PHYS 424 Notes

To go to three dimensions in the Schroedinger equation, simply repeat what we have done before using vectors:

Hy = i hbar dy/dt

H = p_x²/(2m) + V(x)         H = p²/(2m) + V(r)

In general, the potential is V(r), but in most interesting cases the dependence will be on the magnitude of r only. With

p_x = - i hbar d/dx         p = - i hbar Ñ

so that the Schroedinger equation is

From calculus, Ñ = 1_r ¶/¶r + 1_q (1/r) ¶/¶q + 1_f(1/r sinq) ¶/¶f, where each "denominator" is ds along the indicated direction. Unfortunately, Ѳ is not so simply interpreted, although the only really tricky term is ¶/¶q:

There is an overall 1/r², with the placement of factors of r inside derivatives given by units, and the ¶/¶f term is completely natural.

Take y = R(r) Y(q, f), substitute into the Schroedinger equation, and multiply by
-2 m r² / (hbar² R Y) to get

(1/R) d/dr (r² dR/dr) - (2 m r² / hbar²)[V(r) - E] = - (1/Y){(1/sinq)d/dq (sinq dY/dq) + (1/sin²q) d²Y/df²} = l(l+1)

where the last part of the equation is due to the fact that we have a function of r alone equal to a function of q, f alone. The form of the constant is chosen for later convenience. We can't do anything about the radial equation until we make some choices for V(r), but the angular equation can be further simplified by taking the form

Y(q,f) = Q(q) F(f),

substituting into the equation and dividing by QF.

This time we get a function of q equal to a function of f so that each must be a constant. We choose to call the constant m² [watch out, this is the same as the symbol for the mass], so that we have

(1/Q) [sinq (d/dq) (sinq dQ/dq)] + l(l+1)sin²q = - (1/F) d²F/df² = m²

Altogether we have

The f equation gives     F(f) = e^imf,     with m zero or a positive or negative integer.

Because the equation depends on m² instead of m, the solutions with positive and negative m are the required two solutions. The overall constant may be lumped into Q. The quantity m must be an integer so that F will be the same function at f and f + 2p.

The q equation depends on both l and m, so its solution must also. For physical wavefunctions, l must be a nonnegative integer. The properties of the solutions are given in the text. The notation is

Q(q) = A P_l^m(cosq)     Q(q) F(f) = Y_l^m (q, f)

and the immediately important properties are

Y_l^m is more often written Y_lm. The Y_l^m are a complete set of square-integrable angular functions.