Physics 6210/Spring 2007/Lecture 7
Lecture 7
Relevant sections in text:
§
1.6
Observables with continuous and/or unbounded values
We are now ready to turn to the quantum mechanical description of a (nonrelativistic)
particle. We shall define a (spinless) particle as a system that is completely characterized
by the position and linear momentum, which are the basic observables in this model.
This means that all observables are functions of position and momentum.
While it is
possible that the position and momentum variables take a discrete set of values (as angular
momentum and – often – energy do), there is currently no experimental evidence of this.
We therefore create a model in which these observables can take a continuous, unbounded
set of values. Evidently, we need to define a Hilbert space that admits selfadjoint operators
with a continuous, unbounded spectrum. Neither of these features are possible on finite
dimensional vector spaces, and so here we are forced into the infinitedimensional setting
(
i.e.,
spaces of functions). This leads to some mathematical subtleties that we need to be
wary of. I shall not try to be perfectly rigorous in our discussion since that would take us
too far afield. But I will try to give you a reasonably foolproof – if somewhat formal –
treatment. First, let me give you a recipe for dealing with the situation. Then let me give
you a flavor of the underlying mathematics which makes the formal recipe work (and also
which shows where it can become tricky).
Formal recipe
We want to define an observable
A
with a continuous, unbounded set of values
a
∈
R
,
say. We postulate the existence of a selfadjoint linear operator, again denoted
A
, and a
continuous set of vectors

a
such that
A

a
=
a

a .
We say that
A
has a “continuous” spectrum. These vectors are to be “orthonormal in the
deltafunction sense”:
a

a
=
δ
(
a, a
)
.
You can think of this as a continuum generalization of the usual orthonormality expressed
via the Kronecker delta.
The vectors

a
are to form a basis, that is, they provide a
continuous resolution of the identity:
I
=
∞
∞
da

a
a

,
1
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Physics 6210/Spring 2007/Lecture 7
so that we can write

ψ
=
∞
∞
da

a
a

ψ .
Using quotation marks to indicate where the underlying mathematics is considerably
more subtle than the words indicate, we can say the following. We interpret the “eigenval
ues”
a
as the possible values of an outcome of the measurement of
A
, the vectors

a
are
“states” in which
A
has the value
a
with certainty. The scalar

a

ψ

2
da
is the probability
for finding
A
to have the value in the range [
a, a
+
da
] in the state

ψ
. In particular, the
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 Spring '07
 M
 Physics, mechanics, wave function, Hilbert space, Ψ, Kronecker delta

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