square speed. Compare this speed with the
speed of light and with the root-mean-square speed of the
electron in a hydrogen atom from Example 17.9.

17.41If the spin angular momentum is known to lie somewhere
in the cone of possible directions forms 1 /2, and the
orbital angular momentum is known to lie somewhere in
the cone of possible directions form1, what is the

largest angle possible between the two angular

momentum vectors? What is the smallest angle?

17.42The standard model of subatomic particles does not

ascribe any internal structure to the electron. The “string

theory” asserts that all fundamental particles consist of

patterns of vibration of tiny string-like objects that are the

true fundamental particles. These strings have size in the

range of 10−^35 m. Pretend that the electron is an object of

mass equal to the mass of an electron moving in a circular

orbit of radius 1× 10 −^35 m. Find its apparent speed in

this orbit, given the magnitude of the spin angular

momentum. Compare this apparent speed with the speed

of light.

Summary of the Chapter

The time-independent Schrödinger equation for a hydrogen atom was separated into

a one-particle Schrödinger equation for the motion of the center of mass of the two

particles and a one-particle Schrödinger equation for the motion of the electron relative

to the nucleus. The motion of the center of mass is the same as that of a free particle.

The Schrödinger equation for the relative motion was solved by separation of variables

in spherical polar coordinates, assuming the trial function

ψnlm(r,θ,φ)Rnl(r)Ylm(θ,φ)Rnl(r)Θlm(θ)Φm(φ)