Consider a spherical quantum box or dot with an electron inside it. We presume the potential...
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Consider a spherical quantum box or "dot" with an electron inside it. We presume the potential is infinite at the boundury of the dot and zeto within, and that the "dot" has radius ro. OFind an expression for the eigenenergies Ent, where is the usual angular momentum quantum number and n is another integer quantum number (starting at n=1), expressing your result in ternis of the zeros $nt of the spherical Bessel fiunction / (x), where Sut is the th zero for a given . (ii) Find the electron confinement energies for the nine conditionsn-1,2,3 with =0, 1,2 for each n, for the case of a 10 nm diameter semiconductor dot with electron effective mass of 0.2 mo. (Note: You will have to find appropriate zeros of special functions from mathematical tables or otherwise.) Notes (a) The equation dr has solutions y=V[4J,(ax)+ BY, (ax)] where A and B are arbitrary constants, J, 1s the Bessel function of order p, and Y, is the Weber fiunction of order p. Note that the Weber functions tend to infinity as x0, though the Bessel functions remain finite as x→ 0. (b) The spherical Bessel functions are given by 2x ad these functions can also be expressed as 1 d x dx sinx J.(x)-x' Consider a spherical quantum box or "dot" with an electron inside it. We presume the potential is infinite at the boundury of the dot and zeto within, and that the "dot" has radius ro. OFind an expression for the eigenenergies Ent, where is the usual angular momentum quantum number and n is another integer quantum number (starting at n=1), expressing your result in ternis of the zeros $nt of the spherical Bessel fiunction / (x), where Sut is the th zero for a given . (ii) Find the electron confinement energies for the nine conditionsn-1,2,3 with =0, 1,2 for each n, for the case of a 10 nm diameter semiconductor dot with electron effective mass of 0.2 mo. (Note: You will have to find appropriate zeros of special functions from mathematical tables or otherwise.) Notes (a) The equation dr has solutions y=V[4J,(ax)+ BY, (ax)] where A and B are arbitrary constants, J, 1s the Bessel function of order p, and Y, is the Weber fiunction of order p. Note that the Weber functions tend to infinity as x0, though the Bessel functions remain finite as x→ 0. (b) The spherical Bessel functions are given by 2x ad these functions can also be expressed as 1 d x dx sinx J.(x)-x'
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