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In extension of Planck’s discretized resonator energy model, Einstein proposed a quantization
oftheelectromagneticfield. Everyfieldmodeoffrequencyν couldcarryadiscretenumberoflight
quanta of energy hν per quantum.
The present quantum theory is still a continuum theory in many respects: for infinite systems,
there is a continuity of field modes of frequency ω. Also the quantum theoreticalcoefficients char-
acterizing the mixture between orthogonalstates, as well as space and time and other coordinates
remain continuous — all but one: action. Thus, in the old days, discretization of phase space
appeared to be a promising starting point for quantization. In a 1916 article on the structure of
physicalphasespace,Planckemphasizedthatthequantumhypothesisshouldnotbeinterpretedat
the level of energy quanta but at the level of action quanta, according to the fact that the volume
of 2f-dimensionalphase space (f degrees offreedom) is a positive integer of hf (cf. [77], p. 387),2
Es besta¨tigt sich auch hier wieder, daß die Quantenhypothese nicht auf Energieele-
mente, sondern auf Wirkungselemente zu gru¨nden ist, entsprechend dem Umstand,
daß das Volumen des Phasenraumes die Dimension von hf besitzt.
The following is a very brief introduction to quantum mechanics for logicians and computer
scientists.3 To avoid a shock from a too early exposition to ‘exotic’ nomenclature prevalent in
physics–the Dirac bra-ket notation–the notation of Dunford-Schwartz [37] is adopted.4
2 Again it is confirmed that the quantum hypothesis is not based on energy elements but on action elements,
accordingtothefactthatthevolumeofphasespacehasthedimensionhf.
3 Introductions toquantum mechanics canbefoundinFeynman, Leighton&M.Sands [39],Harris[50],Lipkin
[64], Ballentine [2], Messiah [70], Dirac [36], Peres [75], von Neumann [94], and Bell [4], among many other expo-
sitions. The historyof quantum mechanics isreviewed by Jammer [53]. Wheeler & Zurek[95] publishedahelpful
resourcebook.
4 The bra-ket notation introduced by Dirac which is widely used in physics. To translate expressions into the
bra-ket notation, the following identifications have to be made: for the scalar product, “h≡ (”, “i ≡ )”, “,≡ |”.
States arewrittenas|ψi≡ψ,operatorsashi|A|ji≡Aij.
2
All quantum mechanical entities are represented by objects of Hilbert spaces [94]. A Hilbert
space is a linear vector space H over the field Φ of complex numbers (with vector addition and
scalar multiplication), together with a complex function (, ), the scalar or inner product, defined
· ·
on H H such that (i) (x,x) = 0 if and only if x = 0; (ii) (x,x) 0 for all x H; (iii)
× ≥ ∈
(x+y,z) = (x,z)+(y,z) for all x,y,z H; (iv) (αx,y) = α(x,y) for all x,y H,α Φ; (v)
∈ ∈ ∈
(x,y)=(y,x)forallx,y H(αstandsforthecomplexconjugateofα);(vi)Ifx H,n=1,2,...,
n
∈ ∈
andiflim (x x ,x x )=0,thenthereexistsanx Hwithlim (x x,x x)=0.
n,m n m n m n n n
→∞ − − ∈ →∞ − −
The following identifications between physical and theoretical objects are made (a caveat: this
is an incomplete list):
(I) A physical state is represented by a vector of the Hilbert space H. Therefore, if two vectors
x,y H represent physical states, their vector sum z =x+y H represent a physical state
∈ ∈
as well. This state z is called the coherent superposition of state x and y. Coherent state
superpositions will become most important in quantum information theory.
(II) Observables A are represented by self-adjoint operators A on the Hilbert space H such that
(Ax,y) = (x,Ay) for all x,y H. (Observables and their corresponding operators are
∈
identified.)
In what follows, unless stated differently, only finite dimensional Hilbert spaces are consid-
ered.5 Then, the vectors corresponding to states can be written as usual vectors in complex
Hilbert space. Furthermore, bounded self-adjoint operators are equivalent to bounded Her-
mitean operators. They can be represented by matrices, and the self-adjoint conjugation is
just transposition and complex conjugation of the matrix elements.
Elements b ,b H of the set of orthonormal base vectors satisfy (b ,b ) = δ , where δ is
i j i j ij ij
∈
the Kronecker delta function. Any state x can be written as a linear combination of the set
of orthonormal base vectors b ,b , , i.e., x = N β b , where N is the dimension of
{ 1 2 ···} i=1 i i
H and β = (b ,x) Φ. In the Dirac bra-ket notation, unity is given by 1 = N b b .
i i ∈ P i=1| i ih i |
Furthermore, any Hermitean operator has a spectral representation A = N α P , where
i=1Pi i
the P ’sareorthogonalprojectionoperatorsontothe orthonormaleigenvectorsa ofA(non-
i i
P
degenerate case).
As infinite dimensional examples, take the position operator~x = ~x = (x ,x ,x ), and the
1 2 3
momentum operator ~p = ~~ = ~ ∂ , ∂ , ∂ , where ~ = h . The scalar product is
x i∇ i ∂x1 ∂x2 ∂x3 2π
given by (~x,~y) = δ3(~x ~y) = δ(x (cid:16) y )δ(x y(cid:17))δ(x y ). The non-relativistic energy
1 1 2 2 3 3
− − − −
operator (Hamiltonian) is H = ~p~p +V(x)= ~2 2+V(x).
2m −2m∇
Observables are said to be compatible if they can be defined simultaneously with arbitrary
accuracy; i.e., if they are “independent.” A criterion for compatibility is the commutator.
Two observables A,B are compatible, if their commutator vanishes; i.e., if [A,B] = AB
BA = 0. For example, position and momentum operators6 [x,p ] = xp p x = x~ ∂ −
~ ∂ x = i~ = 0 and thus do not commute. Therefore, position ax nd momx − entux m of ai s∂ tx at− e
i ∂x 6
cannot be measured simultaneously with arbitrary accuracy. It can be shown that this
~
property gives rise to the Heisenberg uncertainty relations ∆x∆p , where ∆x and ∆p
x ≥ 2 x
is given by ∆x = x2 x 2 and ∆p = p2 p 2, respectively. The expectation
h i−h i x h xi−h x i
value or averagevalue is defined in (V) below.
p h·i p
Ithasrecentlybeendemonstratedthat(byananalogembodimentusingparticlebeams)every
self-adjointoperatorinafinitedimensionalHilbertspacecanbeexperimentallyrealized[78].
(III) The result of any single measurement of the observable A on a state x H can only be
∈
one of the real eigenvalues of the corresponding Hermitean operator A. If x is in a coherent
superposition of eigenstates of A, the particular outcome of any such single measurement is