Discussion
The Green's function G(x, \(x_{0}\)) satisfying the equation \(\frac{d^{2}G(x, x_{0})}{dx^{2}} = \delta(x - x_{0})\) with boundary conditions G(-L, \(x_{0}\)) = G(L, \(x_{0}\)) = 0 is :
Answer: D
The Green's function is \(\frac{1}{2L}(x_{0} - L)(x + L)\) for \(x \le x_{0}\) and \(\frac{1}{2L}(x_{0} + L)(x - L)\) for \(x \ge x_{0}\).
On each side of \(x_{0}\), G'' = 0, so G is linear. To vanish at x = -L and x = L, take \(G = A(x + L)\) on the left and \(G = B(x - L)\) on the right. G must be continuous at \(x_{0}\), and integrating the delta function shows that its slope jumps by 1:
\[ \begin{aligned} A(x_{0} + L) &= B(x_{0} - L) \\ B - A &= 1 \end{aligned} \]
Solving, \(A = (x_{0} - L)/2L\) and \(B = (x_{0} + L)/2L\), which is the fourth option. The first option has the opposite overall sign (its slope falls by 1 at \(x_{0}\)), the second is not continuous at \(x_{0}\), and the third has no kink at \(x_{0}\) at all.
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