General Uncertainty Principle
Michael Fowler 12/
9/06
Uncertainty and Non-Commutation
As we discussed in the Linear Algebra lecture, if two physical variables correspond to commuting Hermitian operators, they can be diagonalized simultaneously—that is, they have a common set of eigenstates. In these eigenstates both variables have precise values at the same time, there is no “Uncertainty Principle” requiring that as we know one of them more accurately, we increasingly lose track of the other. For example, the energy and momentum of a free particle can both be specified exactly. More interesting examples will appear in the sections on angular momentum and spin.
But if two operators do not commute, in general one cannot
specify both values precisely. Of
course, such operators could still have some common eigenvectors, but
the interesting case arises in attempting to measure A and B
simultaneously for a state 
in which the
commutator 
has a nonzero
expectation value, 
.
A Quantitative Measure of “Uncertainty”
Our task here is to give a quantitative analysis of
how accurately noncommuting variables can be measured together. We found
earlier using a semi-quantitative argument that for a free particle, 
at best. To improve on that result, we need to be
precise about the uncertainty 
in a state 
We define as the root mean
square deviation:
To make the equations more compact, we define 
by
(We’ll put a caret (a hat) on the to remind ourselves
it’s an operator—and, of course, it’s a Hermitian operator, like A.) We also drop the 
bra and ket, on the
understanding that this whole argument is for a particular state. Now
Introduce an operator B in exactly similar fashion, 
having the property
that .
The Generalized Uncertainty Principle
The quantitative measure of how the combined “uncertainty” of measuring two variables relates to their lack of commutativity is most simply presented as a
Theorem:
(Remark: remember that for A, B Hermitian, is antiHermitian:
so 
is real! To make this clear, take adjoints: if H
is hermitian then 
so 
is real. If K
is anti Hermitian, 
then 
from which 
is pure imaginary.)
Proof of the Theorem:
Define
Then
Using Schwartz’s inequality
gives immediately
The operator 
is neither Hermitian nor antiHermitian. To evaluate the mod squared of its
expectation value, we break the amplitude into real and imaginary parts:
.
(The first term on the right-hand side is the expectation value of a Hermitian matrix, and so is real, the second term is the expectation value of an antiHermitian matrix, so is pure imaginary.)
It follows immediately that
.
But
so
Minimizing the Uncertainty
For a particle in one dimension denote
(It important in that last step to understand that the
operator 
operates on everything
to its right, and, as we are always finding matrix elements of operators, there will be a following ket it operates on,
so 
)
We conclude that
Question: Is there a wavefunction for which this
inequality becomes an equality?
That would require 
which can only be true
if the two vectors are parallel, 
or, written
explicitly,
Actually , that’s not enough: we also need 
to be zero. (Look at
the equation above giving 
in terms of its real
and imaginary parts, and how we used it to establish the inequality.)
Writing 
as 
we find
so this will be zero if and only if 
is pure imaginary.
Turning to the differential equation, we first take the
simplest case where 
and 
are both zero. The
first requirement just sets the origin, but the second stipulates that our wave
function has no net momentum.
For this simple case,
becomes
and recalling that is pure imaginary,
this is a Gaussian wave packet! It is
straightforward to check that the solution with and nonzero is
where 
is real, and C
is the usual Gaussian normalization constant.
Exercise: check this.
The conclusion is then that the Gaussian wave packet gives
the optimum case for minimizing the joint uncertainties in position and
momentum.
Note that the condition 
does not mean that is an eigenstate of
either 
, but it is an eigenstate of the nonHermitian
operator 
, with eigenvalue zero.
We shall soon see that this nonHermitian operator and its adjoint play
important roles in the quantum mechanics of the simple harmonic oscillator.