A quantum mechanics problem Schrödinger's equation in the absence of a potential is h² 2m -V²=E, (1) where his Planck's constant divided by 27, m is the mass, E is the energy, and is the wave- function. Consider a particle confined in a sphere of radius a. ("Confined" means that the wavefunction vanishes at r = a.) (a) Determine the possible values of the energy E, considering only states with no dependence on the azimuthal angle o. Also write down the corresponding states (i.e. wavefunctions). Note: Your answer will involve zeros of spherical Bessel functions.
A quantum mechanics problem Schrödinger's equation in the absence of a potential is h² 2m -V²=E, (1) where his Planck's constant divided by 27, m is the mass, E is the energy, and is the wave- function. Consider a particle confined in a sphere of radius a. ("Confined" means that the wavefunction vanishes at r = a.) (a) Determine the possible values of the energy E, considering only states with no dependence on the azimuthal angle o. Also write down the corresponding states (i.e. wavefunctions). Note: Your answer will involve zeros of spherical Bessel functions.
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Transcribed Image Text:A quantum mechanics problem
Schrödinger's equation in the absence of a potential is
²=E,
(1)
2m
where his Planck's constant divided by 27, m is the mass, E is the energy, and is the wave-
function. Consider a particle confined in a sphere of radius a. ("Confined" means that the
wavefunction vanishes at r = a.)
(a) Determine the possible values of the energy E, considering only states with no dependence
on the azimuthal angle o. Also write down the corresponding states (i.e. wavefunctions).
Note: Your answer will involve zeros of spherical Bessel functions.
(b) Now consider only states with no dependence on the polar angle 0. Write down all values
of the energy. You are given that the lowest energy state, which has energy E = Emin, is in
this sector, i.e. has no angular dependence. What is Emin?
Note: We are not dealing with superpositions in this question. We are interested in individual
quantum states, which are specified by a value for I (which is called the angular momentum
quantum number since the square of the angular momentum of the particle is h²l(+1)) and a
zero of the spherical Bessel function ji(x).
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