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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