Orbital angular momentum and spherical harmonics A particle in a spherically sym

Question

Orbital angular momentum and spherical harmonics A particle in a spherically symmetric potential is described by the wavefunction where N is a normalization constant, r = x2+y3+z2 and a > 0. You do need to know N or a to answer the following: Is psi an eigenfunction of the square of the orbital angular momentum L2? If so, what are the eigenvalues? If not, what are the possible results we may obtain when L2 is measured? What are the possible outcomes for a measurement of the z component, Lz, of the orbital angular momentum? What are the probabilities for these results?

Explanation / Answer

In physics, aquantum (plural: quanta) is theminimum unit of any physical entity involved in an interaction. Anexample of an entity that is quantized is the energy transfer ofelementaryparticles of matter (called fermions) and ofphotons and otherbosons. The word comesfrom the Latin "quantus", for "howmuch." Behind this, one finds the fundamental notion that aphysical property may be "quantized", referred to as "quantization".This means that the magnitude can take on only certain discretenumerical values,rather than any value, at least within a range. There is a relatedterm of quantumnumber.

A photon, for example, is asingle quantum of light, and may thus be referred to as a"light quantum".The energy of an electron bound to anatom(at rest) is said to be quantized, which results in the stabilityof atoms, and of matter in general.

As incorporated into the theory of quantummechanics, this is regarded by physicists as part of thefundamental framework for understanding and describing nature atthe infinitesimal level, for the very practical reason that itworks. It is "in the nature of things", not a more or lessarbitrary human preference.

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