A particle in a central potential

Consider a spinless particle of mass m in a central potential V(r).  The Hamiltonian is  , [H,L_i]=0, [H,L²]=0.  The angular momentum L of the particle is a constant of motion.  We can find a common eigenbasis of H, L² and L_z.  We denote these basis states |k,l,m> and the corresponding eigenfunctions by .  We have

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The wavefunction is a product of a radial function and the spherical harmonic  .  The differential equation for u_kl(r) is .

The asymptotic behavior of R_kl(r)

Near the origin the radial behavior of an acceptable wavefunction of a particle moving in a central potential is proportional to r^l, (if  as r®0).

Two interacting particles

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We are often only interested in the relative motion.  If the mutual interaction depends only on the distance between the particles r=|r₁-r₂|, then the eigenvalue equation becomes  . Then  .

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The hydrogen atom

The time-independent Schroedinger equation for the hydrogen atom is , where .  Writing , we find  . 

The ground state energy of the hydrogen atom is -E_I.  where .  a  is the fine structure constant.

 is the energy of the nth excited state.  Here n is called the principal quantum number, n fixes the energy of the eigenstate.  Given n, l can take on n possible values  .  n characterizes an electron shell, which contains n subshells characterized by l.  Each subshell contains 2l+1 distinct states.

We also write .  R_H is the Rydberg constant.

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To find the eigenfunctions and eigenvalues of the Hamiltonian of a hydrogenic atom we replace in the eigenfunctions of the Hamiltonian of the hydrogen atom  , and in the eigenvalues of the Hamiltonian of the hydrogen atom we replace .