The Lagrangian-density (as a function of the "independent fields" $\Psi$ and $\Psi^*$ which leads to the Schrödinger-equation is: $$ \frac{i \hbar}{2}(\Psi^* \dot{\Psi} - \Psi \dot{\Psi^*}) - \frac{\hbar^2}{2 m_0} \nabla\Psi \nabla\Psi^* $$ For this lagrangian density, the Euler-Lagrange-Equations will yield the correct formulas of $\Psi$ and $\Psi^*$. However, calculating the canonical momentum will yield: $$\pi = \frac{\partial L}{\partial \dot{\Psi}} = \frac{i \hbar}{2}\Psi^* $$ And the other way arround for $\pi^*$. This gives me a hard time to calculate the hamiltonian density as a function of $\Psi$, $\Psi^*$,$\pi$, $\pi^*$, $\nabla \Psi$ and $\nabla \Psi^*$. I mean, I want the hamiltonian density to satisfy: $$ H = \pi \dot{\Psi} + \pi^* \dot{\Psi^*} - L $$ But I can satisfy this condition with different choices, since the momentum $\pi$ doesn't differ from the the conjugate field $\Psi^*$. I could choose a hamiltonian that doesn't depend on $\pi$ at all, with every instance of $\pi$ being replaced by $\Psi^*$.
Where is my Mistake? Is there a mistake? If yes, where did the mistake start? Has it allready been wrong to assume $\Psi$ and $\Psi^*$ to be independent.
My maybe most important question here is: Is there an unambigous definition of the link between Lagrangian and Hamiltonian, which will lead to a) the right Euler Lagrange equations, and at the same time to b) the right canonical equations in the hamiltonian-fields formalism?
