Lecture 17.
Office hours *for next week only* will be Monday after class.
Today we discussed symmetries and conservation laws in quantum mechanics; introduced the Heisenberg picture and the classical limit; and began an analysis of the quantum simple harmonic oscillator.
From here on, iff = if and only if.
As in classical mechanics, a transformation is a symmetry iff the
transformed system obeys the same equations of motion as the original
system. In the Schrodinger picture, then, U is a symmetry iff U
obeys the Schrodinger equation. We showed that this
occurs iff U commutes with the Hamiltonian H.
We then showed that U commutes with H iff its generating charge
Q commutes with H. We did this by taking the derivative with
respect to the transformation distance a of 
, which
vanishes iff U commutes with H for all a. We found that the
derivative vanishes iff Q commutes with H.
To consider conservation of the charge operator Q, we introduced the
Heisenberg picture. We have previously studied Schrodinger's version
of quantum mechanics, in which physical states evolve, 
(with U(t) the time evolution operator
determined by Schrodinger's equation); and in
which operators and their eigenstates remain constant in time. We
noted that this theory makes only three kinds of physical predictions:
1) that the inner product of physical states 
is constant
in time; 2) that expectation values vary in time as 
; and 3) that probability amplitudes for measuring A to be
the eigenvalue 
vary in time (since operator eigenstates stay
constant) as 
.
We noted that these three physical predictions had an alternative
explanation. We could assume that, as time evolves, the physical
states 
remain constant, while the operators evolve
according to 
(which implies that their
eigenstates evolve as 
). These
assumptions -- called the Heisenberg picture -- reproduce the same 3
physical predictions above, and so give a physically indistinguishable
version of quantum mechanics.
We derived the Heisenberg equation of motion, a differential equation
for the time dependence of operators in the Heisenberg picture. This
told us that if the operator A has no explicit time-dependence, its
time derivative is given by 
times the commutator 
. Thus operators are constant if and only if they commute
with the Hamiltonian. Combining this result with our discussion of
symmetries above gives the quantum mechanical version of Noether's
theorem: every Hermitian charge operator that commutes with the
Hamiltonian is conserved, and generates a symmetry transformation
.
We then showed how identities you will prove in homework imply that
Heisenberg's equation of motion, when applied to position and momentum
operators, produce an operator version of Hamilton's equations. We
thus know that, in the classical limit where 
goes to zero
and operators give their expectation values, we will always recover
classical mechanics.
Finally, we began discussion of the quantum simple harmonic
oscillator. We noted that the classical Hamiltonian, when promoted
into an operator, is second order. A frequent approach with second
order differential operators is to factor them into first order
operators, to make solving the eigenvalue problem easier. We attempted
a naive factorization into the (non-Hermitian) factors 
and
. We found that noncommutation of 
and
in the cross-terms spoils the exact factorization, giving us
a Hamiltonian proportional to the sum 
.
-- KB