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- Thread starter moriheru
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mfb

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mfb

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First, one would rewrite Schrodinger's equation in spherical coordinates.

The angular variables separate out, so that the energy eigenfunctions are radial functions (of n and L) times spherical harmonics. (of L and Lz)

The radial equation is a bit more challenging, but one can show that the radial solutions (which depend on n and L) are spherical Bessel functions.

Just as with the ordinary particle-in-a-box, one can use the boundary conditions to fix the scale of these spherical Bessel functions, and obtain the energy eigenvalues. Here, the energy eigenvalues don't go up as the square of positive integers, but as the square of the positive zeroes of the spherical Bessel function of order L, where L is the angular momentum quantum number.

Hope this helps:)

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helps lots thanks...yet why replace the radial part with the sign of the density matrix?

First, one would rewrite Schrodinger's equation in spherical coordinates.

The angular variables separate out, so that the energy eigenfunctions are radial functions (of n and L) times spherical harmonics. (of L and Lz)

The radial equation is a bit more challenging, but one can show that the radial solutions (which depend on n and L) are spherical Bessel functions.

Just as with the ordinary particle-in-a-box, one can use the boundary conditions to fix the scale of these spherical Bessel functions, and obtain the energy eigenvalues. Here, the energy eigenvalues don't go up as the square of positive integers, but as the square of the positive zeroes of the spherical Bessel function of order L, where L is the angular momentum quantum number.

Hope this helps:)

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