# Weak solution of Schrödinger Equation

Consider a particle in a box $$\Lambda = [0, L]$$. The wavefunction $$\psi \in L_D^2(\Lambda)$$ where $$D$$ denotes a Dirichlet Condition $$\psi(0)=0=\psi(L)$$. We have, then

$$- \frac{\hbar^{2}}{2m} \frac{d^{2}\psi}{dx^{2}} = E \psi$$

inside de box. Solving this equation,

$$\psi_{m}(x) = \sqrt{2/L} \sin(n\pi x/L).$$

Now consider a weak formulation of this problem:

$$\frac{\hbar^{2}}{2m} \int_{0}^{L} \frac{d\psi}{dx} \frac{d\varphi}{dx} dx = E \int_{0}^L \psi \varphi dx$$

so that $$\varphi \in H^1_0(\Lambda)$$. Is it possible to find an analytical solution for this equation? If so, how can one proceed to find solutions of weak formulations in general?

• What do you mean for $L^2_D(\Lambda)$? Is it different from $L^2(\Lambda)$? Boundary conditions do not change the Hilbert space... Mar 2, 2021 at 18:54
• It's just that the wavefunction is in $L^2(\Lambda)$ and $\psi$ has Dirichlet Conditions $\psi(0)=0=\psi(L)$. So I'm just saying that $\psi$ is in $L^{2}$ in the interval $\Lambda$ and $\psi(0)=0=\psi(L)$. Mar 2, 2021 at 18:56
• Since $\psi$ is defined up to zero measure set it does not mean much. There is no a Hilbert space made of functions vanishing at the boundary of an interval. Boundary conditions instead can be used to define the domain of your Hamiltonian... Mar 2, 2021 at 19:40

Elliptic regularity implies that $$\psi$$ admits weak derivatives of every order which are $$L^2$$ locally. Now, Sobolev's lemma implies that these derivatives must be standard derivatives. In summary, every weak solution is just the standard solution $$\psi_m$$.