What is a Lagrangian density equation? For the linear Schrodinger equation, how can we write this Lagrangian density equation?
This topic is discussed in many places; a particular one is the last chapter of Goldstein's textbook on Classical Mechanics. The Lagrangian density for the for the linear Schroedinger equation is given in problem 4 at the end of chapter 12 (except it has 2 typographical errors). Here it is, in Latex.
\[{\cal L} = \frac{{{\hbar ^2}}}{{2m}}\nabla \psi \cdot \nabla {\psi ^*} + V{\psi ^*}\psi + \frac{\hbar }{{2i}}\left( {{\psi ^*}\dot \psi - \psi {{\dot \psi }^*}} \right)\]
Hi William, my Goldstein book on Classical Mechanicshas only 11 Chapters. Chaper 11 does discuss continuous medium and the problem 4 and 5 at the end of this chapter give equations of Hamiltonian and Lagrangian density. Is it the right chapter. My book is Classical Mechanics, Herbert Goldstein, Addison Wesley, 12 th Printing, 1977.
By the way your answer reminds me my M.Sc. class in Physics where our Professor (long live his soul!) caught some of us and asked him to solve the problems from the Goldstein book on the blackboard. Sometimes it was me of course and my humiliation! We could not go back home unless we could solve the problem. Do you do the same thing with your students!
I found your exercise and equation in the second edition of Goldstein.
- Badges
- Science topic
- Similar topics
- Physical Sciences
- Quantum Physics
- Quantum Mechanics