Fund Quantum Mechanics Lect & HW Solutions 58

Fund Quantum - v v v C r θ e i mφ v v v 2 = | C r θ | 2 is independent of φ So to be in a state of deFnite angular momentum the particle must

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40 CHAPTER 4. SINGLE-PARTICLE SYSTEMS Solution angub-b Question: What is the magnetic quantum number of a macroscopic, 1 kg, particle that is encircling the z -axis at a distance of 1 m at a speed of 1 m/s? Write out as an integer, and show digits you are not sure about as a question mark. Answer: Showing all digits as a question mark is not acceptable, of course. The classical angular momentum is 1 m distance times 1 kg times 1 m/s, or 1 J s. Since that should be m ¯ h with m the magnetic quantum number, you get m = 1 J s 1 . 054 , 57 10 34 J s = 9 , 482 , 3?? , ??? , ??? , ??? , ??? , ??? , ??? , ??? , ??? , ??? Solution angub-c Question: Actually, based on the derived eigenfunction, C ( r, θ ) e i , would any macroscopic particle ever be at a single magnetic quantum number in the Frst 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 Fnding the particle. The square magnitude,
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Unformatted text preview: v v v C ( r, θ ) e i mφ v v v 2 = | C ( r, θ ) | 2 , is independent of φ . So to be in a state of deFnite 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 φ , it is called “axisym-metric around the z-axis”. Note that the wave function itself is only axisymmetric if m = 0, in other words, if the angular momentum in the z-direction is zero. Eigenstates with di±erent angular momentum look the same if you just look at the probability distribution. 4.1.3 Square angular momentum...
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This note was uploaded on 01/06/2012 for the course PHY 3604 taught by Professor Dr.danielarenas during the Fall '11 term at UNF.

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