As usual we consider the cases λ > 0,
λ < 0 and
λ = 0 separately.
Consider the Φ
equation first, since this has the most restrictive explicit boundary conditions (8.10).
λ = −{α}^{2} < 0 We have to
solve
|
Φ'' = {α}^{2}Φ,
| (8.11) |
which has as a solution
|
Φ(ϕ) = A\mathop{cos}\nolimits αϕ + B\mathop{sin}\nolimits αϕ.
| (8.12) |
Applying the boundary conditions, we get
If we eliminate one of the coefficients from the equation, we get
|
A = A\mathop{cos}\nolimits (2απ) − A\mathop{sin}\nolimits {(2απ)}^{2}∕(1 − cos(2απ))
| (8.15) |
which leads to
|
\mathop{sin}\nolimits {(2απ)}^{2} = −{(1 − cos(2απ))}^{2},
| (8.16) |
which in turn shows
|
2\mathop{cos}\nolimits (2απ) = 2,
| (8.17) |
and thus we only have a non-zero solution for α = n, an integer. We have found
|
{λ}_{n} = {n}^{2},\kern 2.77695pt \kern 2.77695pt {Φ}_{
n}(ϕ) = {A}_{n}\mathop{ cos}\nolimits nϕ + {B}_{n}\mathop{ sin}\nolimits nϕ.
| (8.18) |
λ = 0 We have
|
Φ'' = 0.
| (8.19) |
This implies that
|
Φ = Aϕ + B.
| (8.20) |
The boundary conditions are satisfied for A = 0,
|
{Φ}_{0}(ϕ) = {B}_{n}.
| (8.21) |
λ > 0 The solution (hyperbolic sines and cosines) cannot satisfy the boundary conditions.
___________________________
Now let me look at the solution of the R equation for each of the two cases (they can be treated as one),
|
{ρ}^{2}R''(ρ) + ρR'(ρ) − {n}^{2}R(ρ) = 0.
| (8.22) |
Let us attempt a power-series solution (this method will be discussed in great detail in a future lecture)
|
R(ρ) = {ρ}^{α}.
| (8.23) |
We find the equation
|
{ρ}^{α}[α(α − 1) + {α}^{2} − {n}^{2}] = {ρ}^{α}[{α}^{2} − {n}^{2}] = 0
| (8.24) |
If n\mathrel{≠}0 we thus have two independent solutions (as should be)
|
{R}_{n}(ρ) = C{ρ}^{−n} + D{ρ}^{n}
| (8.25) |
The term with the negative power of ρ diverges as ρ goes to zero. This is not acceptable for a physical quantity (like the temperature). We keep the regular solution,
|
{R}_{n}(ρ) = {ρ}^{n}.
| (8.26) |
For n = 0 we find only one solution, but it is not very hard to show (e.g., by substitution) that the general solution is
|
{R}_{0}(ρ) = {C}_{0} + {D}_{0}\mathop{ ln}\nolimits (ρ).
| (8.27) |
We reject the logarithm since it diverges at ρ = 0.