 Problem Statement

Hi,
I am given the following set of trial wave functions depending on some parameter alpha.
[tex]
\psi(x,y) = \begin{cases}
A \left(1  xy/a^2\right) e^{ \alpha} & x \leq a,\, y \leq a \\
A \left(1  x/a\right) e^{ \alpha y / a} & x \leq a,\, y > a \\
A \left(1  y/a\right) e^{ \alpha x / a} & x > a,\, y \leq a \\
0 & \text{else} \\
\end{cases}
[/tex]
where a > 0 and asked to show that the expectation value of the free Hamiltonian reads:
[tex]
<\psi  H  \psi> = \frac{3 \hbar^2}{m a^{2}} \left( \frac{a^{2} + 2 \alpha + 3}{6 + 11 \alpha} \right)
[/tex]
 Relevant Equations

[tex]
H =  \frac{\hbar^{2}}{2m} \left( \frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}} \right)
[/tex]
I have calculated the normalization constant, but I'm struggling with the discontinuities in the derivatives of the wave function. Due to the symmetry, it should suffice to consider the first two cases. The results should be (according to WolframAlpha):
[tex]
\left( \frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}} \right) \left(1  xy/a^2\right) e^{ \alpha} = 2 e^{\alpha} (x^2 + y^2) \delta(x y)/a^2
[/tex]
and
[tex]
\left( \frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}} \right) \left(1  x/a\right) e^{ \alpha y / a} =
(1  x/a) \left( \frac{ \alpha^2 e^{\alpha y/a}}{a^2}  \frac{2 \alpha \delta(y)}{a}\right)  \frac{2 \delta(x) e^{\alpha y/a}}{a}
[/tex]
Then, again for symmetry reasons, I assume that I need to compute three integrals, namely:
[tex]
2 \int_{a}^{a} \mathrm{d}x \int_{a}^{a} \mathrm{d}y \left(1  xy/a^2\right) e^{ 2 \alpha} x^2 \delta(xy)
[/tex]
(first equation, term with y^2 should give the same result)
[tex]
\int_{0}^{a} \mathrm{d}x \int_{a}^{\infty} \mathrm{d}y \left(1  x/a\right)^{2} e^{ 2 \alpha y / a} \frac{\alpha^{2}}{a^{2}}
[/tex]
(first term in the second equation)
[tex]
 \frac{2}{a} \int_{a}^{a} \mathrm{d}x \int_{a}^{\infty} \mathrm{d}y \left(1  x/a\right) e^{ 2 \alpha y / a} \delta(x)
[/tex]
(third term in the second equation, the second should not contribute)
Could someone check if those integrals (including the limits) are correct? I get the wrong result when adding them up (assuming that the first integral contributes twice, the second eight times and the third four times)... I would just like to know if I'm completely on the wrong track or just made a mistake in the calculation.
Thanks a lot!
[tex]
\left( \frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}} \right) \left(1  xy/a^2\right) e^{ \alpha} = 2 e^{\alpha} (x^2 + y^2) \delta(x y)/a^2
[/tex]
and
[tex]
\left( \frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}} \right) \left(1  x/a\right) e^{ \alpha y / a} =
(1  x/a) \left( \frac{ \alpha^2 e^{\alpha y/a}}{a^2}  \frac{2 \alpha \delta(y)}{a}\right)  \frac{2 \delta(x) e^{\alpha y/a}}{a}
[/tex]
Then, again for symmetry reasons, I assume that I need to compute three integrals, namely:
[tex]
2 \int_{a}^{a} \mathrm{d}x \int_{a}^{a} \mathrm{d}y \left(1  xy/a^2\right) e^{ 2 \alpha} x^2 \delta(xy)
[/tex]
(first equation, term with y^2 should give the same result)
[tex]
\int_{0}^{a} \mathrm{d}x \int_{a}^{\infty} \mathrm{d}y \left(1  x/a\right)^{2} e^{ 2 \alpha y / a} \frac{\alpha^{2}}{a^{2}}
[/tex]
(first term in the second equation)
[tex]
 \frac{2}{a} \int_{a}^{a} \mathrm{d}x \int_{a}^{\infty} \mathrm{d}y \left(1  x/a\right) e^{ 2 \alpha y / a} \delta(x)
[/tex]
(third term in the second equation, the second should not contribute)
Could someone check if those integrals (including the limits) are correct? I get the wrong result when adding them up (assuming that the first integral contributes twice, the second eight times and the third four times)... I would just like to know if I'm completely on the wrong track or just made a mistake in the calculation.
Thanks a lot!