Class calendar, week 4 Mon., 9/20:
# Let's get sophisticated and not put hats on operators anymore unless
we really need to to avoid confusion. Also we can denote the X and P operators
by the same symbol as their eigenvalues x and p,
since the context will (almost?) always indicate whether the operator
or the eigenvalue is intended.
# Consider Hamiltonians of the form H = p^2/2m + V(x).
# The Schrodinger eqn in position space is then
i hbar (d/dt)<x |w> = <x|H|w> = [-hbar^2
nabla^2 + V(x)] <x|w>.
# The Schrodinger eqn in momentum space could be obtained
just by Fourier transforming
the position space eqn. Instead, let's start from scratch:
i hbar (d/dt)<p |w> = <p|H|w> = p^2/2m
<p|w> + int dp' <p|V(x)|p'><p'|w>,
where the pp' matrix element of the potential operator is
<p|V(x)|p'> = int dx' <p|V(x)|x'><x'|p'>
= int dx'/(2pi hbar)^3 V(x') exp[i(p'-p)x'/hbar] =
Fourier transform of V evaluated at (p-p')/hbar, divided by ((2pi)^1/2
hbar)^3 (or (2pi)^1/2 hbar
in one dimension). In this form, the momentum space Schrodinger eqn
is an integro-differential eqn.
# The asymmetry is because the kinetic energy term is local in momentum
space, while the potential
is local in position space.
# We could do things more symmetrically by using <p|x|w>= i hbar
(d/dp) <p|w>, which would mean we would replace V(x) by V(i hbar
d/dp) in the momentum space Schrodinger eqn. This is a good idea if the
function V(x) is simple enough, say linear or quadradic in x (or
in some other special cases). Otherwise we wind up with a higher order
differential eqn.
Wed., 9/22:
# <p| exp(ikx)|p'> = exp(ik(i hbar (d/dp))<p|p'> = exp(-hbar
k (d/dp))<p|p'> = <p- hbar k| p'>
= delta(p - hbar k - p'), where the second to last step follows from
Taylor's theorem.
We knew this result anyway, since exp(i p_0 x/hbar) is the momentum
shift operator (by p_0).
# Alternatively, the Fourier transform expression of <p|V(x)|p'>
from Monday's minutes
yields the integral representation of the delta function:
int (dx/2pi) exp(iqx) = delta(q).
In hw3 you show the limiting sense in which this identity in the box
is true, when a convergence factor is inserted in the integral. In class
I explained the justification of this:
Consider int dp' <p|V(x)|p'><p'|v>. The matrix
element <p|V(x)|p'> need only
make sense as an integrand in the p' integral, i.e. it can be a "generalized
function" or "distribution"
(such as a delta function), as the mathematicians call it. If
you try to evaluate it by itself it may
be ill-defined as a function. Suppose you insert a set of x' states:
int dx' int dp' <p|V(x)|x'><x'|p'><p'|v>.
By doing the p' integral first you get int dx' <p|V(x)|x'><x'|v>.
Now you can use what you
know about the wavefunction <x'|v>: you want it to be normalizable,
so it must vanish as |x'| goes
to infinity, so you can insert a factor exp(-bx') into the integrand
without changing anything
as long as b is small enough. Having done so, however, you now see
that you can instead do the x' integral
BEFORE the p' integral provided you keep b nonzero, then do the p'
integral, then take the
limit as b goes to zero. Applying this to V(x)=exp(ikx), the exp(-bx')
provides the needed
convergence factor. When we calculate, we rarely need to operate so
carefully however.
We just use the formula in the box above for the integral representation
of the delta function,
without any convergence factors, and cross our fingers. It almost always
works OK, but every once in a while we get an ill-defined or infinite result.
In those cases we go back to basics and figure out what we really should
have been doing. I don't know of any case where nothing obviously goes
wrong but we still
get the wrong result.
# EPR criterion for an element of reality: If, without in
any way disturbing a system, we can predict with certainty (i.e., with probability equal to unity) the value
of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity. Einstein, Podolsky and Rosen (EPR, Phys. Rev. 47 (1935)
777) argued that quantum mechanics does not contain a counterpart
for every element of physical reality, so it is not complete as
a physical theory.
J.S. Bell showed that quantum mechanics is in fact inconsistent with
the assumption that all such
elements of physical reality exist. He did this by showing that if
the elements of reality exist then
Bell's inequality holds, which restricts the strength of certain
correlation probabilities. Both quantum mechanics and experiments violate
this inequality. Around 1990 a simple example was found in quantum mechanics
showing directly, without probabilities, that quantum mechanics is inconsistent
with the existence of the elements of reality. This is the example of the
GHZ (Greenberger, Horne, Zeilinger) three-spin state.
[For nice discussions see N. David Mermin , Amer. J. Phys.
58 (1990) 731; Physics Today, June 1990, p. 9; H. J. Bernstein,
Found. Phys., 29 (1999) 521.] Bernstein's
derivation is explained in a supplement
I've written.
Fri., 9/24:
# In case you are bothered by our sloppy treatment of infinite
dimensional Hilbert space,
a recent paper discussing the mathematical subtleties and the limitations
or ambiguities of Dirac notation
is quant-ph/9907069, Dirac's
formalism and mathematical surprises in quantum mechanics by F.
Gieres.
I suggest that you look into this now only if you cannot restrain yourself!
# I discussed supplement EPR&GHZ I (Non-locality
of quantum mechanics: the GHZ example) on which
a previous homework problem was based. I mentioned Mermin's point
of view, in which the meaning of
all this is that "Correlations have physical reality; that which they
correlate does not". This viewpoint is
described in his paper, What
is quantum mechanics trying to tell us? ( N. D. Mermin, Am. J. Phys. 66, 753-767 (1998), or quant-ph/9801057).
# I discussed the supplement Teleportation.
The idea is that if two qubits B and C are properly entangled,
the state of a qubit A can be transferred to qubit C by measuring the
state of the joint system (AB) in an
appropriate basis of entangled states. For each of the possible results
of this measurement, there is a
unitary transformation that can be applied to C to put it in the same
state A was originally in. In the
process, the state of A was never measured, and in fact A has become
entangled with B and is no longer in a definite state by itself.
# Ehrenfest's theorem: For a Hamiltonian of the form H= p^2/2m
+ V,
d<x>/dt = <p>/m, and d<p>/dt = - < grad V>.
We derived this by showing that for any time independent
operator A, the Schrodinger eqn implies d<A>/dt = (1/i hbar) <
[A,H] >, and evaluting the commutators
[x^i,H] = i hbar p^i /m, [p^i, H] = -i hbar V,i (where V,i is
the derivative of V wrt x^i).
# In the process I introduced the Einstein summation convention:
repeated indices are assumed summed
over unless otherwise indicated, so, e.g., "p dot p" can be written
p^j p^j.
# Note <V,i> is not the same as the partial derivative
of V(<x>) wrt <x^i>. How different are they?
Let's work in one dimension:
If V = V_0 + V_1 x + V_2 x^2 + V_3 x^3 + ... then
<dV/dx> =
V_1 + 2V_2 <x> + 3V_3 <x^2> + ... , whereas
dV(<x>)/d<x> = V_1 + 2V_2 <x> + 3V_3 <x>^2 + ....
For linear or quadratic potentials they are they same (e.g. for a uniform
force or a harmonic oscillator),
but in general they are different. At lowest order this difference
is determined by the dispersion of the
position, (Delta x)^2 = <x^2> - <x>^2, i.e., the mean square
deviation from the mean.
# Uncertainty relation: (Delta x)(Delta p) >= hbar/2.
Since p= hbar k, this is equivalent to
(Delta x)(Delta k) >= 1/2. A handwaving argument why: if a wavepacket
with wavevectors
between k and k + Delta k is localized in an interval Delta x, then
it must be true that, over a distance
Delta x, exp(ikx) and exp(i(k+Delta k)x) can interfere destructively.
this means that
(k+Delta k)(Delta x) - k(Delta x) must be at least of order unity,
i.e. (Delta x)(Delta k) >~ O(1).
# In fact, we have a very precize lower limit on the product of the
uncertainties (rms deviations).
This is a special case of a general result:
# For two self-adjoint ops A and B in any state, the general uncertainty
relation holds: