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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)
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.
H is the generator of
translation in time.
Example. A particle in a potential...
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
(which must be re-expressed
in terms of p1…pN and q1…qN.)
Classical field theory
(suppress spin for now)
We replaced
q(t) ⟶ { qi(t) ; i ∈ Z } ;
Now replace
q(t) ⟶ { ψ(x,t) ; x ∈ R3 } ; continuum
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.”
Lagrange’s Equations
Thus, the classical field equation is the
Schroedinger equation.
Canonical momenta
The Hamiltonian
So far, this is the classical field theory.
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
[ 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).
Or, replace these by anticommutators for
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.
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’) ,
(a ) Show that in the Heisenberg picture, this
commutation relation holds at all equal times.
(b ) What is the commutation relation for
different times?
(a ) Do
(b ) Do
(c ) Do
problem 2.1
problem 2.2
problem 2.3
in Mandl and Shaw.
in Mandl and Shaw.
in Mandl and Shaw.