text stringlengths 0 8.13M |
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indeterministic; i.e., it cannot be predicted with certainty. As a result of the measurement, |
5 Infinite dimensional cases and continuous spectra are nontrivial extensions of the finite dimensional Hilbert |
space treatment. As a heuristic rule, it could be stated that the sums become integrals, and the Kronecker delta |
function δij becomes the Diracdelta function δ(i−j), which is a generalized function inthe continuous variables |
i,j. IntheDiracbra-ketnotation,unityisgivenby1= −+ ∞∞ |iihi|di. |
6theexpressionsshouldbeinterpretedinthesenseofoperatorequations;theoperatorsthemselvesactonstates. |
R |
3 |
the system is in the state which corresponds to the eigenvector a of A with the associated |
n |
real-valued eigenvalue α ; i.e., Ax=α a (no summation convention here). |
n n n |
This “transition” x a has given rise to speculations concerningthe “collapseof the wave |
n |
→ |
function (state).” But, as has been argued recently (cf. [47]), it is possible to reconstruct |
coherence;i.e.,to“reversethecollapseofthewavefunction(state)”iftheprocessofmeasure- |
ment is reversible. After this reconstruction, no information about the measurement must |
be left, not even in principle. How did Schr¨odinger, the creator of wave mechanics, perceive |
the ψ-function? In his 1935 paper “Die Gegenw¨artige Situation in der Quantenmechanik” |
(“The present situation in quantum mechanics” [83], p. 53), Schr¨odinger states,7 |
Die ψ-Funktion als Katalog der Erwartung: ... Sie [[die ψ-Funktion]] ist jetzt das |
Instrument zur Voraussage der Wahrscheinlichkeit von Maßzahlen. In ihr ist die |
jeweils erreichte Summe theoretisch begru¨ndeter Zukunftserwartung verk¨orpert, |
gleichsam wie in einem Katalog niedergelegt. ... Bei jeder Messung ist man |
gen¨otigt,derψ-Funktion(=demVoraussagenkatalogeineeigenartige,etwaspl¨otzliche |
Ver¨anderung zuzuschreiben, die von der gefundenen Maßzahl abh¨angt und sich |
nicht vorhersehen la¨ßt; woraus allein schon deutlich ist, daß diese zweite Art |
von Ver¨anderung der ψ-Funktion mit ihrem regelma¨ßigen Abrollen zwischen zwei |
Messungen nicht das mindeste zu tun hat. Die abrupte Ver¨anderung durch die |
Messung ... ist der interessanteste Punkt der ganzen Theorie. Es ist genau der |
Punkt, der den Bruch mit dem naiven Realismus verlangt. Aus diesem Grund |
kannmandieψ-FunktionnichtdirektandieStelle desModellsoderdesRealdings |
setzen. Und zwar nicht etwa weil man einem Realding oder einem Modell nicht |
abrupte unvorhergesehene A¨nderungen zumuten du¨rfte, sondern weil vom realis- |
tischen Standpunkt die Beobachtung ein Naturvorgang ist wie jeder andere und |
nicht per se eine Unterbrechung des regelma¨ßigenNaturlaufs hervorrufen darf. |
It therefore seems not unreasonable to state that, epistemologically, quantum mechanics is |
more a theory of knowledge of an (intrinsic) observer rather than the platonistic physics |
“God knows.” The wave function, i.e., the state of the physical system in a particular |
representation (base), is a representation of the observer’s knowledge; it is a representation |
or name or code or index of the information or knowledge the observer has access to. |
(IV) TheprobabilityP (x)tofindasystemrepresentedbystatexinsomestatey ofanorthonor- |
y |
malized basis is given by P (x)= (x,y)2. |
y |
| | |
(V) The average value or expectation value of an observable A in the state x is given by A = |
x |
h i |
N α (x,a )2. |
i=1 i | i | |
(VI)PThe dynamical law or equation of motion can be written in the form x(t) = Ux(t ), where |
0 |
U =U 1 (“ standsfortranspositionandcomplexconjugation)isalinearunitaryevolution |
† − |
† |
operator. |
The Schro¨dinger equation i~∂ ψ(t)=Hψ(t) is obtained by identifying U with U =e iHt/~ , |
∂t − |
where H is a self-adjoint Hamiltonian (“energy”) operator, by differentiating the equation |
of motion with respect to the time variable t; i.e., ∂ ψ(t) = iHe iHt/~ ψ(t ) = iHψ(t). |
∂t − ~ − 0 − ~ |
In terms of the set of orthonormal base vectors B 1,b ,... , the Schr¨odinger equation |
2 |
can be written as i~∂ (b ,ψ(t)) = H (b ,ψ({ t)).− In the c} ase of position base states |
∂t i j ij j |
ψ(x,t) = (x,ψ(t)), the Schr¨odinger equation takes on the form i~∂ ψ(x,t) = Hψ(x,t) = |
P ∂t |
pp +V(x) ψ(x,t)= ~2 2+V(x) ψ(x,t). |
2m −2m∇ |
7 Th(cid:2)e ψ-function(cid:3)as expectatioh n-catalog: ... In iti [[the ψ-function]] is embodied the momentarily-attained sum |
of theoretically based future expectation, somewhat as laid down in a catalog. ... For each measurement one is |
requiredtoascribetotheψ-function(=thepredictioncatalog)acharacteristic,quitesuddenchange,whichdepends |
on the measurement result obtained, and socannot be forseen; from whichalone itis alreadyquite clear that this |
secondkindofchangeoftheψ-functionhasnothingwhateverincommonwithitsorderlydevelopmentbetweentwo |
measurements. The abrupt change [[ofthe ψ-function (=the prediction catalog)]] by measurement ... is the most |
interestingpointoftheentiretheory. Itispreciselythepointthat demands thebreakwithnaiverealism. Forthis |
reasononecannotputtheψ-functiondirectlyinplaceofthemodelorofthephysicalthing. Andindeednotbecause |
onemightneverdareimputeabruptunforseenchangestoaphysicalthingortoamodel,butbecauseintherealism |
point of view observation is a natural process likeany other and cannot per se bringabout an interruption of the |
orderlyflowofnatural events. |
4 |
−(i/~)Entψ |
For stationary ψ n(t)=e n, the Schr¨odinger equation can be brought into its time- |
independent form Hψ = E ψ . Here, i~∂ ψ (t) = E ψ (t) has been used; E and ψ |
n n n ∂t n n n n n |
stand for the n’th eigenvalue and eigenstate of H, respectively. |
Usually, a physical problem is defined by the Hamiltonian H. The problem of finding the |
physically relevant states reduces to finding a complete set of eigenvalues and eigenstates of |
H. Most elegant solutions utilize the symmetries of the problem, i.e., of H. There exist two |
“canonical” examples, the 1/r-potential and the harmonic oscillator potential, which can be |
solvedwonderfully by this methods (and they are presentedover and overagainin standard |
courses of quantum mechanics), but not many more. (See [30] for a detailed treatment of |
various Hamiltonians H.) |
For a quantum mechanical treatment of a two-state system, see appendix A. For a review of |
the quantum theory of multiple particles, see appendix B. |
3 Quantum information theory |
The fundamental atom of information is the quantum bit, henceforth abbreviated by the term |
‘qbit’. As we shall see, qbits feature quantum mechanics ‘in a nutshell.’ |
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