Comments on Problem Set 13, and Mathematic Code for the Harmonic Oscillator.
PART A: What was to be proved was that
HDAL2
= <
A2
> -
H<A>L2
.
This is a general result, for any normalized wave function Y. In particular there is/was no reason to assume anythign special
about Y, and in particular it was not stated that Y was an eigenfuction of A, so there was no reason to assume this. The special
case where Y is an eigenfunction of A is important, but it is what you were asked to consider in PART Bi), below.
Proof:
HDAL2
ª
<HA- < A>L2>
. Expanding the square this becomes
HDAL2
= <(
A2
- <A> A - A<A> + <A> <A>)>. Now < > implies that a definite integral is being performed.
Constants, such as <A> and
H<A>L2
factor out of such integrals, so, using the fact that the integral of a sum is the sum of
the integrals we have:
HDAL2
= <(
A2
- <A> A - A<A> + <A> <A>)> = <
A2
> -<A> <A> - <A> <A> +
H<A>L2
= <
A2
> -
H<A>L2
. QED
A comment on notation: Several students wrote <A> =
ŸtAop PHtL „ t
, where I assume that P(t) =
Y*
(t)Y(t).
This is a dangerous notation, as if
Aop
is a differential operator this expression is NOT CORRECT! In "classical statistics" where
there are "no operators" it's OK, but in QM, a differential operator MUST be placed between
Y*
(t) and Y(t,) or incorrect results
follow. THUS the postulate states:
<A> =
ŸtY*HtL Aop YHtL „ t
.
This is NOT an accident, and so is the form to be used in quantum theory!
PART B i): NOW iff Y is an eigenfunction of A with eigenvalue
a,
we have
HDAL2
= <
A2
> -
H<A>L2
= <
a2
> -
H<a>L2
=
a2
-
a2
= 0. QED This is indeed an important result, as it says, in words: if a
system is in an eigenstate of an observable we can make measurements with "certainty."
Harmonic Oscillator:
Here is an "optional" oscillator program for Mathematica fans. Mathematia may be found in many places on campus, for exam -
ple, the CHEM STUDY CENTER. Note that for intelligible plots which take as the zero of the Wave Function the quantum
energy "n+1/2" the normalalized funtion is multiplied by "n" to keep it visible on this scale. Units are such that alpha, m, and k
and h are all "one" so no dimensions show.
Y@n_, x_D:=
Hn+1L*HH1êpL^H1ê4L*1êSqrt@2^n*Hn!LDL *HermiteH@n, xD*Exp@-x^2ê2D+Hn+1ê2L
