I am attempting to solve the Schrödinger equation with the potential $V = – \delta (x)$. This leads to a differential equation
$$
\alpha \psi''(x) + (E + \delta(x)) \psi(x) = 0
$$
where
$$
\alpha \equiv \frac{\hbar^2}{2m}
$$
My instinct to solve the equation was to use an integral transform as I didn't know what else to do with the delta function. Since I am concerned with all space, I chose the Fourier transform. To get the Fourier transform, I transform each term, $\alpha \psi''(x)$, $E \psi(x)$, $\delta(x) \psi(x)$, and $0$. I get these transforms:
\begin{align}
&\mathcal{F}\{ \alpha \psi''(x) \} = -4\alpha \pi^2 p^2 \hat{\psi}(p)\\
&\mathcal{F}\{ E \psi(x) \} = E \hat{\psi}(p) \\
&\mathcal{F}\{ \delta(x)\psi(x) \} = \hat{\psi}(p) \\
&\mathcal{F} \{0 \} = 0
\end{align}
Where $\hat{\psi}(p)$ is the Fourier transform of $\psi(x)$. Unfortunately, when I combine this with the differential equation above, I get
$$
(1 + E -4 \alpha \pi^2 p^2) \hat{\psi}(p) = 0
$$
which results in the $\hat{\psi}(p)$ term going away and leaving me with a function only of $p$, so I am not able to solve for $\psi(x)$ via an inverse Fourier transform like I expected. I did replace the right hand side with an arbitrary function and work through it, but that results in $\psi(x) = 0$ is that function is $0$.
Is there something that I have done incorrectly in my approach? Is there a better approach to solving this equation? My professor and book both solved this question by solving on $(-\infty, 0)$ and $(0, \infty)$ and determining the condition at the origin by integrating the Schrödinger equation over $(-\epsilon, \epsilon)$, but I was hoping to be able to solve it in a more general way and plan to move to a Dirac comb next, which should be very similar using Fourier transformations.
Best Answer
Fourier transforms is a perfectly fine tool to solve the system. The advantage of using it is that you avoid having to solve two equations, one for $x<0$ and one for $x>0$, and then matching at $x=0$. The drawback is that you will have to compute the inverse transform which is usually a bit more work than solving and performing the matching if you have not done it before. However if you can use Fourier transform tables then this is imo the simplest way to solve it. The only mistake in your approach so far is not using the correct formula for the Fourier transform of $\delta(x)\psi(x)$. If you correct this then the derivation is straight forward and below I show an example of such a derivation for completeness.
For a general delta-function sink, $V(x) = -\lambda\delta(x)$, the Schrödinger equation reads
$$-\alpha\frac{d^2\psi(x)}{dx^2} - \lambda \delta(x)\psi(x) = E\psi(x)$$
where $E<0$ for a bound state (which is the case we are interested in). Taking the Fourier transform, using the convention $\int {\rm d}x\, e^{ipx}$ for simplicity, we obtain
$$\alpha p^2\hat{\psi}(p) - \lambda\psi(0) = E\hat{\psi}(p) \implies \hat{\psi}(p) = \frac{A}{p^2 + p_0^2}$$
where $p_0 = \sqrt{\frac{-E}{\alpha}}$ and $A= \frac{\lambda\psi(0)}{\alpha}$. Transforming back to real space we get the wavefunction
$$\psi(x) = \frac{A}{2\pi}\int_{-\infty}^\infty\frac{e^{-ipx}}{p^2+p_0^2}{\rm d}x = \frac{A}{2 p_0} e^{-p_0|x|}$$
where I simply used a Fourier transform table to compute the integral. If you want to know how to solve it analytically see this answer. The final thing left is to compute the energy which is determined by imposing wave-function normalization
$$\int_{-\infty}^\infty |\psi(x)|^2{\rm d}x = 1\implies \psi(0) = \sqrt{p_0} \implies E = -\frac{\lambda\alpha}{2}p_0 = -\frac{\lambda^2}{4\alpha}$$
so the full solution can be written
$$\psi(x) = \sqrt{p_0}e^{-p_0|x|}~~~\text{where}~~~p_0 = \frac{\lambda}{2\alpha}$$