Commutators and Uncertainty Principle
Discussing two effects of non-commuting operators: on the measurement process, and on the limitation of the wave function description of particles.
This lecture is motivated by the question:
To answer this question, we introduce the commutator of two Hermitian operators and
explore its physical interpretation. We will prove a generalisation of
Heisenberg’s
uncertainty principle, which is a fundamental limitation on the
precision that observables and
can be determined
simultaneously.
The Commutator
Two commuting operators share an (orthonormal) basis of eigenfunctions. This is extremely important, because eigenfunctions and eigenvalues are related to possible measurement outcomes.
Suppose we have two linear operators and , such as position , momentum or the Hamiltonian . The commutator is defined by It has the
following properties:
Anti-symmetry: .
Linear: .
is a derivation:
.
Jacobi identity: .
The commutator plays an important role in quantum mechanics due to
the following theorem:
Theorem: “Two commuting matrices
are simultaneously diagonalizable." If and are Hermitian operators with , it is possible to find an
orthonormal basis of wave functions that are simultaneous eigenfunctions
of and .
Proof: To keep things simple, we will
assume the spectrum of eigenvalues of is discrete and non-degenerate. This
means that up to normalisation there is a unique solution to for each
eigenvalue . The normalisation
can be chosen to make the basis orthonormal, .
We want to prove that
is simultaneously an eigenfunction of . The commutator is equivalent to . This means so is also an eigenfunction
with eigenvalue . But such wave function is unique up
to normalisation and therefore for some . So is simultaneously an
eigenfunction of .
Compatibility
Quantum operators which originate from classical observables which have a vanishing Poisson bracket are called ‘‘compatible''.
This theorem motivates the following definition.
Let us first determine whether the observables we have encountered so
far are compatible. Recall that the position and momentum operators are
while
Position and Momentum. Let us
compute the commutator of position and momentum acting on a generic wave
function, This is the “canonical commutation relation" which
shows position and momentum are not compatible.
Momentum and Energy. A similar
computation shows that So
momentum and energy are compatible only if is constant. We will return to this
result later in the course.
Position and Energy. From the
commutator of position and momentum, so position and energy are never
compatible.
Compatibility and Measurement
To understand the physical significance of compatibility, suppose we
measure and find the eigenvalue
. Then the wave function
immediately after the measurement is the eigenfunction .
If and do not commute, and we measure , immediately after that, the uncertainty of has to be nonzero. This is a statement about subsequent measurements.
What happens if we make another measurement of immediately afterwards? As is an eigenfunction of , the measurement will again find with probability . Correspondingly, the uncertainty of a
normalised eigenfunction vanishes,
What happens if we make a measurement of immediately afterwards?
If , is also an eigenfunction of
with eigenvalue . So a measurement of will find with probability . Correspondingly, the same argument as
above shows that . In
other words, we can simultaneously determine the values of both and .
If , is not an eigenfunction of
and there are multiple potential
outcomes. Suppose the measurement of yields a particular eigenvalue, say
. Then the wave function jumps to
the corresponding eigenfunction of with . But now .
It is important to emphasise that the subsequent measurements
discussed in this section are measurements without time between
them. A generic eigenstate of an operator will not stay an eigenstate
under time evolution, so a measurement of , followed by a delay and then another
measurement of , will typically
still lead to a non-zero spread. Only eigenstates of the Hamiltonian
operator, or operators which commute with it, remain eigenstates under
time evolution.
The Generalised Uncertainty Principle
If , we cannot
necessarily find simultaneous eigenfunctions of and with both and . In fact, there is a fundamental limitation in quantum
mechanics on the how small we can simultaneously make the uncertainties
and . This is quantified by the
“Generalised Uncertainty Principle”:
The generalised uncertainty principle relates the product of uncertainties of two operators to their commutator. It says something about the spread of two observables in the same state, not about subsequent measurements.
Theorem: For any square-normalisable
wave function,
Proof: We will assume here that
for the wave function in question. This will simplify the argument
without losing any of its essence. The translation to , as an exercise
for the interested reader.
With our assumption, the uncertainty in can be expressed, where . There is an identical statement for and therefore we can write We can
now use the Cauchy-Schwarz inequality, This result holds for any Hermitian inner
product. It is analogous to the standard result from real euclidean geometry, which
follows from the formula for the dot product.
The right-hand side of this inequality can be expressed as where is the commutator and is the “anti-commutator". It is straightforward to
check that,
is anti-Hermitian
.
is Hermitian .
so the commutator and anti-commutator provide the imaginary and real
parts of . Recalling the formula for the modulus squared
of a complex number , we
have This concludes the proof.
Position and Momentum. For position
and momentum, which is Heisenberg’s uncertainty
principle.
Momentum and Energy. For momentum
and energy, which vanishes
automatically when is
constant.
Position and Energy. For position
and energy, This implies that
square-normalisable Hamiltonian eigenfunctions must have . In fact,
square-normalisable Hamiltonian eigenfunctions may always be chosen
real, compatible with this statement.
Note that what is derived here is a statement which is different from
the one in the previous section: the generalised uncertainty principle
as derived above says nothing about subsequent measurements (see also
Jacques Distler and Sonia Paban.
Uncertainties in successive measurements.
Physical Review A, Jun 2013.
URL: http://dx.doi.org/10.1103/PhysRevA.87.062112, doi:10.1103/physreva.87.062112.).
The uncertainty relation always contains on the right hand side, and is thus fundamentally ‘‘quantum''.
It should be emphasised that this is a fundamental feature of quantum
mechanics. Only in the classical limit, , can we simultaneously
determine exactly the values of non-compatible observables such as
position and momentum.
Problems
- Generalised uncertainty principle:
Derive the generalised uncertainty principle, where and similar for .
- Energy-position uncertainty relation:
Show that measurements of position and measurements of energy of a
particle in one dimension satisfy the uncertainty relation
TUTORIAL 3
Time-energy uncertainty relation (sort of)
For an operator which does
not depend on time explicitly, we have For
small and small
standard-deviation we can
write What is the meaning of here? Use to derive the
“time-energy uncertainty relation” . What does this ‘uncertainty relation’ express?
Solution ▶
Writing and ditto for we can write Setting
we then have
For the particular we have and similar for
the conjugate. Using that for the difference then gives the requested
result.
This requires computing and thus from which
the relation follows.
Insertion of the Schrödinger equation into the uncertainty
relation gives Then
inserting the definition of and rearranging factors produces the uncertainty relation.
The symbol is the
time it takes the expectation value of the (arbitrary) operator to change by one standard
deviation.
The uncertainty relation expresses the fact that if all
observables change rapidly ( small), then the uncertainty in the energy must be large, and
if all observables change slowly, the uncertainty in the energy is
small.
See Griffiths for more detail and discussion.