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In ], ]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 ]), wavefunctions must be ''normalizable'' so the total probability of all outcomes is unity. The total is an ] over ]s like ], or ]s over descrete variables like ] ].

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 ] to be discarded, if their integral diverges over the relavent interval. For example, this disqualifies ]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 ] ], it has no direct interpretation. However, the quantity

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

is ], and positive definite (always greater than zero), and is the ]. This quantity has the interpretation of the probability the system is in a given state. Here, * (asterisk) indicates the ].

==Position normalization==

For one particle in one ], 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 ]. 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 ]. 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 ] under normalization, and consequently associated properties are unchanged.

== See also ==

* ]
* ]
* ]

==References==
{{reflist}}

==External links==
* Normalization.

]

]
]
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