The Hydrogen Atom
The Schrodinger equation for the hydrogen atom can be solved exactly. Our model of the hydrogen atom consists of a proton fixed at the origin and an electron of mass me interaction with the proton through a Coulombic potential:
where e is the charge on the proton/electron,
e0 is the permittivity of free space, and r is the distance between the electron and the proton. The factor 4pe0 arises because we are using MKS units. The hamiltonian is:
where del-squared is the Laplacian operator in spherical coordinates:
Now we can write down the Schrodinger equation for the hydrogen atom H
Y = EY.
At first this looks exceedingly complicated, however, of we multiply through by 2mer2 we note that the angular and radial terms can be separated.
This suggests that we can write the wavefunction as a product of radial and angular parts.
We substitute this wavefunction into the equation and divide through by the wavefunction to obtain:
We define a variable of separation called
b. Now the Schrodinger equation can be written as two separate equations.
A factor of Planck's constant squared has been absorbed into the constant
b. The bottom equation has solutions that are spherical harmonics as we have seen applied to the rigid rotator equation.The Angular Equation
We can express the angular equation as:
The spherical harmonic wavefunctions can be obtained by a further separation of variables:
The separation is accomplished once again by substituting in for Y(
q,f) and then dividing through by Q(q)F(f). We find
We can define m2 as a separation constant to obtain two equations.
We have seen that the azimuthal wave functions are eim
f or e-imf. This solution imposes the constraint the m be a quantum number and have values m = 0, ±1, ±2, ±3, …The
q equation is a Legendre equation. It can be solved by making the substitution x = cos q. Here, x should not be confused with the Cartesian coordinate x. It is a substitution variable. With this substitution the q equation becomes:
When this equation is solved it is found that
b must equal 1(1+1) with l = 0, 1, 2, 3… and as above m = 0, ±1, ±2, ±3, …We have discussed Legendre polynomials (the m = 0 solutions) and associated Legendre polynomials (the m
¹ 0 solutions) elsewhere.The Radial Equation
Thus far, we have obtained two of the spatial quantum numbers for the hydrogen atom. The principal quantum number n is obtained from the solution of the radial equation.
The energy eigenvalues which result are:
The Radial wavefunctions are:
where the L(r/na0) functions are known as associated Laguerre polynomials. The first few of these are given below:
The radial wavefunctions form an orthonormal set of functions.
