In the book Many-Particle Physics by Gerald D. Mahan, he points out that the Schrodinger equation in the form iℏ∂ψ∂t=[−ℏ2∇22m+U(r)]ψ(r,t)
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 ℏ 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 ℏ are not uncommon. In quantum field theory, one can construct effective actions from computing Feynman diagrams, which may have factors of ℏ, outside of natural units.
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