QUANTUM MECHANICS 2535.9 One-Dimensional Coulomb Potential (Princeton)
a) Since the electron is confined to the right half-space, its wave function
must vanish at the origin. So, an eigenfunction such as exp is un-
suitablesince it does not vanish at The ground statewave function
must be of the form where needs to be determined.
The operator acting on this form gives
so that using this wave function in Schrödinger’s equation yields
For this equation to be satisfied, the first and third terms on the left must
be equal, and the second term on the left must equal the term on the right
of the equals sign:
The answer is one sixteenth of the Rydberg, where is the ground state
energy of the hydrogen atom. The parameter where is the
Bohr radius.
b) Next we find the expectation value The first integral is done to find
the normalization coefficient:
The average value of is 6 Bohr radii.5.10 Two Electrons in a Box (MIT)
a) If the box is in the region then the one-electron orbitals are