Quantum Mechanics Solution Manual 

© Leon van Dommelen 

4.2.2.3 Solution angubc
Question:
Actually, based on the derived eigenfunction, , would any macroscopic particle ever be at a single magnetic quantum number in the first place? In particular, what can you say about where the particle can be found in an eigenstate?
Answer:
The square magnitude of the wave function gives the probability of finding the particle. The square magnitude,
is independent of . So to be in a state of definite angular momentum, the particle must be at all sides of the axis with equal probability. A macroscopic particle will at any given time be at a single angle compared to the axis, not at all angles at once. So, a macroscopic particle will have indeterminacy in angular momentum, just like it has indeterminacy in position, linear momentum, energy, etcetera.
Since the probability distribution of an eigenstate is independent of , it is called axisymmetric around the axis
. Note that the wave function itself is only axisymmetric if 0, in other words, if the angular momentum in the direction is zero. Eigenstates with different angular momentum look the same if you just look at the probability distribution.