Κανονικοποίηση

Κανονικοποίησις

normalization

Κανονικοποίηση

Κανονικοποίηση

Κανονικοποίηση

Κανονικοποίηση

Κανονικοποίηση
Στάσιμο Κύμα

- Μία διαδικασία.

Ετυμολογία

Η ονομασία "Κανονικοποίηση" σχετίζεται ετυμολογικά με την λέξη "κανόνας".

Εισαγωγή

For now, consider the simple case of a non-relativistic single particle, without spin, in one spatial dimension. More general cases are discussed below.

Position-space wave functions

The state of such a particle is completely described by its wavefunction,

$\Psi(x,t)\,,$

where x is position and t is time.

This is a complex-valued function of two real variablesx and t.

For one spinless particle in 1d, if the wave function is interpreted as a probability amplitude, the square modulus of the wave function, the positive real number

$\left|\Psi(x, t)\right|^2 = {\Psi(x, t)}^{*}\Psi(x, t) = \rho(x, t),$

is interpreted as the probability density that the particle is at x. The asterisk indicates the complex conjugate. If the particle's position is measured, its location cannot be determined from the wave function, but is described by a probability distribution.

The probability that its position x will be in the interval a ≤ x ≤ b is the integral of the density over this interval:

$P_{a\le x\le b} (t) = \int\limits_a^b d x\,|\Psi(x,t)|^2$

where t is the time at which the particle was measured.

This leads to the normalization condition:

$\int\limits_{-\infty}^\infty d x \, |\Psi(x,t)|^2 = 1\,,$

because if the particle is measured, there is 100% probability that it will be somewhere.

For a given system, the set of all possible normalizable wave functions (at any given time) forms an abstract mathematical vector space, meaning that it is possible to add together different wave functions, and multiply wave functions by complex numbers (see vector space for details). Technically, because of the normalization condition, wave functions form a projective space rather than an ordinary vector space. This vector space is infinite-dimensional, because there is no finite set of functions which can be added together in various combinations to create every possible function. Also, it is a Hilbert space, because the inner product of two wave functions Ψ₁ and Ψ₂ can be defined as the complex number (at time t)

$( \Psi_1 , \Psi_2 ) = \int\limits_{-\infty}^\infty d x \, \Psi_1^*(x, t)\Psi_2(x, t).$

More details are given below. Although the inner product of two wave functions is a complex number, the inner product of a wave function Ψ with itself,

$(\Psi,\Psi) = \|\Psi\|^2 \,,$

is always a positive real number. The number!!Ψ!! (not !!Ψ!!²}}) is called the norm of the wave function Ψ, and is not the same as the modulus |Ψ|.

If (Ψ, Ψ) = 1, then Ψ is normalized.

If Ψis not normalized, then dividing by its norm gives the normalized function Ψ/||Ψ||. Two wave functions Ψ₁ and Ψ₂ are orthogonal if (Ψ₁, Ψ₂) = 0. If they are normalized and orthogonal, they are orthonormal. Orthogonality (hence also orthonormality) of wave functions is not a necessary condition wave functions must satisfy, but is instructive to consider since this guarantees linear independence of the functions. In a linear combination of orthogonal wave functions Ψ_n we have,

$\Psi = \sum_n a_n \Psi_n \,,\quad a_n = \frac{( \Psi_n , \Psi )}{( \Psi_n , \Psi_n )}$

If the wave functions Ψ_n were nonorthogonal, the coefficients would be less simple to obtain.

In the Copenhagen interpretation, the modulus squared of the inner product (a complex number) gives a real number

$\left|(\Psi_1,\Psi_2)\right|^2 = P\left(\Psi_2 \rightarrow \Psi_1\right) \,,$

which, assuming both wave functions are normalized, is interpreted as the probability of the wave function Ψ₂ "collapsing" to the new wave function Ψ₁ upon measurement of an observable, whose eigenvalues are the possible results of the measurement, with Ψ₁}} being an eigenvector of the resulting eigenvalue. This is the Born rule, and is one of the fundamental postulates of quantum mechanics.

At a particular instant of time, all values of the wave function Ψ(x, t) are components of a vector. There are uncountably infinitely many of them and integration is used in place of summation. In Bra–ket notation, this vector is written

$|\Psi(t)\rangle = \int dx \Psi(x,t) |x\rangle$

and is referred to as a "quantum state vector", or simply "quantum state".There are several advantages to understanding wave functions as representing elements of an abstract vector space:

• All the powerful tools of linear algebra can be used to manipulate and understand wave functions. For example:
  • Linear algebra explains how a vector space can be given a basis, and then any vector in the vector space can be expressed in this basis. This explains the relationship between a wave function in position space and a wave function in momentum space, and suggests that there are other possibilities too.
  • Bra–ket notation can be used to manipulate wave functions.
• The idea that quantum states are vectors in an abstract vector space is completely general in all aspects of quantum mechanics and quantum field theory, whereas the idea that quantum states are complex-valued "wave" functions of space is only true in certain situations.

The time parameter is often suppressed, and will be in the following. The x coordinate is a continuous index. The Πρότυπο:Ket are the basis vectors, which are orthonormal so their inner product is a delta function;

$\langle x' | x \rangle = \delta(x' - x)$

thus

$\langle x' |\Psi\rangle = \int dx \Psi(x) \langle x'|x\rangle = \Psi(x')$

and

$|\Psi\rangle = \int dx |x\rangle \langle x |\Psi\rangle = \left( \int dx |x\rangle \langle x |\right) |\Psi\rangle$

which illuminates the identity operator

$I = \int dx |x\rangle \langle x | \,.$

Finding the identity operator in a basis allows the abstract state to be expressed explicitly in a basis, and more (the inner product between two state vectors, and other operators for observables, can be expressed in the basis).

Momentum-space wave functions

The particle also has a wave function in momentum space:

$\Phi(p,t)$

where p is the momentum in one dimension, which can be any value from −∞ to +∞, and t is time.

Analogous to the position case, the inner product of two wave functions Φ₁(p, t) and Φ₂(p, t) can be defined as:

$(\Phi_1 , \Phi_2 ) = \int\limits_{-\infty}^\infty d p \, \Phi_1^*(p, t)\Phi_2(p, t) \,.$

One particular solution to the time-independent Schrödinger equation is

$\Psi_p(x) = e^{ipx/\hbar},$

a plane wave, which can be used in the description of a particle with momentum exactly p, since it is an eigenfunction of the momentum operator. These functions are not normalizable to unity (they aren't square-integrable), so they are not really elements of physical Hilbert space. The set

$\{\Psi_p(x, t), -\infty \le p \le \infty\}$

forms what is called the momentum basis. This "basis" is not a basis in the usual mathematical sense. For one thing, since the functions aren't normalizable, they are instead normalized to a delta function,

$(\Psi_{p},\Psi_{p'}) = \delta(p - p').$

For another thing, though they are linearly independent, there are too many of them (they form an uncountable set) for a basis for physical Hilbert space. They can still be used to express all functions in it using Fourier transforms as described next.

Relations between position and momentum representations

The x and p representations are

$\begin{align} |\Psi\rangle = I|\Psi\rangle &= \int |x\rangle \langle x|\Psi\rangle dx = \int \Psi(x) |x\rangle dx,\\ |\Psi\rangle = I|\Psi\rangle &= \int |p\rangle \langle p|\Psi\rangle dp = \int \Phi(p) |p\rangle dp.\end{align}$

Now take the projection of the state Ψ onto eigenfunctions of momentum using the last expression in the two equations,

$\int \Psi(x) \langle p|x\rangle dx = \int \Phi(p') \langle p|p'\rangle dp' = \int \Phi(p') \delta(p-p') dp' = \Phi(p).$

Then utilizing the known expression for suitably normalized eigenstates of momentum in the position representation solutions of the free Schrödinger equation

$\langle x | p \rangle = p(x) = \frac{1}{\sqrt{2\pi\hbar}}e^{\frac{i}{\hbar}px} \Rightarrow \langle p | x \rangle = \frac{1}{\sqrt{2\pi\hbar}}e^{-\frac{i}{\hbar}px},$

one obtains

$\Phi(p) = \frac{1}{\sqrt{2\pi\hbar}}\int \Psi(x)e^{-\frac{i}{\hbar}px}dx\,.$

Likewise, using eigenfunctions of position,

$\Psi(x) = \frac{1}{\sqrt{2\pi\hbar}}\int \Phi(p)e^{\frac{i}{\hbar}px}dp\,.$

The position-space and momentum-space wave functions are thus found to be Fourier transforms of each other. The two wave functions contain the same information, and either one alone is sufficient to calculate any property of the particle. As representatives of elements of abstract physical Hilbert space, whose elements are the possible states of the system under consideration, they represent the same state vector, hence identical physical states, but they are not generally equal when viewed as square-integrable functions.

In practice, the position-space wave function is used much more often than the momentum-space wave function. The potential entering the relevant equation (Schrödinger, Dirac, etc.) determines in which basis the description is easiest. For the harmonic oscillator, x and p enter symmetrically, so there it doesn't matter which description one uses. The same equation (modulo constants) results. From this follows, with a little bit of afterthought, a factoid: The solutions to the wave equation of the harmonic oscillator are eigenfunctions of the Fourier transform in L².

Εσωτερική Αρθρογραφία

• κυματοσυνάρτηση
• διακριτοποίηση
• Γραμμικοποίηση
• Ομαλοποίηση
• Επανακανονικοποίηση, ορθοκανονικοποίηση
• Κανονικότητα
• Κβαντική Φυσική
• Όριο, κανονικοποίηση

Ιστογραφία

• Ομώνυμο άρθρο στην Βικιπαίδεια
• Ομώνυμο άρθρο στην Livepedia
• Normalize-a-Vector
• [ ]

Κίνδυνοι Χρήσης

Αν και θα βρείτε εξακριβωμένες πληροφορίες σε αυτήν την εγκυκλοπαίδεια ωστόσο, παρακαλούμε να λάβετε σοβαρά υπ' όψη ότι η "Sciencepedia" δεν μπορεί να εγγυηθεί, από καμιά άποψη, την εγκυρότητα των πληροφοριών που περιλαμβάνει. "Οι πληροφορίες αυτές μπορεί πρόσφατα να έχουν αλλοιωθεί, βανδαλισθεί ή μεταβληθεί από κάποιο άτομο, η άποψη του οποίου δεν συνάδει με το "επίπεδο γνώσης" του ιδιαίτερου γνωστικού τομέα που σας ενδιαφέρει." Πρέπει να λάβετε υπ' όψη ότι όλα τα άρθρα μπορεί να είναι ακριβή, γενικώς, και για μακρά χρονική περίοδο, αλλά να υποστούν κάποιο βανδαλισμό ή ακατάλληλη επεξεργασία, ελάχιστο χρονικό διάστημα, πριν τα δείτε. Επίσης, Οι διάφοροι "Εξωτερικοί Σύνδεσμοι (Links)" (όχι μόνον, της Sciencepedia αλλά και κάθε διαδικτυακού ιστότοπου (ή αλλιώς site)), αν και άκρως απαραίτητοι, είναι αδύνατον να ελεγχθούν (λόγω της ρευστής φύσης του Web), και επομένως είναι ενδεχόμενο να οδηγήσουν σε παραπλανητικό, κακόβουλο ή άσεμνο περιεχόμενο. Ο αναγνώστης πρέπει να είναι εξαιρετικά προσεκτικός όταν τους χρησιμοποιεί.

- Μην κάνετε χρήση του περιεχομένου της παρούσας εγκυκλοπαίδειας
αν διαφωνείτε με όσα αναγράφονται σε αυτήν

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>>Διαμαρτυρία προς την wikia<<

- Όχι, στις διαφημίσεις που περιέχουν απαράδεκτο περιεχόμενο (άσεμνες εικόνες, ροζ αγγελίες κλπ.)

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