Simple Nature - Light and Matter

Exercise 13C: Rotation around different axes

This exercise refers to the example at the beginning of section 13.4.3 on p. 923, which analyzes

the figure below:

To simplify the writing:

• Ψ− 1 means the state withz=−1, Ψ 0 hasz= 0, etc.


• States with definite values ofxare notated as Ψx=0, etc.

1. The wavefunctions are shown with values of the wavefunction written at the north pole and
   at two points on the equator. Fill in the south poles. Why do the results make sense physically
   for the`z= 1 and−1 wavefunctions?


2. By rotating the pictures 90 degrees counterclockwise, we can make states of definitex. We now want to express the states of definitexin terms of the states of definite`z.

Ψ`x=0= 1 /

√

2 Ψ− 1 + 0 Ψ 0 + 1 /

√

2 Ψ 1 Done on p. 923.

Ψ`x=1= Ψ− 1 + Ψ 0 + Ψ 1 Demonstrated by the instructor.

Ψ`x=− 1 = Ψ− 1 + Ψ 0 + Ψ 1 Done by the students.

3. Fred takes a molecule known to have= 1, and measures itsx. (This can be done by
   passing it through a magnetic field, as described in more detail in section 14.1.) If the molecule
   is not prepared in any particular orientation, then the result is random, and can bex=−1, 0, or 1. (The probabilities are all 1/3, although this is not obvious.) Suppose he measures x= 0, so thataftermeasurement, he is sure that the wavefunction is Ψx=0. (Fred may now be superimposed with other versions of himself who sawx=−1 or 1, but we stop keeping track
   of them now.)
   Now suppose that Fred follows up with a second measurement, on the same molecule, but this
   time he orients the magnetic field so that he’s measuring`z. What are the probabilities of the
   three possible results? Check normalization.

954 Chapter 13 Quantum Physics