(2002-11-01) The Quantum Substitute for Logic
How is the probability of an outcome computed in quantum theory?
-
If you are not completely confused by
quantum
mechanics,
you do not understand it.
John Archibald Wheeler (b. 1911)
First, let's consider how probabilities are ordinarily
computed:
When an event consists of two mutually exclusive events,
its probability is the sum of the probabilities of those two events.
Similarly, when an event is the conjunction of two statistically independent events,
its probability is the product of the probabilities of those two events.
For example, if you roll a fair die, the probability of obtaining a multiple of 3 is
1/3 = 1/6+1/6; it's the sum of the probabilities (1/6 each)
of the two mutually exclusive events "3" and "6".
You add probabilities when the
component events can't happen together (the outcome of the roll cannot be
both "3" and "6").
On the other hand, the probability of rolling two fair dice without obtaining a 6
is 25/36 = (5/6)(5/6); it's the product of the probabilities
(5/6 each)
of two independent events,
each consisting of not rolling a 6 with one throw of a die.
Quantum Logic and [Complex] Probability Amplitudes :
In the quantum realm, as long as two logical possibilities are not actually observed,
they can be neither exclusive nor independent
and the above does not apply.
Instead, so-called probability amplitudes are defined as
complex numbers whose
absolute values squared correspond to ordinary probabilities.
The angular components of such complex numbers have no classical equivalents
(although they happen to provide a deep explanation for the existence
of the conserved classical quantity known as electric charge).
To obtain the amplitude of an event with two unobserved logical components:
- For EITHER-OR components, the two amplitudes are added.
- For AND components, the two amplitudes are multiplied.
In practice, "AND components" are successive steps that would logically lead to the desired
outcome and make up what can be called an acceptable history for that outcome.
The "EITHER-OR" components, whose amplitudes are to be added,
are thus all the possible histories logically leading up to the same outcome.
Following Richard Feynman,
the whole thing is therefore called a "sum over histories".
These algebraic manipulations are a mind-boggling substitute for statistical logic,
but that's the way the physical universe appears to work.
The above quantum logic normally applies only at the microscopic level,
where "observation" of individual components is either impossible or would
introduce an unacceptable disturbance.
At the macroscopic level, the observation of a combined outcome usually implies that
all relevant components are somehow "observed" as well (and the ordinary
algebra of probabilities applies).
For example, in our examples involving dice, you cannot tell
if the outcome of a throw is a multiple of 3 unless you actually observe
the precise outcome and will thus know if it's a "3" or a "6",
or something else.
Similarly, to know that you haven't obtained a "6"
in a double throw, you must observe separately the outcome of each throw.
Surprisingly enough, when the logical components of an event are only imperfectly
observed (with some remaining uncertainty), the probability of the outcome
is somewhere between what the quantum rules say
and what the classical rules would predict.
(2002-11-01) The Infamous Measurement Problem
What does a quantum observation entail?
This is arguably the most fundamental unsolved question in quantum mechanics.
According to the above, one should deal strictly
with amplitudes between observations (or measurements),
but another recipe holds when measurements are made.
That would be fine if we knew exactly what a measurement entails,
but we don't...
Should we really assume that a system can only be measured by some outside agency
(the observer)?
If we do, nothing prevents us from considering a larger system that includes this
observer as well, and that system's evolution would involve only
measurement-free quantum rules.
If we don't assume that, we can't avoid the conclusion that a system can
observe itself, in some obscure sense.
Either way, the simple quantum rules outlined above would have to be smoothly modified
to account for a behavior which can be nearly classical for a large enough system.
In other words, current quantum ideas must be incomplete, because they fail to describe
any bridge between a quantum system waiting only to be observed,
and an entity capable of observation.
Our current quantum description of the world has proven its worth and reigns supreme,
just like Newtonian mechanics reigned supreme
before the advent of Relativity Theory.
Relativity consistently bridged the gap between the slow and the fast,
the massive and the massless (while retaining the full applicability of
Newtonian theories to the domain of ordinary speeds).
Likewise, the gap must ultimately be bridged
between observer and observed,
between the large and the small,
between the classical world and the quantum realm,
for there is but one single physical reality in which everything
is immersed...
This bothers [or should bother] everybody who deals with quantum mechanics:
The so-called Schrödinger's Cat theme is often used to discuss the
problem, in the guise of a system that includes a cat (a "qualified" observer)
in the presence of a quantum device which could trigger a lethal device.
It seems silly to view the whole thing as a single quantum system,
which would only exist (until observed) in some superposition of states,
where the cat would be neither dead nor alive, but both at once.
Something must exist which collapses the quantum state of a large enough system
frequently enough to make it appear "classical".
It stands to reason that Schrödinger's Cat must be dead
very shortly after being killed...
Doesn't it?
(2005-07-03)
Matrix Mechanics (1925)
Physical quantities are multiplied like matrices... Order matters.
In 1925, Werner Heisenberg
(1901-1976; Nobel 1932)
discovered that observable physical quantities
obey noncommutative rules similar to those governing
the multiplication of algebraic matrices.
If the measurement of a physical quantity would disturb the measurement of the
other, then a noncommutative circumstance exists which disallows even the
possibility of two separate sets of experiments yielding
the values of these two quantities with arbitrary precision (read this again).
This delicate connection between
noncommutativity and uncertainty
is now known as Heisenberg's uncertainty principle.
In particular, the position and momentum of a particle can only be measured with respective
uncertainties
(i.e., standard deviations in repeated experiments)
Dx and
Dpx satisfying the following inequality :
Dx Dpx
³
h/4p
[where h is
Planck's constant]
The early development of Heisenberg's Matrix Mechanics
was undertaken by M. Born and P. Jordan.
The theory was given its current form by Paul Dirac.
In March 1926, Erwin Schrödinger established that Heisenberg's
viewpoint was equivalent to his own "undulatory" approach
(Wave Mechanics, January 1926) for which he
would share the 1933 Nobel prize with Paul Dirac.
Heisenberg's Viewpoint [skip on first reading]
For the record, we'll give a brief summary of Heisenberg's approach
in terms of the so-called Schrödinger viewpoint which is adopted
here, following Dirac and virtually all modern scholars...
In the modern Schrödinger-Dirac perspective, a
ket |y> is
introduced which describes a quantum state
varying with time.
Since it remains of unit length, its value at time t is obtained
from its value at time 0 via a unitary
operator Û.
| yt >
=
Û (t,0)
| y0 >
The unitary operator Û so defined is called the
evolution operator.
Heisenberg's viewpoint consists in considering that a given system is represented
by the constant ket
Û* |y>.
Operators are modified accordingly...
A physical quantity which is associated with the operator Â
in the Schrödinger viewpoint
(possibly constant with time) is then associated with the following
time-dependent operator in the Heisenberg viewpoint.
Û* Â Û
=
Û-1 (t,0)
Â Û (t,0)
(2002-11-02)
The Schrödinger Equation (1926)
The dance of a single nonrelativistic particle
in a classical force field.
The Schrödinger equation governs the
probability amplitude y
of a particle of mass m and energy E
in a space-dependent potential energy V.
Strictly speaking, E is the total relativistic
mechanical energy
(starting at mc2 for the particle at rest).
However, the final stationary Schrödinger equation
(below) features only the
difference E-V with respect to the potential V,
which may thus be shifted to incorporate the rest energy
of a single particle.
For several particles, the issue cannot be skirted so easily
(in fact, it's partially unresolved) and it's one of several reasons
why the quantum study of multiple particles takes the form of an
inherently relativistic theory (Quantum Field Theory,
which also accounts for the creation and anihilation of particles).
In 1926, when the Austrian physicist
Erwin
Schrödinger
(1887-1961; Nobel 1933)
worked out the equation now named after him, the notion of probability amplitude
presented in the above article was not yet clear.
At first, Schrödinger wrongly thought that the quantity
y
he introduced for the electron might indicate the density of its electric charge.
The modern interpretation of y
in terms of probability amplitudes
was proposed by
Max Born
(1882-1970; Nobel 1954):
The squared length of y
is proportional to the probability
of actually finding the electron at a particular position in space, if you were to
locate it with utmost precision.
This hindered neither the early development of Schrödinger's theory of
"Wave Mechanics",
nor the derivation of the nonrelativistic equation at its core.
A Derivation of the Schrödinger Equation :
We may start with the expression of the phase-speed, or celerity
u = E/p of a matter wave, which comes directly from
de Broglie's principle, or less directly from other
more complicated analogies between particles and waves.
The nonrelativistic defining relations
E = V + ½ mv 2 and
p = mv imply:
|  |
| p = Ö |
2m (E-V) |
The standard way to solve the general wave equation in 3-dimensional
space is to first obtain solutions j which are
products of a time-independent space function y
by a sinusoidal function of the time (t) alone.
The general solution is simply a linear superposition of these
stationary waves :
j =
y exp (
-2pin t )
For a frequency n, the stationary amplitude
y thus defined must satisfy:
Dy +
( 4pn2 / u2 )
y = 0
Using n = E/h
(Planck's formula) and the above for
u = E/p we obtain...
The Schrödinger Equation :
| Dy +
(8 p2 m / h2 )
(E - V) y
= 0
|
This superb equation is best kept in its nonrelativistic context,
where it does determine allowed levels of
energy in relative terms (within an additive constant).
A frequency may only be associated with a Schrödinger solution
at energy E if E is the total relativistic energy (including rest energy)
and V has been ajusted accordingly, against the usual nonrelativistic freedom,
as discussed in this article's introduction.
In the above particular stationary case, we have:
E j
=
( i h / 2p )
¶j/¶t
This relation turns the previous equation into
a more general linear equation :
|
( i h / 2p ) ¶j/¶t
=
V j -
( h2 / 8p2 m )
Dj
|
Signed Energy and the Arrow of Time
Historically, Erwin Schrödinger associated an equally valid
stationary function with the positive (relativistic) energy
E = hn and obtained a
different equation :
j =
y
exp ( 2pin t )
( -i h / 2p ) ¶j/¶t
=
V j -
( h2 / 8p2 m )
Dj
Formally, a reversal of the direction of time turns one equation into the other.
We may also allow negative energies and/or frequencies in
Planck's formula E = hn
and observe that a particle may be described by the same wave function
whether it carries energy E in one direction of time, or energy
-E in the other.
To retain only one version of the Schrödinger equation and
one arrow of time (the term was coined by Eddington)
we must formally allow particles to carry a signed energy
(typically, E = ± mc2 ).
If the wave function j
is a solution of one version of the Schrödinger equation, then its
conjugate j* is a solution of the other.
However, time-reversal and conjugation need not result in the same wave
function whenever Schrödinger's equation has
more than one solution at a given energy.
Principle of Superposition :
The linearity of the Schrödinger equation implies that
a sum of satisfactory solutions is a satisfactory solution.
This is the
principle of superposition which is at the root of the more
general Hilbert space formalism introduced by Dirac:
Until it is actually measured,
a quantum state may contain
(as a linear superposition)
several acceptable realities at once.
(2002-11-02) Hamilton's Analogy: Paths to the
Schrödinger Equation
Equating the principles of Fermat and Maupertuis
yields the celerity u.
This is only of historical interest
(we don't recommend teaching the details of what follows)
but it's enlightening to see what type of arguments
Schrödinger developed when he had to
introduce the subject in the early years.
Here are some of the ideas which made the revolutionary concepts of wave mechanics acceptable
to physicists of a bygone era, including Erwin Schrödinger himself.
Schrödinger took seriously an analogy
attributed to William Rowan Hamilton (1805-1865)
which bridges the gap between well-known features
of two aspects of physical reality,
classical mechanics and wave theory.
Hamilton's analogy states that, whenever waves conspire to create the illusion
of traveling along a definite path (like "light rays" in geometrical optics),
they are analogous to a classical particle:
The Fermat principle for waves may then be equated with
the Maupertuis principle for particles.
Equating also the velocity of a particle with
the group speed of a wave, Schrödinger drew the
mathematical consequences of combining it all with Planck's quantum hypothesis
(E = hn).
These ideas were presented (March 5, 1928)
at the Royal Institution of London, to start a course of
"Four Lectures on Wave Mechanics"
which Schrödinger dedicated to his late teacher,
Fritz Hasenöhrl.
Maupertuis' Principle of Least "Action" (1744, 1750)
When a point of mass m moves at a speed v in a force field described
by a potential energy V (which depends on the position),
its kinetic energy is
T = ½ mv2
(the total energy E = T+V remains constant).
The actual trajectory from a point A to a point B turns out to be
such as to minimize the quantity that Maupertuis
(1698-1759) dubbed action, namely the integral
ò 2T dt.
(Maupertuis' Principle is thus also called the
Least Action Principle.)
Now, introducing the curvilinear abscissa (s) along the trajectory, we have:
2T = mv2 = m (ds/dt)2
= 2(E-V)
Multiply the last two quantities by m and take their square roots to obtain an expression
for m (ds/dt) , which you may plug back into the whole thing
to get an interesting value for 2T:
| | | | | |
| 2T = (ds/dt) Ö |
2m (E-V) | so the action
is: | ò Ö | 2m (E-V) | ds |
The time variable (t) has thus disappeared from the integral to be minimized,
which is now a purely static function of the spatial path from A to B.
Fermat's Principle: Least Time (c. 1655)
When some quantity j propagates in
3 dimensions at some celerity u
(also called phase speed), it verifies the well-known wave equation:
| 1 | |
¶ 2 j |
= |
¶ 2 j |
+ |
¶ 2 j |
+ |
¶ 2 j |
|
|
|
|
|
|
| u 2 |
¶ t 2 |
¶ x 2 |
¶ y 2 |
¶ z 2 |
| |
| | = | Dj |
[D is the Laplacian operator] |
The speed u may depend on the properties of the medium in which the "thing" propagates,
and it may thus vary from place to place.
When light goes through some nonhomogeneous medium with a varying
refractive index (n>1), it propagates at a speed
u = c/n and will travel along
a path (a "ray", in the approximation of geometrical optics) which is always such
that the time (òdt)
it takes to go from point A to point B is minimal
[among "nearby" paths].
This is Fermat's Principle, first stated by
Pierre
de Fermat (1601-1665) for light in the context of geometrical optics,
where it implies both the
law of reflection
and Snell's law
for refraction.
This principle applies quite generally to any type of wave,
in those circumstances where some path of propagation can be defined.
If we introduce a curvilinear abscissa s for a wave that follows some path in
the same way light propagate along rays [in a smooth enough medium],
we have u = ds/dt. This allows us to express the time it takes to go
from A to B as an integral of ds/u.
The conclusion is that a wave will [roughly speaking]
take a "path" from A to B along which the following integral is minimal:
ò 1/u ds
Hamilton's Analogy :
The above shows that, when a wave appears to propagate along a path,
this path satisfies a condition of the same mathematical form as that
obeyed by the trajectory of a particle.
In both cases, a static integral along the path has to be minimized.
If the same type of "mechanics" is relevant,
it seems the quantities to integrate should be proportional.
The coefficient of proportionality cannot depend on the position, but it may very well
depend on the total energy E (which is constant
in the whole discussion).
In other words, the proportionality between the integrand of
the principle of Maupertuis and its Fermat counterpart (1/u)
implies that the following quantity is a function of the energy E alone:
| |
| f (E) = u Ö |
2m (E-V) |
Combined with Planck's formula, the next assumption implies
f (E) = E ...
Schrödinger's Argument :
Schrödinger assumed that the wave equivalent of the speed v of
a particle had to be the so-called group velocity,
given by the following expression:
We enter the quantum realm by postulating Planck's formula :
E = hn.
This proportionality of energy and frequency turns the previous equation into:
On the other hand, since ½ mv2 = E-V,
the following relation also holds:
Recognizing the square root as the quantity we denoted f (E) / u
in the above context of Hamilton's analogy
[it's actually the momentum p, if you must know]
the equality of the right-hand sides
of the last two equations implies that the following quantity C
does not depend on E:
| |
| ( f (E) - E ) / u =
C =
[ 1 - E / f (E) ]
Ö |
2m (E-V) |
This means f (E) = E / ( 1 - C
[ 2m (E-V) ] -1/2 which is, in
general, a function of E alone only if C vanishes
(as V depends on space coordinates).
Therefore f (E) = E, as advertised,
which can be expressed by the relation:
| |
| u = E / Ö |
2m (E-V) |
In 1928, Schrödinger quoted only as "worth mentioning" the fact that
this boils down to u = E/p,
without identifying it as the nonrelativistic counterpart of
the formally identical relation for
the celerity u = ln
obtained from the 1923 expression
of a de Broglie wave's momentum
(p = h/l)
as used above.
English translations of the 9 papers and 4 lectures that Erwin Schrödinger published
about his own approach to Quantum Theory ("Wave Mechanics")
between 1926 and 1928 have been compiled in:
" Collected Papers on Wave Mechanics " by E. Schrödinger
(Chelsea Publishing Co., NY, 1982)
(2003-05-26) Noether's Theorem
In 1905, the German mathematician Emmy Noether
proved the following deep result,
which has since been named after her (Noether's Theorem):
For every continuous symmetry of the laws of physics,
there's a conservation law.
For every conservation law, there's a continuous symmetry.
(2005-06-27) Formalism of Hilbert Spaces:
Dirac's <bras| & |kets>
A nice symbolic notation with
built-in simplification features...
The standard vocabulary for the Hilbert spaces used in quantum mechanics
started out as a joke:
P.A.M. Dirac (1902-1984;
Nobel 1933)
decided to call < j | a bra
and | y > a ket,
to make it natural to call
< j | y >
a bracket.
Hilbert Space and "Hilbertian Basis" :
A Hilbert space is a vector space over the field of
complex numbers
(its elements are kets) endowed with an inner hermitian
product (Dirac's "bracket").
This is to say that the following properties hold
(z* being the complex conjugate of z):
- Hermitian symmetry:
< y | j >
=
< j | y >*
- Semilinearity:
< j | (
x | x > + y | y >)
=
x < j | x > +
y < j | y >
- For any nonzero ket | y >,
the real < y | y >
is positive
(= ||y|| 2 ).
A Hilbert space is also required to be separable and complete,
which means that its dimension is either finite or countably infinite.
It's customary to use raw indices for the kets of an agreed-upon
hilbertian basis :
| 1 >, | 2 >, | 3 >, | 4 > ...
Such a basis is a maximal
set of unit kets which are pairwise orthogonal :
< i | i > = 1 and < i | j > = 0
if i ¹ j
The so-called closure relation
Î = å
| n > < n |
is a nice way to state that any ket is a
generalized linear combination of kets from the basis.
(It need not be a proper linear combination,
since infinitely many of the coefficients
< n | y >
could be nonzero: A Hilbertian basis need not be a
Hamel basis.)
| y >
=
Î | y >
=
å
| n > < n | y >
=
å
< n | y > | n >
Operators :
A linear operator is a "square matrix"
 = [ aij ]
which we may express as:
 = å
aij | i > < j |
alternately,
aij =
< i | Â | j >
To the left of a ket or the right of a
bra, Â yields another like vector.
Hermitian Conjugation
(Conjugates, Duals, Adjoints) :
Hermitian conjugation generalizes to vectors and operators the
complex conjugation of scalars.
We prefer to use the same notation X* for the hermitian conjugate
of any object X, regardless of its dimension
We use interchangeably the terms which are preferred, respectively,
for scalars, vectors (bras and kets) and operators
namely "conjugate", "dual" and "adjoint".
On the other hand,
many authors use an overbar for the conjugate of a scalar and an obelisk
for the adjoint
A
of an operator A.
In other words,
A º A*
Loosely speaking, conjugation consists in replacing all coordinates by
their complex conjugates and
transposing (i.e., flipping about the main diagonal).
| y >* = < y |
and
< y |* = | y >
< j | Â* | y >
=
( < y | Â | j > )*
The adjoint of a product is the product of the adjoints in reverse order.
For an inner product, this merely restates
the axiomatic hermitian symmetry.
( X Y )* = Y* X*
< y | j >*
=
< j | y >
An operator  is self-adjoint or
hermitian if  = Â*.
All eigenvalues of an hermitian operator are real, and
two eigenvectors for distinct eigenvalues are orthogonal.
(In finitely many dimensions, such operators are diagonalizable.)
An hermitian operator multiplied by a real
scalar is hermitian.
So is a sum of hermitian operators,
or the product of two commuting hermitian operators.
The following combinations of two hermitian operators are always hermitian:
1/2
( Â Ê + Ê Â )
1/2i
( Â Ê - Ê Â )
Unitary Transformations Preserve Length :
A unitary
operator Û is a Hilbert isomorphism:
Û Û* = Û* Û = Î.
It turns
| y >,
< j | and
 (respectively) into
Û | y >,
< j | Û* and
Û Â Û*.
Û* transforms an orthonormal Hermitian basis into another such basis.
For an infinitesimal e,
Û = Î + ieÊ
is unitary (only) when Ê is hermitian.
State Vectors, Observables and the Measurement Postulate :
A quantum state, state vector, or microstate is a ket
| y > of unit length :
< y | y >
= 1
Such a ket | y >
is associated with the density operator
| y >
< y |
(whose entropy is zero) which determines it back,
within some phase factor exp(iq).
An observable physical quantity corresponds to an hermitian
operator  whose eigenvalues are the possible values of a
measurement. The average value of a measurement of
this observable from a pure microstate
| y > is:
< y | Â | y >
This is a corollary of the following measurement postulate,
which states the consequence of a measurement,
in terms of the eigenspace projector matching each possible outcome
(necessarily an eigenvalue a
of  = åa
a
Pa ).
|
| y > becomes
|
|
Pa
| y >
| | |
|| Pa
| y > ||
|
|
with probability < y |
Pa | y >
|
|
The above statement is often called the
principle of spectral decomposition.
(Note that, since P2 = P = P*,
we have || P | y > || 2
=
< y | P | y >.)
Vocabulary:
The principle of quantization limits the observed
values of a physical quantity to the eigenvalues of
its associated operator.
The principle of superposition
asserts that a pure quantum state is represented by a ket...
Nonrelativistic Postulate of Evolution with Time :
In nonrelativistic quantum theory, time (t) is not an observable in the
above sense, but a parameter with which things
evolve between measurements,
according to the following substitute for
Schrödinger's equation,
involving the hamiltonian operator H
(associated with the system's total energy) :
| i h |
|
d |
| y >
|
|
H
| y >
|
|
|
|
| 2p |
dt |
|
This is completely wrong unless Hamiltonians
are properly adjusted to incorporate rest energies
(see our discussion of Schrödinger's
equation).
(2005-07-03) Operators Corresponding to
Physical Quantities
Building on 6 operators for the
coordinates of position and momentum.
Only scalar physical quantities correspond to basic
observables (hermitian square matrices)
within the relevant
Hilbert space L.
For convenience, physical vectors may also be considered, which
correspond to operators mapping a ket into a vector of kets
(an element of some cartesian power of L ).
The following table embodies the so-called
principle of correspondence,
for those physical quantities which have a classical equivalent.
The so-called orbital angular momentum of a
pointlike particle does; its spin doesn't.
The historical
equation of Schrödinger
is retrieved from the postulated evolution of kets involving
the Hamiltonian H, in the following special case :
E = V(r) + ||p||2 / 2m
and
H(j) =
V j -
( h2/8p2m )
Dj
(2005-07-03) Uncertainty Relations
& Commutator Algebra
The commutator of two operators
A and B is :
[A,B] = AB - BA.
It's worth noting that if A and B are hermitian,
then so is i[A,B].
When the two observables A and B
are repeatedly measured from the same quantum state
| y >
the expected standard deviations are
Da
and Db.
( Da )2
=
< y |
A2
| y >
-
< y |
A
| y >2
( Db )2
=
< y |
B2
| y >
-
< y |
B
| y >2
The following inequality then holds
( Heisenberg's uncertainty relation ).
|
Da
Db
³
½ |
< y |
[A,B]
| y >
|
|
Proof:
Assuming, without loss of generality, that both
observables have zero averages (so the second terms
vanish in the above defining equations) this may be
identified as a type of Schwartz inequality, which may be proved
with the remark that the following quantity is nonnegative
for any real number x :
| || ( A + i x B )
| y > || 2
| = |
< y |
( A - i x B )
( A + i x B )
| y >
|
| = |
< y | (
x 2 B 2
+
i x AB
-
i x BA
+
A2
) | y >
|
| = |
x 2 ( Db )2
+
x < y |
i[A,B]
| y >
+
( Da )2
|
So, the discriminant of this real
quadratic function of x
can't be positive.
As the known expressions
of the observables for the position and momentum along the
same axis yield a commutator equal to
( i h / 2p ) Î,
we have:
Contrary to popular belief, the above doesn't simply state that two quantities
can't be pinpointed simultaneously (supposedly because "measuring one would
disturb the other").
Instead, it expounds that no experiments can be made on
identically prepared systems to determine separately both quantities
with arbitrary precision... At least whenever the following
holds, which asserts that the average of the product would
depend on the order in which the quantities are considered.
< y | AB | y >
¹
< y | BA | y >
Algebraic Rules for Commutators :
A few useful relations hold about commutators, which are easily verified :
|
[B,A] | = |
- [A,B]
|
|
[A,B]* | = |
[B*, A*]
|
|
[A,B+C] | = |
[A,B] + [A,C]
|
|
[A,BC] | = |
[A,B]C + B[A,C]
|
|
Ô | = |
[A,[B,C]] +
[B,[C,A]] +
[C,[A,B]]
|
The following relation holds for two operators whose commutator
commutes with both of them
(as is the case if the commutator is a scalar times
Î ).
[ A, f (B) ] =
[A,B] f ' (B)
Proof: As usual,
f is an analytic function,
of derivative f '.
The relation being linear with respect to f,
it holds generally if it holds for
f (z) = z n...
The case n = 0 is trivial
(zero on both sides) and an induction on n completes the proof.
[A,Bn+1] =
[A,Bn]B + Bn[A,B]
=
[A,B]nBn + [A,B]Bn
=
[A,B](n+1)Bn
(2005-06-30) Density operators characterize macrostates
A quantum representation of systems in
imperfectly known states.
A microstate (or pure quantum state)
is represented by a ket
from the relevant Hilbert space.
A more realistic macrostate is a statistical mixture
represented by a [hermitian] density operator
r
with positive eigenvalues adding up to 1.
r =
å
pn | n > < n |
The trace of an operator is the sum of the elements in its main diagonal
(this doesn't depend on the base).
All density operators have a trace equal to 1.
Tr ( Â ) =
ån < n | Â | n >
The measurement of any observable Â
yields the eigenvalue a
with the following probability, involving the
projector onto the relevant eigenspace:
p ( a ) =
Tr ( r Pa )
Thus, systems are experimentally different if and only
if they have different density operators.
We may as well talk about r as
being a system's macrostate.
Mere interaction with a measuring instrument turns the
macrostate r into
åa
Pa r
Pa
Recording the measure a
makes it
Pa r
Pa /
Tr ( r Pa )
An [analytic] function of an operator, like the logarithm of an operator, is
defined in a standard way:
In a base where the operator is diagonal, its image is the
diagonal operator whose eigenvalues are the images of its eigenvalues.
The statistical entropy
S of r is
defined in units of a positive constant k :
S ( r ) =
-k Tr ( r
Log ( r ) )
S is positive, except for a pure state
r = | y >
< y |
for which S = 0.
Algebraically, the following strict inequality holds, unless
r = r'.
S ( r ) <
-k Tr ( r
Log ( r' ) )
An isolated nonrelativistic system evolves according to its
hamiltonian H :
( ih / 2p )
dr/dt =
H r
- r H
With thermal contacts, a quasistatic evolution has different rules (T and H vary):
Z = Tr exp ( - H / kT )
and
r = exp ( - H / kT ) / Z
The variation of the internal energy
U = Tr ( r H ) may be expressed as
dU =
Tr ( dr H ) +
Tr ( r dH )
=
dQ + dW
U - TS = -kT Log Z
