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Transcript

Forget about particles. Review of Lagrangian dynamics For a single coordinate q(t) : What are the fields? What equations govern the fields? ★ We always start with a classical field theory. Lagrangian L = L ( q, dq/dt ) ; and Action A = ∫t1t2 L( q , dq/dt ) dt . The equation of motion for q(t) comes from the requirement that δA = 0 (with endpoints fixed); i.e., the action is an extremum. For a variation δq(t) ★ The field equations come from Lagrangian dynamics. Today’s example: The Lagrangian for the Schroedinger equation. (Lagrange’s equation) 1 Canonical momentum and the Hamiltonian Canonical Quantization (Dirac) Rules to convert classical dynamics to a quantum theory: ★ q and p become operators; they operate on the Hilbert space of physical states. ★ [q,p]=iħ ★ H is the generator of translation in time. Example. A particle in a potential... 2 Theorem. H is the generator of translation in time for the quantum theory. Suppose L = ½ M (dq/dt)2 − V(q). So far, we have considered only one degree of freedom. Now consider a system with many degrees of freedom; { qi : i = 1 2 3 … D } For many degrees of freedom… q(t) ⟶ Q(t) ≡ { qi(t) ; i = 1 2 3 … i … N } ◾ ⇒ Lagrange’s equations L = L(Q, dQ/dt) for i = 1 2 3 … N ◾ Canonical momentum... ◾ and Hamiltonian... = dq/dt = dp/dt Q.E.D. (which must be re-expressed in terms of p1…pN and q1…qN.) 3 Classical field theory (suppress spin for now) We replaced q(t) ⟶ { qi(t) ; i ∈ Z } ; discrete Now replace q(t) ⟶ { ψ(x,t) ; x ∈ R3 } ; continuum THE LAGRANGIAN FOR SCHROEDINGER WAVE MECHANICS L = L( ψ(x,t), ∂ψ(x,t)/∂t ) L = ∫ L( ψ(x,t), ∇ψ(x,t) , ∂ψ(x,t)/∂t ) d3x Lagrange’s equation --- this is the “classical field theory.” 4 Lagrange’s Equations Thus, the classical field equation is the Schroedinger equation. Canonical momenta The Hamiltonian 5 Quantization So far, this is the classical field theory. Now... Dirac’s canonical commutation relation [ q , p ] = iħ is valid for Hermitian operators q and p. We need to change that (because ψ is complex) to Summary [ ψ(x) , ψ(x’) ] = 0 [ ψ(x) , ψ♱(x’) ] = δ3(x-x’) ; or, use anticommutators for fermions; This is precisely the NRQFT that we have been using, but with a 1-body potential V(x) and without a 2-body potential V2 (x,y). Therefore Or, replace these by anticommutators for fermions. Exercise: Figure out the Lagrangian that would include a 2-body potential. Hint: The Lagrangian must include a term quartic in the field. Exercise: Verify that H is the generator of translation in time, in the quantum theory. 6 Homework Problems due Wednesday March 2 Problem 30. Equal time commutation relations. We have, in the Schroedinger picture, [ ψ(x) , ψ♰(x’) ] = δ3(x-x’) , etc. (a ) Show that in the Heisenberg picture, this commutation relation holds at all equal times. (b ) What is the commutation relation for different times? Problem (a ) Do (b ) Do (c ) Do 31. problem 2.1 problem 2.2 problem 2.3 in Mandl and Shaw. in Mandl and Shaw. in Mandl and Shaw. 7