Quantum Mechanics Solution Manual |
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© Leon van Dommelen |
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4.2.2.3 Solution angub-c
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.