In science, one tries to tell people something that no one everknew before, in such a way as to be understood by everyone. But in poetry, it's the exact opposite. Paul A.M. Dirac (1902-1984;Nobel 1933)
(2002-11-01) How is the probability of an outcome computed in quantum theory?
If you're not completely confused byquantum mechanics,you do not understand it. John Archibald Wheeler (1911-2008)
First, let's consider howprobabilities are ordinarilycomputed: When an event consists of twomutually exclusive events,its probability is thesum of the probabilities of those two events. Similarly, when an event is the conjunctionof twostatistically independent events,its probability is theproduct of the probabilities of those two events.
For example, if you roll afair die,the probability of obtaining a multiple of 3 is1/3 = 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 thecomponent events can't happen together (the outcome of the roll cannot beboth "3" and "6").
On the other hand, the probability of rolling two fair dice without obtaining a 6is 25/36 = (5/6)(5/6); it's the product of the probabilities(5/6 each) of twoindependent events,each consisting ofnot rolling a 6 witheach throw.
Quantum Logic and [Complex] Probability Amplitudes :
In the quantum realm, as long as two logical possibilities are not actuallyobserved,they can be neitherexclusive norindependentand the above doesnot apply. Instead, quantum mechanical probability amplitudes are defined ascomplex numbers whoseabsolute valuessquared correspond to ordinary probabilities. The phases (the angular directions) of such complex numbershave no classical equivalents (although they happen to provide adeep explanation for the existenceof the conserved classical quantity known as electric charge).
To obtain theamplitude of an event with two unobserved logical components:
For (exclusive) components,the amplitudes areadded.
For (independent) components,the amplitudes aremultiplied.
In practice, " components" are successive steps that could logically lead to the desiredoutcome, forming what's called an acceptablehistory for that outcome. The " components", whose amplitudes are to be added,are thus all the possible histories logically leading up to the same outcome. FollowingRichard 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 abovequantum logic normally applies only at the microscopic level,where "observation" of individual components is either impossible or wouldintroduce an unacceptable disturbance. At the macroscopic level, the observation of a combined outcome usually implies thatall relevant components are somehow "observed" as well (and the ordinaryalgebra of probabilities applies). For example, in our examples involving dice, you cannot tellif the outcome of a throw is a multiple of 3 unless you actually observethe 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 imperfectlyobserved (with some remaining uncertainty), the probability of the outcomeis somewhere between what the quantum rules sayand what the classical rules would predict.
(2007-07-19) A direct consequence of quantum logic: Pauli's Exclusion Principle
In very general terms, you may call "particle" some part of a quantumsystem. Swapping (or switching) a pair of particles is making oneparticle take the place of the other and vice versa, while leaving everything elseunchanged. Although swapping particles may deeply affect a quantum system,swappingtwice will certainly not change anything since, by definition,this is like doing nothing at all.
So, according to theabovequantum logic,theamplitude associated with one swappingmust have a square of 1. Therefore (assuming that amplitudes are ordinarycomplex numbers) the swapping amplitude iseither +1 or -1.
In the mathematical description of quantum states, swapping is well-defined only for particles ofthe same "nature". Whether swapping involves a multiplicative factor of+1 or -1 depends on that "nature". Particles for which swapping leaves the quantum state unchanged are called bosons, those for which swapping negates the quantum state are called fermions.
, whereas the spin of a fermion isan odd multiple of the "half quantum" h/4.
With the concepts so defined, let's consider a quantum state where twofermions would be absolutely indistinguishable. Not only would they be particles of the same kind (e.g., two electrons)but they would have the same position, the same state of motion, etc. So, the quantum state is clearly unchanged by swapping. Yet, swapping fermions must negate the quantum state... Therefore, it's equal to its own opposite and can only be zero ! The probability associated to a zero quantum state is zero;following this corresponds to something impossible. In other words, two different fermions can't "occupy" the exact same state.
This result is called Pauli's exclusion principle. It's the reason why all the electrons around a nucleus don't collapseto the single state of lowest energy. Instead, they occupy successively different "orbitals", according to ruleswhich explain the entireperiodic table ofchemical elements.
-
(2002-11-01) What does a quantumobservation entail?
There are no things, only processes. David Bohm (1917-1992)
That's the most fundamental unsolved question in quantum mechanics.
According to theabove, one should deal strictly with amplitudes between observations (ormeasurements),butanother recipe holds when measurements are made (according to the Copenhagen interpretation).
Whatever a measurement entails (which nobody has precisely defined yet) it includes at the very least a distinction between the observer and the observed and a transfer of information from the latter to the former.
Nothing prevents us from considering a larger system which includes both theobserver and the observed (possibly the whole Universe) where nosuch information leak takes place. The state of that larger system obeys the unaltered Schrödinger equation (or any relevant generalization thereof) whose unitarity actually expresses the conservation of information. (The classical counterpart of that is Liouville's theorem.)
For a system which isn't measured by any outside agency, it's thus difficult to avoid the conclusion that a large enough system canobserve itself, in some obscure sense. Information gets concentrated at some locations and depleted at others.
Either way, the simple quantum rules outlined above would have to be smoothly modifiedto account for a behavior which can be nearly classical for a large enough system. In other words, the above quantum ideas (which constitute the Copenhagen interpretation) 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 supremebefore the advent ofRelativity Theory. Relativity consistently bridged the gap between theslow and thefast,themassive and themassless (while retaining the full applicability ofNewtonian theories to the domain of ordinary speeds). Likewise, the gap must ultimately be bridged betweenobserver andobserved, between thelarge and thesmall, between theclassical world and thequantum realm, for there is but one single physical reality in which everything is immersed...
This bothers, orshould bother, everybody who deals with quantum mechanics: The so-called Schrödinger's Cat theme is often used to discuss theproblem, in the guise of a system that includes a cat (a "qualified" observer)in the presence of a quantum device whichcould trigger a lethal device. It seems silly to view the whole thing as a single quantum system,which would only exist (until observed) in somesuperposition of states, where the cat wouldbe neither dead nor alive, butboth at once. Something must exist which collapses the quantum state ofa large enough system frequently enough to make itappear "classical". It stands to reason thatSchrödinger's Cat must be deadvery shortly after being killed... Doesn't it?
Otherwise, we're led to the bizarre conclusion that severalversions of reality can somehow co-exist indefinitely. That's a respectable viewpoint, which was introduced in 1957 by Hugh Everett (1930-1982) and is now known as Everett's many-worlds interpretation.
(2005-07-03) Physical quantities are multiplied likematrices... Order matters.
In June 1925,Werner Heisenberg(1901-1976;Nobel 1932)discovered that observable physical quantitiesobeynoncommutative rules similar to those governingthe multiplication of algebraic matrices.
If the measurement of a physical quantitywould disturb the measurement of theother, then anoncommutative circumstance exists which disallows even thepossibility of twoseparate sets of experiments yieldingthe 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 respectiveuncertainties (i.e., standard deviations in repeated experiments) x and px satisfying the following inequality :
The early development of Heisenberg's Matrix Mechanics was undertaken byM. Born andP. Jordan.
In March 1926, Erwin Schrödinger showed 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, who gave basic Quantum Theory its current form.
Heisenberg's picture [skip on first reading ] :
Here is a terse summary of Heisenberg's original viewpointin terms of the Schrödinger picture which we adoptelsewhere on this page, following Paul Dirac and almost all modern scholars:
In the modern nonrelativistic Schrödinger-Dirac picture, a ket |> isintroduced 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.
|t> = Û (t,0) |0>
The unitary operatorÛ so defined is called the evolution operator.
Heisenberg's picture consists in considering that a given system is representedby the constant ket Û* |>. Operators are modified accordingly...
A physical quantity which is associated with the operator  in the Schrödinger picture(possibly constant with time) is then associated with the followingtime-dependent operator in the Heisenberg picture.
Û*Â Û = Û-1(t,0) Â Û (t,0)
(2002-11-02) The dance of a single nonrelativistic particle in a classical force field.
TheSchrödinger equation governs theprobability amplitude 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 thedifference E-V with respect to the potential V, which may thus be shifted to incorporate the rest energyof a single particle.
In 1926, when the Austrian physicistErwin Schrödinger(1887-1961;Nobel 1933)worked out the equation now named after him, he thought that the relevant quantity was something like a density of electric charge...
Instead, is now understood to be a probability amplitude, asdefined in theabove article,namely a complex number whose squared length is proportional to the probabilityof actually finding the electron at a particular position in space. That interpretation of was proposed by Max Born(1882-1970;Nobel 1954) the very person who actually coined the term quantum mechanics (Max Born also happens to be the maternal grandfather ofOlivia Newton-John).
The controversy about the meaning of hindered neitherthe early development of Schrödinger's theory of "Wave Mechanics",nor the derivation of thenonrelativistic equation at its core:
A Derivation of Schrödinger's Equation :
We may start with the expression of the phase-speed, orcelerity u = E/p of a matter wave, which comes directly fromde Broglie's principle,or less directly from othermore complicated analogiesbetween particles and waves.
The nonrelativistic (defining) relations E = V + ½ mv 2 and p = mv imply:
p =
2m (E-V)
Therefore, the wave celerity u = E/p is simply:
u = E /
2m (E-V)
Now, the general 3-dimensionalwave equationof some quantity propagatingat celerity u is:
1
j
=
j
+
j
+
j
u
t
x
y
z
=
[ is theLaplacian operator]
The standard way to solve this (mathematically)is to first obtain solutions which areproducts of a time-independent space function by a sinusoidal function of the time (t) alone. The general solution is simply a linear superposition of these stationary waves :
= exp (-2i t )
For a frequency , the stationary amplitude thus defined must satisfy:
+( 42 / u2 ) = 0
Using = E/h (Planck's formula) and the above for u = E/p we obtain...
The Schrödinger Equation :
Schrödinger's Stationary Equation
+ (82m / h2)(E V) = 0
This equation is best kept in its nonrelativistic context,where it determines allowed levels ofenergy up to an additive constant.
In the above particular stationary case, we have: E = ( i h / 2 ) t This relation turns the previous equation intoa more general linear equation :
Schrödinger's Wave Equation
( i h / 2 ) t = V ( h2 / 82m )
Signed Energy and the Arrow of Time
Historically, Erwin Schrödinger associated an equally valid stationary function with the positive (relativistic) energy E = h and obtained adifferent equation :
= exp (2i t ) ( -i h / 2 ) t = V ( h2 / 82m )
Formally, a reversal of the direction of time turns one equation into the other. We may also allow negative energies and/or frequencies inPlanck's formula E = h and observe that a particle may be described by thesame wave functionwhether it carries energy E in one direction of time, or energy -E in the other.
To retain only one version of the Schrödinger equation andone arrow of time (the term was coined by Eddington) we must formally allow particles to carry a signed energy (typically, E = mc ).
If the wave function is a solution of one version of the Schrödinger equation, then itsconjugate * is a solution of the other. However, time-reversal and conjugation need not result in the same wavefunction whenever Schrödinger's equation hasmore than one solution at a given energy.
Principle of Superposition :
The linearity of Schrödinger's equation means thatany sum of satisfactory solutions is also a solution. This principle of superposition justifies thegeneralHilbert space formalism introduced by Dirac:
Until it is actuallymeasured,a quantum state may contain (as a linear superposition) several acceptable realities at once.
This is, of course, mind-boggling. Schrödinger and many others have argued thatthis cannot beentirely true: Something in the ultimate quantum rules mustescapeany linear description to defeat this principle of superposition, which is unacceptableas an overall rule for everything observed and anything observing.
(2003-05-26)
The German mathematician Emmy Noether (1882-1935)established this deep result (Noether's Theorem) in 1915:
For every continuous symmetry of the laws of physics, there's a conservation law,and vice versa.
This result was first established in the context of classicalrational mechanics but it remainstrue (and even more meaningful) in the quantum realm.
Quantum Formalism
(2005-06-27) A nice notation with built-in simplification features.
The standard vocabulary for the Hilbert spaces used in quantum mechanicsstarted out as a pun: P.A.M. Dirac (1902-1984;Nobel 1933)decided to call < | abraand | > aket,because < | > is clearly a bracket...
Hilbert Space and "Hilbertian Basis" :
A Hilbert space is a vector space over the field ofcomplex numbers(its elements are called kets ) endowed with an inner hermitianproduct (Dirac's "bracket", of which the left half is a "bra"). That's to say that the following properties hold (z* being the complex conjugate of z):
Semilinearity: < | (x | > + y | >) = x < > + y < >
For anynonzero ket | >, the real < > ispositive (= ||||).
A Hilbert space is also required to beseparable andcomplete,which implies 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 ofunit kets which are pairwiseorthogonal :
< i | i > = 1 and < i | j > = 0 if i j
The so-called closure relationÎ = | n > < n | is a nice way to state that any ket is a generalized linear combination of kets from the "basis":
| > = Î | > = | n > < n | > = < n | > | n >
> could be nonzero: An Hilbertian basis is not a properlinear basis unless it's finite (cf.Hamel basis).
Operators :
A linear operator is asquare matrix = [a ij ] which we may express as:
 = a ij | i > < j | alternately, a ij = < i | | j >
To the left of a ket or the right of a bra, Â yields another like vector.
Hermitian conjugation generalizes to vectors and operators thecomplex conjugation of scalars. We prefer to use the same notation X* for the hermitian conjugateof any object X, regardless of its dimension. We use interchangeably the terms which other authors prefer touse for specific dimensions, namely conjugate for scalars, dual for vectors (bras and kets) and adjoint for operators (theadjugate of a matrix is somethingentirely different).
A operator equal to its own Hermitian conjugate is said to be self-adjoint or Hermitian (French: auto-adjoint).
Many authors (especially in quantum theory) use an overbar for the conjugate of a scalar and an obelisk ("dagger") for the adjointA of an operator A. In other words, AA*
Loosely speaking, conjugation consists in replacing all coordinates bytheir complex conjugates and transposing (i.e., flipping about the main diagonal). Theconjugate transpose is also calledadjoint, Hermitianadjoint,Hermitiantranspose, Hermitianconjugate, etc. The word conjugate can also be used by itself,since conjugation of the complex coordinatesof a vector or matrix is rarely used, if ever, without a simultaneous transposition.
The adjoint of a product is the product of the adjointsin reverse order. For an inner product, this restatestheaxiomatic hermitian symmetry.
( X Y )* = Y* X* < >* = < >
An operator  is self-adjoint or hermitian if  = Â*. All eigenvalues of an hermitian operator are real.
Two eigenvectors of an hermitian operatorfor distinct eigenvalues are necessarily orthogonal (seeproof below). 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 ÊÊ Â )
The first operation is a commutative product whichendows the Hermitian operators with the structure of a Jordan algebra (it's not an associative algebra butit's apower-associative one).
Unitary Transformations Preserve Length :
A unitary operator Û is a Hilbert isomorphism: Û Û* =Û* Û =Î. It transforms anHilbertian basisinto another Hilbertian basis and turns| >, < | and  respectively into Û | >, < |Û* and ÛÂÛ*.
For an infinitesimal , Û =Î + iÊ isunitary (only) when Ê ishermitian.
State Vectors, Observables and the Measurement Postulate :
A quantum state, state vector, ormicrostate is a ket | > of unit length :
< | > = 1
Such a ket | > is associated with thedensity operator | >< | (whoseentropy is zero) which determines it back,within some physically irrelevant phase factor exp(i).
Anobservable physical quantity corresponds to an hermitianoperator  whose eigenvalues are the possible values of ameasurement. Theaverage value of a measurement of  from a pure microstate | > is:
< |Â | >
This is a corollary of the following measurement postulate (von Neumann's projection postulate) which states the consequence of ameasurement,in terms of the eigenspace projector matching each possible outcome
Any outcomeis necessarily an eigenvalue of =P
| > becomes
P | >
||P | > ||
with probability < |P | >
The above is also often called the principle of spectral decomposition. Note that, since P2 =P =P*, we have:
|| P | > || 2 = < |P | >
Vocabulary: The principle of quantization limits the observedvalues of a physical quantity to the eigenvalues ofits associated operator. The principle of superposition asserts that a pure quantum state is represented by a ket... A quantum state represented by an eigenvector of an observableis called an eigenstate. It always yields the same measurement of that observable.
Orthogonality of Eigenspaces :
Two kets |> and |> that are eigenstates of an hermitianoperator  associated with distinct eigenvalues and are necessarily orthogonal.
Proof : If  > = > and < = < with , then we have: <  > = <> = <>. Therefore, <> = 0
(2020-10-04) It's a tensor product of Hilbert spaces (not a direct product).
The Hilbertian basis for the composite space is indexed by two independent sets of labels inherited from labels which would denote the two parts.
| i > | j > = | i, j >
In the finite-dimensional case where we have m labels for thefirst part and n labels for the second one, we have m×n labels for the composite space (as opposed to the m+n labels we'd have for an irrelevant direct product).
(2005-07-03) Thecommutator of two operators A and B is : [A,B] =AB -BA.
Algebraic Rules for Commutators :
A few general relations hold about commutators, which are easily verified :
[B,A]
=
- [A,B] (anticommutativity)
[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]]
This last relation is known as the Jacobi identity. It's the relations an anticommutative bilinear map must satisfy to be called a Lie bracket. The commutator bracket thus turns the vector space of quantum operatorsinto a Lie algebra. However, the commutator of two hermitian operators isn't hermitian. To fix this and turn the hermitian operators by themselves into a Lie algebra, just modify the definition of the bracket by multiplyingthe above into some real multiple of the imaginary unit i. For example:
[A,B] = i (AB -BA)
Derivative of an analytic function defined on operators :
The following relation holds for two operators whose commutatoris a scalar times Î (or, at least, iftheir commutator commutes with the operator B ).
[A,f (B) ] = [A,B] f ' (B)
Proof:As usual, f is ananalytic function,of derivative f '. The relation being linear with respect to f, it holds generally if it holds for f (z) = z... The case n = 0 is trivial (zero on both sides) and an induction on n completes the proof:
(2005-07-03) Building on 6 operators for thecoordinates of position and momentum.
Bohr's correspondenceprinciple was a fuzzy set of recipes invoked by Bohr in his old quantum theory, beforeDirac's formulation. It was based on the idea that a quantization of reality should yield backclassical laws in the limit of large quantum numbers. In the new quantum theory, Dirac could achieve thatdesirable goal in a more systematic way, by matching classical quantitieswith quantum operators in the way described next.
Onlyscalar physical quantities correspond to basicobservables (hermitian square matrices) within the relevantHilbert spaceL. Physical vectors may also be considered, whichcorrespond to operators mapping a ket into a vector of kets (i.e., an element of some Cartesian power of L ).
Canonical Quantizations (Dirac, 1925) :
The key was a revelation which Paul Dirac had during his Sunday walk on 20 September 1925. Vaguely recalling the beautiful construct known as Poisson brackets in the Hamiltonian formulation of classical mechanics, Dirac guessed those could well be the classical counterparts of quantum commutators. The library was closed. He had to wait until the next morning to refresh his memory and confirm his hunch:
To the Hamiltonian description of a classical system using generalized positions qj and associated momenta pj correspondquantum observables Qj and Pj such that (using Kronecker delta notation):
[ Qj , Pk ] = ijk
The following table embodies Dirac's correspondence principle for those physical quantities which have such a classical analog... The orbital angular momentum of apointlike particle does; its spin doesn't.
According to theabove expressions,the commutator of the two operators respectively associated withthe position x and the momentum px along the same axis is the operator forwhich the image of is:
x ( h / 2i ) x ( h / 2i ) xx = ( i h / 2 )
That commutator is thus ( i h / 2 )Î (where Î is the identity operator).
Similarly, we obtain the following expression forthe operators Âx and Ây associated with components Lx and Ly of the orbital angular momentum:
[Âx ,Ây ] = ( i h / 2 )Âz
Proof : Let's evaluate Âx (Ây()) :
( h / 2i ) 2 y
[ z x- x z ] - z
[ z x- x z ]
z
y
= ( h / 2i ) 2 y x + yz zx yx z2 z2 yx + zx yz
All the second-order terms also appear in the like expression for Ây (Âx()) (which is obtained by swapping x and y). So, they cancel in the difference:
[Âx Ây -Ây Âx ]() = ( h / 2i ) 2y xx y = ( i h / 2 ) Âz()
 =
Âx Ây Âz
For the 3-component column operator  associated with the ("orbital") angular momentum L, this can besummarized:
ÂÂ = ( i h / 2 )Â
(2005-07-03) The link between commutators and expected standard deviations.
When two observables A andB are repeatedly measured from thesame quantum state| > the expected standard deviations are a and b.
The following inequality then holds (Heisenberg's uncertainty relation ).
ab ½ |< |[A,B]| >|
Proof: Assuming, without loss of generality, that bothobservables have zero averages (so the trailing termsvanish in the above defining equations) this may beidentified as a type of Schwartz inequality, which may be provedwith the remark that the following quantity is nonnegativefor any real number x :
|| (A + i xB )| > || 2
=
< |(A i xB )(A + i xB )| >
=
< | (x 2B 2 + i xAB i xBA + A2) | >
=
x 2 (b )2 + x < |i[A,B]| > + (a )2
The discriminant of this real quadratic function of x can't be positive.
As we have established thatthe observables for the position and momentum along the same axis yield a commutator equal to (ih/2) Î, we have:
x pxh/4
Contrary to popular belief, the above doesn't simply state that two quantitiescan't be pinpointed simultaneously (supposedly because "measuring one woulddisturb the other"). Instead, it expounds that no experiments can be made on identically prepared systems to determine separately both quantitieswith arbitrary precision... At least whenever the following noncommutativitycondition holds:
< |AB | > < |BA | >
For a given quantum state, the uncertainty in the measurement of the momentumalong x always has some definite nonzero value. No experiment can be devisedwhich could achieve a better precision, even if the experimenter does notcare at all about estimating the position along x. Likewise, for that same quantum state, there's a definite limit on the precisionwith which the position along the x-axis can be determined,even if we do not care at all about the momentum along x.
What Heisenberg's uncertainty relation specifies is that no quantum states exists for which the product ofthose two separate uncertainties is below h/4. This has absolutely nothing to do with one type of measurement"disturbing" the other...
The uncertainty principle goes much deeper than that. In particular, it says that there's no way to create a perfectly focused beamof identical particles with the same lateral velocity. Even if you measure only either the lateral position or the lateral momentum of any given particlefrom the beam, your many measurements of both quantities will featurestandard deviations which cannot be better than what's imposed by the aboveuncertainty relation. That's the way it is.
(2012-07-10)
Physical quantities whose commutator is a scalar (i.e., the identity operator multiplied into some complex number) are said to be conjugate of each other and thedispersion in the measurement of one is inversely proportional tothe dispersion in the measurement of the other. This is illustrated by the position and the momentum of a particle along the same axis.
Conversely, when the observables commute, the eigenstate of one isan eigenstate of the other and both quantities can be measuredsimultaneously, without any dispersion, for all possiblevalues of either quantity.
Otherwise, some quantum states are eigenstates of one observablebut not the other, while others may be eigenstates of both. For example, the magnitude of the impulsion (but not its direction) can be measured with zero dispersion if the particle is found tobe at a location where the magnitude || of the wave function is either zero or maximum:
/ x = / y = / z = 0
That's because the commutator between the operators associated to the coordinateposition x and || p2 || vanish at such positions (the same being true for other coordinates):
[ x, ||p||2] | = x (-h2/42)(-h2/42)(x) = (h2/22)/ x || [ x, ||p||2] | = (h2/22)/ x
(2005-06-27) Between measurements, a quantum state obeys Schrödinger's equation.
In nonrelativistic quantum theory, time (t) is not anobservable in theabove sense, but a parameter along which thingsevolve betweenmeasurements,according to the following generalization of Schrödinger's equation, using the hamiltonian operatorH (associated with the system's total energy) :
i h
d
| > =H | >
2
dt
This is completely wrong unless Hamiltoniansare properly adjusted to incorporate rest energies (see ourdiscussion of Schrödinger'sequation).
As an important example of the above general postulate,we may retrieve the original equation of Schrödinger for anonrelativistic particle subjected to a scalar potential. In that case, the total mechanical energy is given by:
The time-derivative we seek is simply obtained from theproduct rule:
d
< | A | > = < | A
d
( | > ) +
d
( < | ) A | >
dt
dt
dt
The two terms on the right-hand-side are given either by the above version ofSchrödinger's equation or by itsHermitian conjugate:
i h
d
< | = < | H
2
dt
Therefore:
i h
d
< | A | > = < | AH | > < | HA | >
2
dt
i h
d
< | A | > = < | [A,H] | >
2
dt
As a consequence, if an observable commutes with the Hamiltonian,then its expected value, in any quantum state, doesn't change with time...
(2015-10-03) The relation between time and energy in nonrelativistic quantum theory.
In nonrelativistic quantum mechanics, time is just a parameter, not an observable with its ownuncertainty (equal to itsstandard deviation).
It makes sense to use some observable A as a clock only if its average value < | A | > changes with time, in which case we may define the time-uncertainty t as the time in which that expected value changes by an amount equal to the clock'sstandard deviation a. That's to say:
We may also apply the previously established uncertainty relation to the observables A and H (knowing that, by definition, the standard deviation of the Hamiltonian H isthe uncertainty in energy E). This gives:
aE ½ |< |[A,H]| > |
If we assume < | [A,H] | > to be nonzero, those three relations yield:
E t h/4
So, even in the framework of nonrelativistic quantum mechanics we can obtainrigorously an uncertainty relation which is the perfect counterpart of the well-known uncertainty relation between position and momentum along one spatial direction. Time is to energy what spatial position is to linear momentum (that's consistent with the tenets ofSpecialRelativity).
The above is established without invoking an observable whose commutator with H would be a scalar multiple of Î (advanced considerations, related to the Stone-von Neumann theorem, would show that there's no such thing).
Mercifully, it's enough to have, for any given quantum state | > , some observable with a non-vanishing time-derivative. We don't need to assume the dubious existence of a single clock whichwould be valid in that sense for every possible quantum state...
(2023-05-03) They achieve the smallest possible product of uncertainties allowed by Heisenberg's principle.
(2007-07-16) Spin is a form of angular momentumwithout a classical equivalent.
Let's investigate the properties of a vectorial observable  which satisfies the fundamental propertypreviouslyestablishedin the case of the quantum operator associated with a classical (orbital) angular momentum, namely:
ÂÂ = ( i h / 2 )Â
This pretty equation is merely a mnemonic for 3 commutation relations:
 =
Âx Ây Âz
[Ây,Âz] = ( i h / 2 )Âx [Âz,Âx] = ( i h / 2 )Ây [Âx,Ây] = ( i h / 2 )Âz
The 3 components Âx ,Ây and Âz arescalar observables (i.e., square matrices with hermitian symmetry). We introduce another such observable:
Therefore, those two things add up to zero, which means: [Â2,Âz] = 0
The above definition of Â2 ensuresthat < |Â2| > is nonnegative for any ket |> (: this is the sum of 3 real squares). Therefore, this operator can only have nonnegative eigenvalues, which (for the sake of future simplicity) we may as well put in the following form, for some nonnegative number j.
j (j+1) (h/2)2
The punch line will be that j is restrictedto integer or half-integer values. For now however, we may just accept this expressionbecause it spans all nonnegative values once and only once when j goes from zero to infinity.
So, j can be used to index every eigenvalueofÂ2.Similarly, we may use another index m to identify theeigenvalue m (h/2) of Âz . For now,nothing is assumed about m (we'll show later that 2m is an integer).
Since those two observables commute, there's an orthonormal Hilbertian basis consisting entirely of eigenvectors common to both of them. We may specify it by introducing a third index n (neededto distinguish between kets having identical eigenvaluesfor both of our observables). Those conventions are summarized by the following relations,which clarify the notation used for base kets:
Â2
| n, j, m > =
j (j+1)
(h/2)2
| n, j, m >
Âz
| n, j, m > =
m
(h/2)
| n, j, m >
Cartan's proof of quantization, by finite descent (1913) :
To determine the restrictions that j and m must obey,we introduce two non-hermitian operators,conjugate of each other. They are collectively known as ladder operators and are respectively called lowering operator (orannihilation operator)and raising operator (orcreation operator)because it turns out that each transforms an eigenvector intoanother eigenvector corresponding to a lesser or greater eigenvalue, respectively.
 = Âx iÂy and Â+ = Âx + iÂy
Both commute with Â2 (becauseÂx andÂy do). The following holds:
|| Â+ | n, j, m > || 2 = < n, j, m | ÂÂ+ | n, j, m >
Where
ÂÂ+
=
ÂxÂx + ÂyÂy + i[Âx,Ây]
=
Â2ÂzÂz ( h / 2 )Âz
So, || Â+ | n, j, m > || 2 = [ j(j+1) m2 m ] ( h / 2 )2
As the nonnegative square bracket is equal to j (j+1) m(m+1) we see that m cannot exceed j. We would find that (-m) cannot exceed j by performing the same computation for || Â | n, j, m > ||. All told:
-j ≤ m ≤ j
Note that the above also proves that the ket Â+ | n, j, m > vanishes only when m = j. Likewise, Â | n, j, m > is nonzero unless m = -j.
Except in the cases where they vanish,such kets are eigenvectors of Âz associated with the eigenvalue of index m 1. Let's prove that:
So, if | > is an eigenvector of Âz forthe eigenvalue m (h/2), then:
ÂzÂ+ | > = (m+1) (h/2) Â+ | >
Thus, Â+ | > is either zero or an eigenvector of Âz associated with the value (m+1) (h/2). The same is true of  | > with (m-1) (h/2).
Since we know that m is between -j and +j , we see that both j-m and j+m must be integers (or else iterating one of the two constructions abovewould yield a nonzero eigenvector with avalue of m outside of the allowed range). Thus, 2j and 2m must be integers (they are thesum and the difference of the integers j+m and j-m). If j is an integer, so is m. If j is an half-integer, so is m (by definition, an "half-integer" is half the value of an odd integer).
The above demonstration is quite remarkable: It shows how a 3-component observable is quantizedwhenever it obeys the samecommutation relationas an orbital angular momentum. Although half-integer values of the numbers j and m are allowed, those do not correspond to an orbital momentum. Indeed, let's show that orbital momenta can only lead to whole values of j and m.
(2008-08-24) Three traceless anticommuting Hermitian matrices with unit squares.
In 1927, Wolfgang Pauli (1900-1958) introduced three matrices for use in the theory of electron spin. Their eigenvalues are +1 and -1.
x =
0 1
1 0
y =
0 i
-i 0
z =
1 0
0 -1
They have unit squares and anticommute: jk = kj when j k. They combine into a 3-vector of matrices verifying the crucial equation:
= 2i
Therefore, they provide an explicit representation of theabove type of "angular momentum" observablesin the simplest case of only two values (eigenvalues). This is meant to describea lone fermion of spin ½, of which the electron is the primary example. The above discussion and notations apply directly to:
 = (h/4) (i.e., Âx = (h/4)x , etc. )
In this simple case, we have Â2 = Âx2 + Ây2 + Âz2 = 3 (h/4)2Î The square of the spin of any electron;is thus always equal to 3 (h/4)2.
The observable corresponding to the projection of the electron spinalong the direction of the unit vector u of Cartesian coordinates (x,y,z) is
Âu = xÂx + yÂy + zÂz = (h/4)
z x + iy
x iy -z
Since x2+ y2+ z2 = 1, theeigenvalues of Âu are indeed always (h/4).
Note that any Hermitian matrix with such opposite eigenvalues can be put in thisform. Thus, any quantum state is associated with an observable which willconfirm its orientation with certainty (probability 1).
(2023-03-08) Matrices quartered into either Pauli matrices or trivial blocks (0, ).
Let's build four anticommuting matrices of dimension 4,using 2×2 blocks equal either Pauli matrices or the 2×2 identity I multipled into 0, +1 or -1.
There are essentially just two nondegenerate ways to do so, which we may respectively identify by the common names of themetric signatures they inducein the Clifford algebra of dimension 4 generated by them:
0 =
I O
O -I
i =
O j
j O
for j = 1,2,3
0 =
I O
O -I
i =
O -j
j O
for j = 1,2,3
Although the former case has seen some usage, The latter case is the star of the show in relativistic quantum mechanic. The gamma notation is standard, the beta one is not, but one reduces to the other in a simple way:
0 = 0 and j = 0j j = 1,2,3
The key propery in either case is that the matrices have unit squares (with a change of sign when the metric signature calls for it) and that the four matrices anticommute pairwaisefor distinct indices. (: so do the 3 Pauli matrices). That makes all cross-products cancel pairwise when expanding a square, so we have nice relations:
( x00 + x11 +x22 + x33) 2
=
x02 + x12 + x22 + x32
( ct0 + x1 + y2 + z3) 2
=
c2t2 x2 y2 z2
Both right-hand sides are understood to be the stated scalars multiplied into the identity matrix, which may be omitted by convention.
The gamma (resp. beta) matrices are not closed under multiplication. They generate an algebra of dimension 16.
Iterating Dirac's Trick
In the above construction, we went from n=3 mutually anticommutaive Pauli matrices of unit squaresin m=2 dimensions to n+1 = 4 gamma matrices which are likewise mutually anticoutativewith unit squares (up to a change of sign). (Actually, we get a fifth gamma matrix for free in the samespace of dimension 4 as the product of the first four.) This can be iterated tospaces of square matrices of order 2 (Pauli), 4 (Dirac's gamma), 8, 16, 32, etc.
Original Form of Dirac's Equation (Paul Dirac, 1927)
(2008-08-26) The singlet and triplet states of two entangled electrons.
According to theprevious article, a pure quantum state for the spin of a lone electron is representedby a ket which isa linear combination of the two eigenvectors of z which we shall henceforth call "up" and "down":
| u > =
1 0
| d > =
0 1
This involves a priori two complex coefficients. However, two kets that are complex multiples of each other represent the same quantum state,so the specification of a state actually depends on just two real numbers.
The juxtaposition of two such spins is represented by a linear combinationof four pairwise orthogonal unit kets in a 4-dimensional Hilbert space :
| u,u > | u,d > | d,u > | d,d >
In that space, a quantum state is describedby 6 independent real numbers (4 complex coefficients modulo one complex scalar) which is 2 more "degrees of freedom" thanwhat might be expected for the separate description of two spins. The extra possibilities are called entangled states.
Consider the same observables as before forthe measurement of the first spin only. Those operators do not change at all the components of the ket whichdescribe the second spin.
With a single spin, we saw that any given pure quantum statewas always a +1 eigenstate of a certain linear combination ofx,y andz.
In particular, as all measurements of the corresponding quantity werealways equal to +1 so was their average. Surprisingly, this no longer holds for the measurement ofa single spin in a two-spin system. In particular, the following two states both yield a zeroaverage for the measurement of the first spin along any direction :
( | u,d > | d,u > ) /2 = | s > (Singlet State)
( | u,d > | d,u > ) /2 = | t > (Triplet State)
Similarly, in either of those two quantum states,the average measurement of the second spin along any directionis also zero.
We may also consider a combined observable which gives the sum of the two spins along some direction. The result can only be +2, 0 or -2 and the average is zero for both the singlet and triplet states.
However, much more is true for the singlet state,since any measurement of the sum of the spins along anydirection always gives zero for the singlet state. Not just a zero average but an actual zero measurement every time !
Thus, if you measure the spin of one of the two electrons entangledin a singlet state, you will know for sure that a measurement of the spin of the other electronalong the same direction will give the opposite result. Always.
(2008-08-31) (John S. Bell, 1964) Statistical relations which are violated in quantum mechanics.
Classically, the probabilities of eventscan be broken down as sums of mutually exclusive events. Such a decomposition implies the following inequality betweenvarious joint probabilities of three events A, B and C:
P ( A & [not B] ) + P ( B & [not C] ) ≥ P ( A & [not C] )
The picture shows that the event of the right-hand-side iscomposed of the two mutually exclusive events shadedin red, which also appear as components of the two events fromthe left-hand-side. So, their probabilities add upto something no greater than the left-hand-side sum. This is known as Bell's Inequality.
In quantum mechanics, there are no such things as mutually exclusiveevents (unless actual observations take place whichturn thequantum logic of virtualpossibilities into the more familiar statistics ofobserved realities). Thus, there's no reason why Bell's inequality should applyto the calculus of virtual quantum possibilities. Indeed, it doesn't in the above case of a singlet state.
(2015-01-25) Definite values would violate the relations between physical quantities.
Using atopos perspective in1998, Chris J. Isham (1944-) and Jeremy Butterfield (1954-) have stated the KS theorem thusly: It's impossible to assign values to all physical quantities whilst preserving the functional relations between them.
(2020-02-20) Determining the relative phase shift between two split beams.
(2008-08-25) Equivalents of the Pauli matrices beyond spin ½.
A particle of spin j issomething which allows a measurement of its spin along anydirection to have 2j+1 values (according toCartan's argument).
The relevantHilbert space has dimension 2j+1 and the observablesfor the three projections of angular momentum on three orthogonaldirections (in a right-handed configuration) can be expressedas in theabove special case (j=½) using the counterparts of Pauli matricesin a Hilbert space of 2j+1 dimensions, namelythree 2j+1 by 2j+1 matrices which combine into a "vector" verifyingthe following compact commutation relation:
= 2i
).
Here's how we may construct such a thing: First we impose wlg that z is diagonal (that simply means we decide to use eigenvectors of z to form a basisfor ourHilbert space). We do know the eigenvalues of z fromCartan's argument, so z is entirely specified upto the ordering of the (real) elements in the diagonal. We choose (arbitrarily)to order our base kets so that those (distinct) eigenvalues appear indecreasing order on the diagonal of z. So, z is simply the diagonalmatrix whose 2j+1 elements are (2j, 2j-2, 2j-4, ... -2j).
We are looking for the hermitian matrix x in terms of (j+1)(2j+1)3 scalar unknowns (including 2j+1 real ones). In the case j = 3/2 this would mean a total of 10 unknowns (6 complex and 4 real ones) in a 4 by 4 matrix:
x =
a b* c* d*
b e f* g*
c f h k*
d g k m
Now, y is obtaineddirectly from the equation:
[z ,x ] = 2 iy
This yields an expression of y where each entry is proportionalto the corresponding (unknown) entry of x. Next, we may use the relation:
[y ,z ] = 2 ix
This tells us that all terms of x (and, therefore, also those of y ) must vanish except at positionsadjacent to the main diagonal. Now, we're faced with only 2j unknowncomplex coefficients (which are unconstrained, at this point) and just one more commutation relation to satisfy, namely:
[x ,y ] = 2 iz
It turns out that this final equation gives us the squaresof the absolute values of the aforementioned remaining 2j unknowns. Each of them is thus determinedup to an arbitrary phase factor (for a total of 2j arbitrary multipliers of unit length). In the following tabulation, we have chosen a "standard" conventionfor those phase factors which makes all the coefficients of x realand positive.
x =
0 1
1 0
y =
0 i
-i 0
z =
1 0
0 -1
x =
2
0 1 0
1 0 1
0 1 0
y =
2
0 i 0
-i 0 i
0 -i 0
z =
2 0 0
0 0 0
0 0 -2
x =
0 3 0 0
3 0 2 0
0 2 0 3
0 0 3 0
y =
0 i3 0 0
-i3 0 2i 0
0 -2i 0 i3
0 0 -i3 0
z =
3 0 0 0
0 1 0 0
0 0 -1 0
0 0 0 -3
0 2 0 0 0
2 0 6 0 0
0 6 0 6 0
0 0 6 0 2
0 0 0 2 0
0 2i 0 0 0
-2i 0 i6 0 0
0 -i6 0 i6 0
0 0 -i6 0 2i
0 0 0 -2i 0
4 0 0 0 0
0 2 0 0 0
0 0 0 0 0
0 0 0 -2 0
0 0 0 0 -4
Relativistic arguments (beyond the scope of this discussion) do not allow elementary particles beyond spin 2. Composite objects with higher spins do not have a fixed value of j. However, if their possible decay into things of lower spin is ignored,they would behave like fictional high-spin objects, starting with:
The same pattern holds for any spin j : The nth coefficient down the upper subdiagonal of x for spin j is simply given by the expression:
(x)n,n+1 =
n ( 2j + 1 n )
exp ( in) [ e.g., n = 0 ]
If used, each phase factor applies to two matching elements inx andy which are above thediagonal. The conjugate phase applies to the transposed elements (below the diagonal). This would turn the ordinary (2x2) Pauli matrices into:
x =
0 ei
ei 0
y =
0 i ei
-i ei 0
z =
1 0
0 -1
The eigenvectors of those three matrices arerespectively proportional to:
1 ei
and
-1 ei
1 i ei
and
-1 i ei
1 0
and
0 1
We may call twists the 2j such phase factors which arepart ofxandy. For spin ½, the single twist can be eliminated by redefiningwhich axis (perpendicular to the z-axis) is associated with the twist-free versionofx. This works for a single spin but cannot be done simultaneously for severalspins... It's as if a spin possessed an internal phasewhich indicates, so to speak, the actual angular position in a "rotation" around a given axis.
The same trick can always be used to make the sum of the twistsvanish in a single higher spin, but what is the physical significance of the2j-1 remaining degrees of freedom? They seem to determine, in a nontrivial way, the relative positionalphases in the "rotations" around each direction of space. In particular, what does meanin the following observables for spin 1 ?
x =
2
0 ei 0
ei 0 ei
0 ei 0
y =
2
0 i ei 0
-i ei 0 i ei
0 -i ei 0
z =
2 0 0
0 0 0
0 0 -2
The columns in the following [unitary and hermitian] matrices are eigenvectors:
1 2
1 2 ei 1
2 ei 0 -2 ei
1 -2 ei 1
1 2
1 i 2 ei -1
-i 2 ei 0 -i 2 ei
-1 i 2 ei -1
1 0 0
0 1 0
0 0 1
(2005-06-30) (von Neumann, 1927) Quantum representation of systems in imperfectly known mixed states.
A microstate (or pure quantum state) is represented by a normedketfrom the relevant Hilbert space, up to an irrelevant phase factor. A more realistic macrostate is a statistical mixture (called mixed state or Gemischt) which can be represented by a unique [hermitian] density operator with positive eigenvalues that add up to 1.
= pn | n > < n |
In particular, the unique density operator representing the pure quantum stateassociated with the normed ket | > is given by the following expression, which is unaffected by phase factors (since multiplying | > by a complex number of unit norm will multiply < | by the reciprocal).
= | > < |
A statistical mixture consisting of a proportion u of the macrostaterepresented by 1 and a proportion 1-u of the macrostaterepresented by 2 is represented by the following density operator:
= u1 + (1-u)2
Thetrace of an operatoris 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. Conversely, all operators of trace 1 can be construed as density operators.
Tr (Â ) = n < n |Â | n >
The measurement of anyobservable yields the eigenvalue with the following probability, involving theprojector onto the relevant eigenspace:
p ( ) = Tr ( P )
Thus, systems are experimentally different if and onlyif they have different density operators. We may as well talk about as being a macrostate.
The average value resulting from a measurement of  = P is:
<Â > = p() = Tr (P ) = Tr ( Â )
Mere interaction with a measuring instrument turns themacrostate into P P Recording the measure makes it P P /Tr ( P )
This is known as "Lüder's rule" or Lüders' projection postulate. It was first discussed in 1951 by Gerhart Lüders, in "Über die Zustandsanderung durch den Messprozess"(On the state-changedue to the measurement process) which appeared in Annalen der Physik, 8 (6) 322-328.
An [analytic] function of an operator, like the logarithm of an operator,is defined in astandard way: In a base where the operator is diagonal, its image is thediagonal operator whose eigenvalues are the images of its eigenvalues.
The Von Neumann entropy S ( ) is what Shannon's statistical entropy becomesin this context. It isdefined in units of a positive constant k :
S ( ) = -k Tr ( Log ( ) )
S ispositive, except for apure state = | >< | for which S = 0. Algebraically, the following strict inequality holds, unless ='.
S ( ) < -k Tr ( Log (' ) )
Anisolated nonrelativistic system evolves according tothe Schrödinger-Liouville equation, involving itshamiltonian H :
( ih / 2 ) d/dt = HH
With thermal contacts, a quasistatic evolution has different rules (T andH vary). Introducing the partition function (Z) :
Z = Trexp ( -H / kT ) and = exp ( -H / kT ) / Z
The variation of the internal energy U = Tr ( H ) may be expressed as
of an operator A. In other words, A
