As the previous section discussed, the standard deviation is
a measure of the uncertainty of a property of a quantum system. The
larger the standard deviation, the farther typical measurements stray
from the expected average value. Quantum mechanics often requires a
minimum amount of uncertainty when more than one quantity is involved,
like position and linear momentum in Heisenberg's uncertainty
principle. In general, this amount of uncertainty is related to an
important mathematical object called the commutator
,
to be discussed in this section.
First, note that there is no fundamental reason why several quantities cannot have a definite value at the same time. For example, if the electron of the hydrogen atom is in a eigenstate, its total energy, square angular momentum, and -component of angular momentum all have definite values, with zero uncertainty.
More generally, two different quantities with operators and have definite values if the wave function is an eigenfunction of both and . So, the question whether two quantities can be definite at the same time is really whether their operators and have common eigenfunctions. And it turns out that the answer has to do with whether these operators “commute”, in other words, on whether their order can be reversed as in .
In particular, {D.18}:
Iff two Hermitian operators commute, there is a complete set of eigenfunctions that is common to them both.(For more than two operators, each operator has to commute with all others.)
For example, the operators and of the harmonic oscillator of chapter 4.1.2 commute:
The same way, commutes with and , and that means that commutes with them all, since is just their sum. So, these four operators should have a common set of eigenfunctions, and they do: it is the set of eigenfunctions derived in chapter 4.1.2.
Similarly, for the hydrogen atom, the total energy Hamiltonian , the square angular momentum operator and the -component of angular momentum all commute, and they have the common set of eigenfunctions .
Note that such eigenfunctions are not necessarily the only game in town. As a counter-example, for the hydrogen atom , , and the -component of angular momentum also all commute, and they too have a common set of eigenfunctions. But that will not be the , since and do not commute. (It will however be the after you rotate them all 90 degrees around the -axis.) It would certainly be simpler mathematically if each operator had just one unique set of eigenfunctions, but nature does not cooperate.
Key Points
- Operators commute if you can change their order, as in .
- For commuting operators, a common set of eigenfunctions exists.
- For those eigenfunctions, the physical quantities corresponding to the commuting operators all have definite values at the same time.
The pointer state
Two quantities with operators that do not commute cannot in general have definite values at the same time. If one has a definite value, the other is in general uncertain.
The qualification in general
is needed because there
may be exceptions. The angular momentum operators do not commute, but
it is still possible for the angular momentum to be zero in all three
directions. But as soon as the angular momentum in any direction is
nonzero, only one component of angular momentum can have a definite
value.
A measure for the amount to which two operators and do not
commute is the difference between and ; this difference
is called their “commutator” :
(4.45) |
A nonzero commutator demands a minimum amount of uncertainty
in the corresponding quantities and . It can be shown,
{D.19}, that the uncertainties, or standard
deviations, in and in are at least so
large that:
Key Points
- The commutator of two operators and equals and is written as .
- The product of the uncertainties in two quantities is at least one half the magnitude of the expectation value of their commutator.
This section will work out the uncertainty relationship (4.46) of the previous subsection for the position and linear momentum in an arbitrary direction. The result will be a precise mathematical statement of the Heisenberg uncertainty principle.
To be specific, the arbitrary direction will be taken as the -axis,
so the position operator will be , and the linear momentum
operator . These
two operators do not commute, is simply not the same as
: means multiply function by
to get the product function and then apply on that
product, while means apply on and then
multiply the resulting function by . The difference is found
from writing it out:
Comparing start and end shows that the difference between and
is not zero, but . By definition, this
difference is their commutator:
Because the commutator is nonzero, there must be nonzero uncertainty
involved. Indeed, the generalized uncertainty relationship of the
previous subsection becomes in this case:
(4.48) |
It implies that when the uncertainty in position is narrowed down to zero, the uncertainty in momentum must become infinite to keep their product nonzero, and vice versa. More generally, you can narrow down the position of a particle and you can narrow down its momentum. But you can never reduce the product of the uncertainties and below , whatever you do.
It should be noted that the uncertainty relationship is often written
as or even as
where and
are taken to be vaguely described
uncertainties
in momentum and position, rather than
rigorously defined standard deviations. And people write a
corresponding uncertainty relationship for time,
, because relativity suggests that time
should be treated just like space. But note that unlike the linear
momentum operator, the Hamiltonian is not at all universal. So, you
might guess that the definition of the uncertainty
in time would not be universal either, and you would be
right, chapter 7.2.2.
Key Points
- The canonical commutator equals .
- If either the uncertainty in position in a given direction or the uncertainty in linear momentum in that direction is narrowed down to zero, the other uncertainty blows up.
- The product of the two uncertainties is at least the constant .
This sounds serious! If I am driving my car, the police requires me to know my speed (linear momentum). Also, I would like to know where I am. But neither is possible according to quantum mechanics.
It is a fact of life in quantum mechanics that commutators pop up all over the place. Not just in uncertainty relations, but also in the time evolution of expectation values, in angular momentum, and in quantum field theory, the advanced theory of quantum mechanics used in solids and relativistic applications. This section can make your life easier dealing with them. Browse through it to see what is there. Then come back when you need it.
Recall the definition of the commutator of any two operators
and :
Everything commutes with itself, of course:
The commutator is antisymmetric
; or in simpler words,
if you interchange the sides; it will change the sign,
{D.20}:
To deal with commutators that involve products of operators, the
rule to remember is: “the first factor comes out at the front
of the commutator, the second at the back”. More precisely:
Now from the general to the specific. Because changing sides in a
commutator merely changes its sign, from here on only one of the two
possibilities will be shown. First the position operators all
mutually commute:
The linear momentum operators all mutually commute:
A generalization that is frequently very helpful is:
Unlike linear momentum operators, angular momentum operators do
not mutually commute. The commutators are given by the
so-called “
fundamental commutation relations:”
The angular momentum components do all commute with the square angular
momentum operator:
Just the opposite of the situation for linear momentum, position and
angular momentum operators in the same direction commute,
The commutators between linear and angular momentum are very similar
to the ones between position and angular momentum:
The following commutators are also useful:
Commutators involving spin are discussed in a later chapter, 5.5.3.
Key Points
- Rules for evaluating commutators were given.
- Return to this subsection if you need to figure out some commutator or the other.