Sunday, February 5, 2017

quantum mechanics - The Schrödinger equation as an Euler-Lagrange equation


In the book Many-Particle Physics by Gerald D. Mahan, he points out that the Schrodinger equation in the form iψt=[222m+U(r)]ψ(r,t)

can be obtained as the Euler-Lagrange equation corresponding to a Lagrangian of the form L=iψ˙ψ22mψψU(r)ψψ.


I have a discomfort with this derivation. As fas as I know a Lagrangian is a classical object. Is it justified in constructing a Lagrangian that has built into it?




Answer



Firstly, one may think of this as a mathematical rather than physical procedure. In the end one is simply constructing a functional,


S=dtL


whose extremisation, δS=0 leads to the Schrodinger equation. However, Lagrangians containing are not uncommon. In quantum field theory, one can construct effective actions from computing Feynman diagrams, which may have factors of , outside of natural units.


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