The schrodinger equation in 3-d is:

H Psi = i hb d/dt (Psi)

with H=1/2m*p^2+V(r_x,r_y,r_z)=1/2m * (p_x^2+p_y^2+p_z^2) + V(r_x,r_y,r_z)

p_i=hb/i * d/dx

More concisely, H=-hb^2/2m del^2 + V(r_x,r_y,r_z)

Typically, the differential equation for Psi will quantize as:

H Psi_n = E_n Psi_n

with E_n= quantized energy states indexed by n and Psi_n = an eigenstate.

If E<0 => discrete spectrm while E>0 => continuous spectrum.

If V-> 0 as r -> infinity => Psi -> 0 as r -> infinity, something necessary for normalization.

An analysis of the schrodinger equation in spherical coordinates indicates that it can be written as:

H=p_r^2/2m + L^2/2mr^2 +V(r)
This is solved by the method of seperation of variables, first seperation into Psi(r,theta,phi)=R(r)Y(theta,phi)

Then into Y(theta,phi)=P(theta) Phi(phi)

The BC on phi is that Phi(phi)=Phi(phi+2 PI) => Phi(phi)=1/(2 PI)^.5 e^(i m phi) for m in Z

P(theta) is transformed into P(cos(theta)). Demanding non-infinite solutions => associated Legendre polynomials are a solution.

The solution of the radial equation is Hermite polynomials.

source jl@crush.caltech.edu index
variational_method
WKB_bound_state
TDPT
WKB
TDSE