(7, 38) The potential energy for the isotropic three-dimensional harmonic oscillator in Cartesian coordinates is U(x, y, z) = ka? + ky² + ¿kz², where the force constant k is the same in all directions. (a) Guided by the discussions of the one-dimensional harmonic oscillator and the two-dimensional infinite well in Chapter 5, show that the energies of the three-dimensional oscillator are given by Enpn,n; = (nx +Ny +nz +)hw. (b) What are the quantum numbers and corresponding degeneracies of the ground state and the first two excited states? (c) We can also solve this problem in spherical polar coordinates, with U(r) = }kr² and r² = x² + y² + z². The solution is of the form of Eq. 7.11 with radial functions different from those of the hydrogen atom but the same angular functions. The energy is En = (n +)hw independent of l and m. The relationship between n and l is different from what it is for hydrogen: 1 < n, and n and l are either both odd or both even. Explain how the degeneracies of the ground state and the first two excited states arise in the spherical polar solutions.
(7, 38) The potential energy for the isotropic three-dimensional harmonic oscillator in Cartesian coordinates is U(x, y, z) = ka? + ky² + ¿kz², where the force constant k is the same in all directions. (a) Guided by the discussions of the one-dimensional harmonic oscillator and the two-dimensional infinite well in Chapter 5, show that the energies of the three-dimensional oscillator are given by Enpn,n; = (nx +Ny +nz +)hw. (b) What are the quantum numbers and corresponding degeneracies of the ground state and the first two excited states? (c) We can also solve this problem in spherical polar coordinates, with U(r) = }kr² and r² = x² + y² + z². The solution is of the form of Eq. 7.11 with radial functions different from those of the hydrogen atom but the same angular functions. The energy is En = (n +)hw independent of l and m. The relationship between n and l is different from what it is for hydrogen: 1 < n, and n and l are either both odd or both even. Explain how the degeneracies of the ground state and the first two excited states arise in the spherical polar solutions.
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Transcribed Image Text:(7, 38) The potential energy for the isotropic three-dimensional harmonic
oscillator in Cartesian coordinates is U(x,y,
where the force constant k is the same in all directions. (a) Guided
by the discussions of the one-dimensional harmonic oscillator and the
two-dimensional infinite well in Chapter 5, show that the energies of the
three-dimensional oscillator are given by Ennynz
(b) What are the quantum numbers and corresponding degeneracies of
the ground state and the first two excited states? (c) We can also solve
this problem in spherical polar coordinates, with U(r)
x² +y? + z?. The solution is of the form of Eq. 7.11 with radial functions
different from those of the hydrogen atom but the same angular functions.
z) = }kx² + ky? + kz2,
= (nx + Ny +nz +)hw.
+ nz
2
= }kr² and p2
is En = (n + hw independent of l and m. The relationship
between n and l is different from what it is for hydrogen: 1 < n, and n
and l are either both odd or both even. Explain how the degeneracies
of the ground state and the first two excited states arise in the spherical
polar solutions.
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