Παρατηρήσιμον Μέγεθος

Μέτρηση
Φυσικό Μέγεθος


Άλγεβρα Jordan
Παρατηρήσιμο Μέγεθος
Απειροστός Γεννήτορας

Ορμή
Στροφορμή
In many cases the molecular rotation spectra of molecules
can be described successfully with
the assumption that they rotate as rigid rotors.
In these cases the energies can be modeled in a manner parallel to
the classical description of the rotational kinetic energy of a rigid object.
From these descriptions, structural information can be obtained (bond lengths and angles).
The most straightforward examples are those of diatomic molecules.
Energy calculations in quantum mechanics
involve the solution of the Schrodinger equation with
a properly formulated Hamiltonian to represent the energy operator.
The form of the Hamiltonian can often be implied from
the nature of the classical energy of such a physical system.
The process involves finding the quantum mechanical operators
associated with the constituents of the system energy.
The energy of a freely rotating rigid rotor
is simply the rotational kinetic energy,
which can be expressed in terms of the angular momentum.
The general form of operators associated with momenta are these.
- Ένα είδος Φυσικών Μεγεθών
Ετυμολογία[]
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Είδη
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Α. Κινηματική |
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Β. Δυναμική |
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Γ. Κυματική |
ύπαρξη «πρόκλησης» |
Δ. Ελαστική Δυναμική |
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Η ονομασία "Μέγεθος" σχετίζεται ετυμολογικά με την λέξη "μεγέθυνση".
Ορισμός[]
Παρατηρήσιμο μέγεθος ενός συστήματος είναι εκείνο το Φυσικό Μέγεθος που μπορεί να μετρηθεί καβαντικά (θέση, ορμή, ενέργεια, στροφορμή κλπ)
Εισαγωγή[]
In physics, particularly in quantum physics, a system observable is a measurable operator, or gauge, where the property of the system state can be determined by some sequence of physical operations. For example, these operations might involve submitting the system to various electromagnetic fields and eventually reading a value off some gauge. In systems governed by classical mechanics, any experimentally observable value can be shown to be given by a real-valued function on the set of all possible system states.
Physically meaningful observables must also satisfy transformation laws which relate observations performed by different observers in different frames of reference. These transformation laws are automorphisms of the state space, that is bijective transformations which preserve some mathematical property.
Φάσμα μεγέθους[]
Φάσμα (spectrum) ενός παρατηρήσιμου μεγέθους είναι το σύνολο των τιμών μίας μέτρησης για του μεγέθους σε δεδομένο Κβαντικό Σύστημα.
π.χ.
- το φάσμα της θέσης (x) ελεύθερου σωματίου στο χώρο είναι απο –∞ έως + ∞ ενώ
- το φάσμα της κινητικής ενέργειας (Ε) είναι απο 0 έως +∞.
- η προβολή της στροφορμής κατα μήκος οποιουδήποτε άξονα έχει διακριτό φάσμα (δεν μπορεί να πάρει συνεχείς τιμές):
Quantum mechanics[]
In quantum physics, the relation between system state and the value of an observable requires some basic linear algebra for its description. In the mathematical formulation of quantum mechanics, states are given by non-zero vectors in a Hilbert space V (where two vectors are considered to specify the same state if, and only if, they are scalar multiples of each other) and observables are given by self-adjoint operators on V. However, as indicated below, not every self-adjoint operator corresponds to a physically meaningful observable. For the case of a system of particles, the space V consists of functions called wave functions or state vectors.
In the case of transformation laws in quantum mechanics, the requisite automorphisms are unitary (or antiunitary) linear transformations of the Hilbert space V. Under Galilean relativity or special relativity, the mathematics of frames of reference is particularly simple, and in fact restricts considerably the set of physically meaningful observables.
In quantum mechanics, measurement of observables exhibits some seemingly unintuitive properties. Specifically, if a system is in a state described by a vector in a Hilbert space, the measurement process affects the state in a non-deterministic, but statistically predictable way. In particular, after a measurement is applied, the state description by a single vector may be destroyed, being replaced by a statistical ensemble. The irreversible nature of measurement operations in quantum physics is sometimes referred to as the measurement problem and is described mathematically by quantum operations. By the structure of quantum operations, this description is mathematically equivalent to that offered by relative state interpretation where the original system is regarded as a subsystem of a larger system and the state of the original system is given by the partial trace of the state of the larger system.
In quantum mechanics each dynamical variable (e.g. position, translational momentum, orbital angular momentum, spin, total angular momentum, energy, etc.) is associated with a Hermitian operator that acts on the state of the quantum system and whose eigenvalues correspond to the possible values of the dynamical variable. For example, suppose is an eigenket (eigenvector) of the observable , with eigenvalue , and exists in a d-dimensional Hilbert space. Then
- =
This eigenket equation says that if a measurement of the observable is made while the system of interest is in the state , then the observed value of that particular measurement must return the eigenvalue with certainty. However, if the system of interest is in the general state , then the eigenvalue is returned with probability (Born rule). One must note that the above definition is somewhat dependent upon our convention of choosing real numbers to represent real physical quantities. Indeed, just because dynamical variables are "real" and not "unreal" in the metaphysical sense does not mean that they must correspond to real numbers in the mathematical sense.
To be more precise, the dynamical variable/observable is a (not necessarily bounded) Hermitian operator in a Hilbert space and thus is represented by a Hermitian matrix if the space is finite-dimensional. In an infinite-dimensional Hilbert space, the observable is represented by a Symmetric operator, which may not be defined everywhere (i.e. its domain is not the whole space - there exist some states that are not in the domain of the operator). The reason for such a change is that in an infinite-dimensional Hilbert space, the operator becomes unbounded, which means that it no longer has a largest eigenvalue. This is not the case in a finite-dimensional Hilbert space, where every operator is bounded - it has a largest eigenvalue. For example, if we consider the position of a point particle moving along a line, this particle's position variable can take on any number on the real-line, which is uncountably infinite. Since the eigenvalue of an observable represents a real physical quantity for that particular dynamical variable, then we must conclude that there is no largest eigenvalue for the position observable in this uncountably infinite-dimensional Hilbert space, since the field we're working over consists of the real-line. Nonetheless, whether we are working in an infinite-dimensional or finite-dimensional Hilbert space, the role of an observable in quantum mechanics is to assign real numbers to outcomes of particular measurements; this means that only certain measurements can determine the value of an observable for some state of a quantum system. In classical mechanics, any measurement can be made to determine the value of an observable.
Incompatibility of observables in quantum mechanics[]
A crucial difference between classical quantities and quantum mechanical observables is that the latter may not be simultaneously measurable. This is mathematically expressed by non-commutativity of the corresponding operators, to the effect that
This inequality expresses a dependence of measurement results on the order in which measurements of observables and are performed. Observables corresponding to non-commutative operators are called incompatible.
η ιδιότητα των μητρών
να μην έχουν την μεταθετικότητα (commutativity) στον πολλαπλασιασμό τους
είχε τεράστια σημασία για την Κβαντομηχανική
καθώς οι σκαπανείς της
ήταν οι πρώτοι που διαπίστωσαν ότι
τα κβαντικά μεγέθη δεν συμπεριφερόταν όπως τα κλασσικά μεγέθη
(στα οποία ίσχυε η μεταθετικότητα, ολικά,
δηλ. ab = ba ή με φυσικά μεγέθη, xp = px)
αλλά μερικές φορές
εμφανιζόταν ένας μη-μηδενικός όρος (π.χ. το ih/2π)
Οπότε έτσι
αναγκάσθηκαν να δεχθούν την αρχή της αβεβαιότητας
Έτσι, στην συνέχεια,
σκέφτηκαν ότι ενώ
1) τα κλασσικά μεγέθη
ήταν απλές συναρτήσεις (όπου ίσχυε η μεταθετικότητα)
2) τα κβαντικά μεγέθη
δεν μπορούν να είναι το ίδιο
άρα θα ήταν μήτρες (στις οποίες δεν ισχύει η μεταθετικότητα)
Οπότε
αφού όλα αυτά
αποδείχθηκαν να δουλεύουν σωστά
στα χιλιάδες πειράματα σωματιδίων
θεμελιώθηκε η σύγχρονη Κβαντική Φυσική
Υποσημειώσεις[]
Εσωτερική Αρθρογραφία[]
- Φυσικό Μέγεθος, Μαθηματικό Μέγεθος
- Παρατηρήσιμο Μέγεθος
- Μονόμετρο Μέγεθος, Διανυσματικό Μέγεθος, Τανυστικό Μέγεθος
- Εντατικό Μέγεθος, Εκτατικό Μέγεθος
- Δυναμικό Μέγεθος, Δυνητικό Μέγεθος
- Παρατηρήσιμο Μέγεθος (observable)
- Κβαντική Κατάσταση
- Observable universe
- Φυσικός Παρατηρητής
- Όργανο Καταμέτρησης
- Μονάδα Μέτρησης
Βιβλιογραφία[]
- S. Auyang, How is Quantum Field Theory Possible, Oxford University Press, 1995.
- G. Mackey, Mathematical Foundations of Quantum Mechanics, W. A. Benjamin, 1963.
- V. Varadarajan, The Geometry of Quantum Mechanics vols 1 and 2, Springer-Verlag 1985.
- Leslie E. Ballentine, "Quantum Mechanics: A Modern Development", World Scientific, 1998
- R. Blume-Kohout, "Lecture 14: and Hilbert space. Wavefunctions, unbounded operators, and rigged Hilbert space.", www.am473.ca, 10/26/08
- Hermann Weyl, "Quantum Physics and Causality", Appendix C in "Philosophy of Mathematics and Natural Science", Princeton University Press, 1949.
Ιστογραφία[]
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Αν και θα βρείτε εξακριβωμένες πληροφορίες "Οι πληροφορίες αυτές μπορεί πρόσφατα Πρέπει να λάβετε υπ' όψη ότι Επίσης, |
- Μην κάνετε χρήση του περιεχομένου της παρούσας εγκυκλοπαίδειας
αν διαφωνείτε με όσα αναγράφονται σε αυτήν
- Όχι, στις διαφημίσεις που περιέχουν απαράδεκτο περιεχόμενο (άσεμνες εικόνες, ροζ αγγελίες κλπ.)