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(2002-11-01)   The Quantum Substitute for Logic
How is the probability of an outcome computed in quantum theory?

 

If you're not completely confused by quantum
mechanics,  you do not understand it.
John Archibald Wheeler  (1911-2008)

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 phases  (the angular directions)  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 (exclusive) components, the amplitudes are added.
• For AND (independent) components, the amplitudes are multiplied.

In practice, "AND components" are successive steps that  could  logically lead to the desired outcome, forming what's 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.

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(2007-07-19)   On the "Statistics" of Elementary Particles
A direct consequence of quantum logic:  Pauli's Exclusion Principle

In very general terms, you may call "particle" some part of a quantum system.  Swapping (or switching) a pair of particles is making one particle take the place of the other and vice versa, while leaving everything else unchanged.  Although swapping particles may deeply affect a quantum system, swapping twice will certainly not change anything since, by definition, this is like doing nothing at all.

So, according to the above quantum logic, the amplitude associated with one swapping must have a square of 1.  Therefore (assuming only that amplitudes are ordinary complex numbers) the swapping amplitude is either +1 or -1.

In the mathematical description of quantum states, this swapping of particles is well-defined only for particles of the 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.

A deep consequence of Special relativity is that spin determines which "statistics" a given type of particles obeys  (Bose-Eintein statistics for bosons, Fermi-Dirac statistics for fermion).  Part of the angular momentum of a fermion cannot be explained in classical terms  (it must include a nonorbital "pointlike" component).  The spin of a boson is a whole multiple of the quantum of angular momentum  h/2p,  whereas the spin of a fermion is an odd multiple of the "half quantum" h/4p.

With the concepts so defined, let's consider a quantum state where two fermions would be absolutely undistinguishable.  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; 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 collapse to the single state of lowest energy.  Instead, they occupy successively different "orbitals", according to rules which explain the entire periodic table of chemical elements.

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(2002-11-01)   The Infamous Measurement Problem
What does a quantum observation entail?

There are no things, only processes.
David Bohm  (1917-1992) 

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?

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(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  Dp_x  satisfying the following inequality :

Dx  Dp_x   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]

Here is a terse summary of Heisenberg's approach in terms of the Schrödinger viewpoint which we adopt here, following Dirac and almost 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.

| y_t >   =   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    in the Schrödinger viewpoint (possibly constant with time)  is then associated with the following time-dependent operator in the Heisenberg viewpoint.

Û*   Û   =   Û^-1 (t,0)    Û (t,0)

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(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, he thought that the quantity  y  which appears in it was somehow related to a density of electric charge...

Instead,  y  is now understood to be a  probability amplitude,  as defined in the above article, namely a complex number whose squared length is proportional to the probability of actually finding the electron at a particular position in space.  That interpretation of  y  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 of Olivia Newton-John).

The controversy about the meaning of  y 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 Schrödinger's 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)

Therefore, the wave celerity  u = E/p  is simply:

u   =   E  /  Ö	2m (E-V)

Now, the general 3-dimensional wave equation of some quantity  j  propagating at celerity  u  is:

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 standard way to solve this (mathematically) 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.

This is, of course, mind-boggling.  Schrödinger and many others have argued that this cannot be  entirely  true:  Something in the ultimate quantum rules must escape any linear description to defeat this  principle of superposition, which is unacceptable as an overall rule for everything observed and anything observing.

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(2003-05-26)   Noether's Theorem   (1915)

In 1915, 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, and vice versa.

E. Noether's 1918 paper, translated by M.A. Tavel (1971)   |   Noether's biography
Wikipedia   |   Noether's Discovery   |   Noether's Theorem in a Nutshell by John Baez

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(2005-06-27) Hilbert Spaces:  Dirac's  <bras|  and  |kets>
A nice 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 >*   is a complex scalar.
• 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"  Â = [ a_ij ]  which we may express as:

   =   å  a_ij | i > < j |       alternately,       a_ij   =   < i |  | j >

To the left of a  ket  or the right of a  bra,    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".  Many authors (especially in quantum theory) 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).  The conjugate transpose is also called adjoint or adjugate, Hermitian adjoint, Hermitian transpose, Hermitian conjugate, etc.  For short, it's best to use the word "conjugate" by itself, since conjugation of the complex coordinates of a vector or matrix is rarely used, if ever, without a transposition.

| 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  (von Neumann's projection postulate)  which states the consequence of a measurement, in terms of the eigenspace projector matching each possible outcome  (necessarily an eigenvalue  a  of   Â = å_a a P_a ).

| 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).

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(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

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(2005-07-03)   Commutators
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].

For example  (according to the above expressions)  the commutator of the two operators respectively associated with the position  x  and the momentum  p_x  along the same axis is the operator for which the image of  y  is:

x ( h / 2pi )  ¶y/¶x  -  ( h / 2pi )  ¶(xy)/¶x     =     ( i h / 2p ) y

So, that commutator is simply  ( i h / 2p ) Î   (where  Π is the identity operator).

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Similarly, we obtain the following expression for the operators  Â_x   and  Â_y  associated with the components  L_x  and  L_y  of the orbital angular momentum:

[ Â_x , Â_y ]   =   ( i h / 2p ) Â_z

Proof :   Let's evaluate  Â_x (Â_y (y)) :

( h / 2pi ) 2   (  y	¶	[ z ¶y/¶x - x ¶y/¶z ]  - z	¶	[ z ¶y/¶x - x ¶y/¶z ]  )
	¶z		¶y

=     ( h / 2pi ) ²   (  y ¶y/¶x  +  yz ¶²y/¶z¶x  -  yx ¶²y/¶z²      
 -  z² ¶²y/¶y¶x  +  zx ¶²y/¶y¶z  )

All the second-order terms also appear in the like expression for  Â_y (Â_x (y))  (which is obtained by swapping x and y).  So, they cancel in the difference:

[ Â_x Â_y - Â_y Â_x ] (y)   =   ( h / 2pi ) ²   ( y ¶y/¶x - x ¶y/¶y )
                 =   ( i h / 2p )  Â_z (y)    

  =

Âx Ây Âz

For the 3-component  column  operator    associated with the ("orbital") angular momentum  L,  this can be summarized thusly:

  ´    =   ( i h / 2p )  

Algebraic Rules for Commutators :

A few general 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]]

This last relation is known as the  Jacobi identity.  It's one of two relations a  bilinear map  must satisfy to be called a  Lie bracket  (the other, which is also satisfied here, is simply:  [A,A] = Ô ).  The commutator bracket thus endows the vector space of quantum operators with the structure of a  Lie algebra.

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  Π).

[ 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

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(2005-07-03)   Noncommutativity and Uncertainty Relations
The link between commutators and expected  standard deviations.

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 trailing 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  we have established  that the observables for the position and momentum along the  same  axis yield a commutator equal to  ( i h / 2p ) Î, we have:

Dx  Dpx   ³   h/4p

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 noncommutative condition holds:

< y | AB | y >     ¹     < y | BA | y >

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For a given quantum state, the uncertainty in the measurement of the momentum along x always has some definite nonzero value.  No experiment can be devised which could achieve a better precision, even if the experimenter does not care  at all  about estimating the position along x.  Conversely, for that same quantum state, there's a definite limit on the precision with which the position x 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 of those two separate uncertainties is below  h/4p.  This has absolutely nothing to do with one type of measurement "disturbing" the other...

This is clearly true, but it's a completely  different  issue:  A precise momentum measurement will leave the system in a new quantum state where the inherent uncertainty in position may very well be much greater than it originally was.

The uncertainty principle goes much deeper than that.  In particular, it says that there's no way to create a perfectly focused beam of identical particles with the same lateral velocity.  Even if you measure only  either  the lateral position  or  the lateral momentum of any given particle from the beam, your many measurements of both quantities will feature standard deviations which cannot be better than what's imposed by the above uncertainty relation.  That's the way it is.

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(2007-07-16)   Orbital Angular Momentum and Spin
Spin is a form of angular momentum without a classical equivalent.

The following argument was essentially fully developed by Elie Cartan (1869-1951) in 1913 from a purely geometrical standpoint  (not involving Planck's constant as such)  as he investigated the  Lie algebra of the group of three-dimensional rotations.  Cartan thus demonstrated, ahead of his time, how the idea of quantified spin is a deep consequence of three-dimensional geometry.  The pioneers of quantum mechanics rediscovered those things in the 1920's.  In 1935, Cartan himself published a remarkable textbook on his  Theory of Spinors.

Let's investigate the properties of a  vectorial  observable    which satisfies the fundamental property previously established in the case of the quantum operator associated with a classical  (or orbital)  angular momentum, namely:

  ´    =   ( i h / 2p ) Â

This pretty equation is  merely  a mnemonic for  3  commutation relations:

  =

Âx Ây Âz

[ Â_y , Â_z ]   =   ( i h / 2p ) Â_x
[ Â_z , Â_x ]   =   ( i h / 2p ) Â_y
[ Â_x , Â_y ]   =   ( i h / 2p ) Â_z

The 3 components  Â_x , Â_y  and  Â_z   are scalar observables (i.e., square matrices with hermitian symmetry).  Let's introduce another scalar observable:

²   =   Â_x Â_x  + Â_y Â_y  + Â_z Â_z          [ Note that:   ²   =   (Â*)    ]

Unlike  Â,  ²  is just a  square  matrix.  It would be classically associated with the square of the length of a classical 3-vector associated with    (if there's one).

The above definition of  ²  ensures that  < y | ² | y >  is  nonnegative  for any ket  |y>  (HINT:  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/2p)²

The punch line  will be  that  j  is restricted to integer or half-integer values.  For now however, we may just accept this expression because it spans all nonnegative values  once and only once  when  j  goes from zero to infinity.

So, we may use  j  as an index to denote each eigenvalue of ².   Similarly, we may use another index  m  to identify the eigenvalue  m (h/2p)  of  Â_z .  For now, nothing special 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  (needed to distinguish between kets having identical eigenvalues for each of our two 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/2p)2	| n, j, m >
Âz	| n, j, m >    =	m	(h/2p)	| n, j, m >

To determine the restrictions that  j  and  m  must obey, we introduce the following two  non-hermitian  operators, which are conjugate of each other.  They are collectively known as ladder operators;  and are respectively called  lowering operator  (or anihilation operator) and  raising operator (or creation operator) because it turns out that each transforms an eigenvector into another eigenvector corresponding to a lesser or greater eigenvalue, respectively.

Â_-   =   Â_x  -  i Â_y         and         Â₊   =   Â_x  +  i Â_y

Both commute with  ²  (because Â_x and Â_y do).  The following holds:

||  Â₊  | n, j, m >  || ²     =     < n, j, m |  Â_-  Â₊  | n, j, m >

Where	Â-  Â+	=	Âx Âx  +  Ây Ây  +  i [ Âx , Ây ]
		=	Â2  -  Âz Âz  -  ( h / 2p ) Âz

So,   ||  Â₊  | n, j, m >  || ²   =   [ j(j+1) - m² - m ]  ( h / 2p )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 >  ||.  Therefore, 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 aforementioned cases where they vanish, such kets are eigenvectors of  Â_z  associated with the eigenvalue of index  m ± 1.  Let's prove that:

Â_z Â₊  -  Â₊ Â_z   =   [ Â_z , Â_x ]  +  i [ Â_z , Â_y ]   =   (ih/2p)  (Â_y  -  i Â_x )
Therefore,     Â_z Â₊   =   Â₊ Â_z  +  (h/2p) Â₊

So, if  | y >  is an eigenvector of  Â_z  associated with the value  m (h/2p), then:

Â_z  Â₊  | y >  =  (m+1) (h/2p)   Â₊  | y >

Thus, the ket  Â₊  | y >  is either zero or an eigenvector of  Â_z  associated with the value  (m+1) (h/2p).  The same is true of  Â_-  | y >  with  (m-1) (h/2p).

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 above would yield a nonzero eigenvector with a value of  m  outside of the allowed range).  Thus, 2j and 2m must be integers  (they are the sum 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 quantized whenever it obeys the same commutation relation as 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.

Wikipedia:   Spin quantum number   |   Spin

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(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   p_n  | 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 )   =   å_n   < n | Â | n >

The measurement of any observable    yields the eigenvalue  a  with the following probability, involving the projector onto the relevant eigenspace:

p ( a )   =   Tr ( r P_a )

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.

The  average value  resulting from a measurement of    = å_a a P_a   is:

<  >   =   å_a  a p(a)   =   å_a  Tr ( r a P_a )   =   Tr ( r  )

Mere interaction with a measuring instrument turns the macrostate  r  into
å_a P_a r P_a     Recording the measure  a  makes it    P_a r P_a / Tr ( r P_a )

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-change due to the measurement process)  which appeared in  Annalen der Physik, 8 (6) 322-328.

In 1957, Gerhart Lüders (1920-1995) proved the CPT theorem  (discovered independently by Wolfgang Pauli and John S. Bell)  using Lorentz symmetry.  In 1958 Lüders also helped establish the relativistic connection between spin and statistics.  He received the Max Planck Medal in 1966.

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' )  )

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An isolated nonrelativistic system evolves according to the Schrödinger-Liouville equation, involving 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