Normalizable wave function - Revision history

Bernanke's Crossbow: Further specificity

2021-07-17T22:09:40Z

Further specificity

← Previous revision		Revision as of 22:09, 17 July 2021
Line 1:		Line 1:
	#REDIRECT [[Wave function]] {{R from merge}}		#REDIRECT [[Wave function#Normalization condition]] {{R from merge}}

RockMagnetist: Corrected redirect (wave -> Wave)

2012-01-21T17:40:42Z

Corrected redirect (wave -> Wave)

← Previous revision		Revision as of 17:40, 21 January 2012
Line 1:		Line 1:
	#REDIRECT [[~~wave~~ function]]		#REDIRECT [[Wave function]] {{R from merge}}

F=q(E+v^B): ←Redirected page to Wave function

2012-01-21T17:26:27Z

←Redirected page to Wave function

← Previous revision		Revision as of 17:26, 21 January 2012
Line 1:		Line 1:
			#REDIRECT [[wave function]]
	{{Merge to\|Wave function \|discuss=talk:Wave function#Merger proposal\|date=January 2012}}
	In [[quantum mechanics]], [[wave function]]s are probability amplitudes containing information about the quantum system. The outcomes of measuring a quantum state, such as the location of a particle, are prescribed by the wavefunction, and since probabilities sum to 1 (one of the [[Probability axioms\|axioms]]), wavefunctions must be ''normalizable'' so the total probability of all outcomes is unity. The total is an [[integral (calculus)\|integral]] over [[continuous variable]]s like [[position (vector)\|position]], or [[summation\|sum]]s over descrete variables like [[spin (physics)\|spin]] [[angular momentum]].

	For the case of the position of a particle, the total probability of finding the particle is 1 - if the particle is to exists somewhare<ref>{{Cite book
	\| last = Griffiths
	\| first = David J.
	\| authorlink = David J. Griffiths
	\| title = Introduction to Quantum Mechanics
	\| publisher = Benjamin Cummings
	\| date = April 10, 2004
	\| page = 11
	\| isbn = 0131118927}}</ref>. For given boundary conditions, this enables solutions to the [[Schrödinger equation]] to be discarded, if their integral diverges over the relavent interval. For example, this disqualifies [[periodic function]]s as wave function solutions for infinite intervals, while those functions can be solutions for finite intervals.

	==Probability and probability density==

	In general, Ψ is a [[complex number\|complex]] [[function (mathematics)\|function]], it has no direct interpretation. However, the quantity

	:<math>\rho \psi^* \psi = \mid \psi \mid ^2</math>

	is [[real number\|real]], and positive definite (always greater than zero), and is the [[probability density function]]. This quantity has the interpretation of the probability the system is in a given state. Here, * (asterisk) indicates the [[complex conjugate]].

	==Position normalization==

	For one particle in one [[dimension]], the normalization condition is:

	:<math>\int_{-\infty}^{\infty} \left \| \Psi(x,t) \right \|^2 dx=1</math>

	where the integration is in the interval (meaning "from <math>\scriptstyle -\infty</math> to <math>\scriptstyle \infty</math>") indicates that the probability that the particle exists ''somewhere'' is unity.

	For three dimensions, the integral is over all of space

	:<math>\iiint_{{\rm all \, space}} \left \|\psi(\bold, t) \right \|^2 d^3 \bold{r} =1 </math>

	==Derivation of normalization==

	This means that

	:<math>p(-\infty < x < \infty) = \int_{-\infty}^{\infty} \mid \psi \mid ^2 dx. \quad (1)</math>

	where <math>p(x)</math> is the probability of finding the particle at <math> x </math>. Equation (1) is given by the definition of a [[probability density function]]. Since the particle exists, its probability of being anywhere in space must be equal to 1. Therefore we integrate over all space:

	:<math>p(-\infty < x < \infty) = \int_{-\infty}^{\infty} \mid \psi \mid ^2 dx = 1. \quad (2)</math>

	If the integral is finite, we can multiply the wave function, Ψ, by a constant such that the integral is equal to 1. Alternatively, if the wave function already contains an appropriate arbitrary constant, we can solve equation (2) to find the value of this constant which normalizes the wave function.

	==Plane-waves==

	Plane waves are normalized in a box or to a Dirac delta in the continuum approach. They are not normalizable over all space, since the integral doesn't converge.

	==Example of normalization==

	A particle is restricted to a 1D region between <math>x=0</math> and <math>x=l</math>; its wave function is:

	:<math>\psi (x,t) = \begin{cases} Ae^{i(kx-\omega t)}, \quad 0 \le x \le l \\ 0, \quad \text{elsewhere}. \end{cases}</math>

	To normalize the wave function we need to find the value of the arbitrary constant <math>A</math>; i.e., solve

	:<math> \int_{-\infty}^{\infty} \mid \psi \mid ^2 dx = 1 </math>

	to find <math>A</math>.

	Substituting Ψ into <math> \mid \psi \mid ^2 </math> we get

	:<math> \mid \psi \mid ^2 = A^2 e^{i(kx - \omega t)} e^{-i(kx - \omega t)} =A^2 </math>

	so,

	:<math> \int_{-\infty}^{0} 0 dx + \int_{0}^{l} A^2 dx + \int_{l}^{\infty} 0 dx = 1 </math>

	therefore;

	:<math>A^2 l = 1 \Rightarrow A = \left ( \frac{1}{\sqrt{l}} \right ).</math>

	Hence, the normalized wave function is:

	:<math> \psi (x,t) = \begin{cases} \left ( \frac{1}{\sqrt{l}} \right )e^{i(kx-\omega t)}, \quad 0 \le x \le l \\ 0, \quad \text{elsewhere.} \end{cases}</math>

	==Normalization invariance==

	It is important that the properties associated with the wave function are invariant under normalization. If normalization of a wave function changed the properties associated with the wave function, the process becomes pointless as we still cannot yield any information about the properties of the particle associated with the un-normalized wave function.

	All properties of the particle such as probability distribution, momentum, energy, expectation value of position etc.; are solved from the [[Schrödinger equation]]. Since the Schrödinger equation is linear, it is simple to see that the properties unchanged wave function is normalized.

	The Schrödinger equation is:

	:<math> \frac{-\hbar^2}{2m} \frac{d^2 \psi}{d x^2} + V(x) \psi (x) = E \psi (x). </math>

	If Ψ is normalized and replaced with <math>A\psi</math>, then the equation becomes:

	:<math> \frac{-\hbar^2}{2m} A\frac{d^2 \psi}{d x^2} + V(x) A \psi (x) = E A \psi(x)</math>

	:<math> \rightarrow A \left ( \frac{-\hbar^2}{2m} \frac{d^2 \psi}{d x^2} + V(x) \psi (x) \right ) = A \left ( E \psi (x) \right )</math>

	:<math> \rightarrow \frac{-\hbar^2}{2m} \frac{d^2 \psi}{d x^2} + V(x) \psi (x) = E \psi (x) </math>

	which is the original Schrödinger wave equation. That is to say, the Schrödinger wave equation is [[invariant (mathematics)\|invariant]] under normalization, and consequently associated properties are unchanged.

	== See also ==

	* [[Quantum mechanics]]
	* [[Schrödinger equation]]
	* [[Wave function]]

	==References==
	{{reflist}}

	==External links==
	* [http://cat.middlebury.edu/~chem/chemistry/class/physical/quantum/help/normalize/normalize.html] Normalization.

	[[Category:Quantum mechanics]]

	[[es:Función de onda normalizable]]
	[[ja:規格化]]
	[[fi:Normitettu aaltofunktio]]
	[[zh:歸一條件]]

AnomieBOT: Dating maintenance tags: {{Merge to}}

2012-01-21T16:53:44Z

Dating maintenance tags: {{Merge to}}

← Previous revision		Revision as of 16:53, 21 January 2012
Line 1:		Line 1:
	{{Merge to \|Wave function \|discuss=talk:Wave function#Merger proposal}}		{{Merge to\|Wave function \|discuss=talk:Wave function#Merger proposal\|date=January 2012}}
	In [[quantum mechanics]], [[wave function]]s are probability amplitudes containing information about the quantum system. The outcomes of measuring a quantum state, such as the location of a particle, are prescribed by the wavefunction, and since probabilities sum to 1 (one of the [[Probability axioms\|axioms]]), wavefunctions must be ''normalizable'' so the total probability of all outcomes is unity. The total is an [[integral (calculus)\|integral]] over [[continuous variable]]s like [[position (vector)\|position]], or [[summation\|sum]]s over descrete variables like [[spin (physics)\|spin]] [[angular momentum]].		In [[quantum mechanics]], [[wave function]]s are probability amplitudes containing information about the quantum system. The outcomes of measuring a quantum state, such as the location of a particle, are prescribed by the wavefunction, and since probabilities sum to 1 (one of the [[Probability axioms\|axioms]]), wavefunctions must be ''normalizable'' so the total probability of all outcomes is unity. The total is an [[integral (calculus)\|integral]] over [[continuous variable]]s like [[position (vector)\|position]], or [[summation\|sum]]s over descrete variables like [[spin (physics)\|spin]] [[angular momentum]].

RockMagnetist: Proposed merger into Wave function

2012-01-21T16:35:12Z

Proposed merger into Wave function

← Previous revision		Revision as of 16:35, 21 January 2012
Line 1:		Line 1:
			{{Merge to \|Wave function \|discuss=talk:Wave function#Merger proposal}}
	In [[quantum mechanics]], [[wave function]]s are probability amplitudes containing information about the quantum system. The outcomes of measuring a quantum state, such as the location of a particle, are prescribed by the wavefunction, and since probabilities sum to 1 (one of the [[Probability axioms\|axioms]]), wavefunctions must be ''normalizable'' so the total probability of all outcomes is unity. The total is an [[integral (calculus)\|integral]] over [[continuous variable]]s like [[position (vector)\|position]], or [[summation\|sum]]s over descrete variables like [[spin (physics)\|spin]] [[angular momentum]].		In [[quantum mechanics]], [[wave function]]s are probability amplitudes containing information about the quantum system. The outcomes of measuring a quantum state, such as the location of a particle, are prescribed by the wavefunction, and since probabilities sum to 1 (one of the [[Probability axioms\|axioms]]), wavefunctions must be ''normalizable'' so the total probability of all outcomes is unity. The total is an [[integral (calculus)\|integral]] over [[continuous variable]]s like [[position (vector)\|position]], or [[summation\|sum]]s over descrete variables like [[spin (physics)\|spin]] [[angular momentum]].

F=q(E+v^B): extend scope

2012-01-21T15:08:46Z

extend scope

Show changes

Yakeyglee: /* Derivation of normalization */

2011-11-03T01:01:49Z

Derivation of normalization

← Previous revision		Revision as of 01:01, 3 November 2011
Line 22:		Line 22:
	:<math>\psi^* \psi = \mid \psi \mid ^2</math>		:<math>\psi^* \psi = \mid \psi \mid ^2</math>

	is [[real number\|real]], greater than or equal to zero, and is known as a [[probability density function]].		is [[real number\|real]], greater than or equal to zero, and is known as a [[probability density function]]. Here, <math>\psi^*</math> indicates the complex conjugate.

	This means that		This means that

92.13.142.193 at 09:01, 13 October 2011

2011-10-13T09:01:50Z

← Previous revision		Revision as of 09:01, 13 October 2011
Line 15:		Line 15:
	where the integration from <math>{-\infty}</math> to <math>{\infty}</math> indicates that the probability that the particle exists ''somewhere'' is unity.		where the integration from <math>{-\infty}</math> to <math>{\infty}</math> indicates that the probability that the particle exists ''somewhere'' is unity.

	All wave functions which represent real particles must be normalizable, that is, they must have a total probability of one - they must describe the probability of the particle existing as 100%. ~~This~~ trait enables anyone who solves the [[Schrödinger equation]] ~~for certain boundary conditions~~ to discard solutions which do not have a finite integral at a given interval. For example, this disqualifies [[periodic function]]s as wave function solutions for infinite intervals, while those functions can be solutions for finite intervals.		All wave functions which represent real particles must be normalizable, that is, they must have a total probability of one - they must describe the probability of the particle existing as 100%. For certain boundary conditions, this trait enables anyone who solves the [[Schrödinger equation]] to discard solutions which do not have a finite integral at a given interval. For example, this disqualifies [[periodic function]]s as wave function solutions for infinite intervals, while those functions can be solutions for finite intervals.

	==Derivation of normalization==		==Derivation of normalization==

90.216.193.249 at 14:32, 19 January 2011

2011-01-19T14:32:19Z

← Previous revision		Revision as of 14:32, 19 January 2011
Line 11:		Line 11:

	:<math>\int_{-\infty}^{\infty} \psi^*(x)\psi(x)\ dx=1</math>		:<math>\int_{-\infty}^{\infty} \psi^*(x)\psi(x)\ dx=1</math>
			Or identically:

			:<math>\int_{-\infty}^{\infty} \left \|\psi(x) \right \|^2 dx=1</math>
	where the integration from <math>{-\infty}</math> to <math>{\infty}</math> indicates that the probability that the particle exists ''somewhere'' is unity.		where the integration from <math>{-\infty}</math> to <math>{\infty}</math> indicates that the probability that the particle exists ''somewhere'' is unity.

Larryisgood: cleaning up wording

2010-11-12T16:33:19Z

cleaning up wording

← Previous revision		Revision as of 16:33, 12 November 2010
Line 10:		Line 10:
	\| isbn = 0131118927}}</ref> [[Mathematics\|Mathematically]], in one [[dimension]] this is expressed as:		\| isbn = 0131118927}}</ref> [[Mathematics\|Mathematically]], in one [[dimension]] this is expressed as:

	<math>\int_{-\infty}^{\infty} \psi^*(x)\psi(x)\ dx=1</math>		:<math>\int_{-\infty}^{\infty} \psi^*(x)\psi(x)\ dx=1</math>

	~~in which~~ the integration ~~parameters~~ ~~''A''~~ ~~and~~ ~~''B''~~ ~~indicate~~ the ~~interval~~ ~~in which~~ the particle ~~must~~ ~~exist~~.		where the integration from <math>{-\infty}</math> to <math>{\infty}</math> indicates that the probability that the particle exists ''somewhere'' is unity.

	All wave functions which represent real particles must be normalizable, that is, they must have a total probability of one - they must describe the probability of the particle existing as 100%. This trait enables anyone who solves the [[Schrödinger equation]] for certain boundary conditions to discard solutions which do not have a finite integral at a given interval. For example, this disqualifies [[periodic function]]s as wave function solutions for infinite intervals, while those functions can be solutions for finite intervals.		All wave functions which represent real particles must be normalizable, that is, they must have a total probability of one - they must describe the probability of the particle existing as 100%. This trait enables anyone who solves the [[Schrödinger equation]] for certain boundary conditions to discard solutions which do not have a finite integral at a given interval. For example, this disqualifies [[periodic function]]s as wave function solutions for infinite intervals, while those functions can be solutions for finite intervals.