| Version current |
Version 11 |
| A \emph{unit vector} is a unit-length element of Euclidean space. |
A \emph{unit vector} is a unit-length element of Euclidean space. |
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Equivalently, one may say that the norm of a unit vector is equal
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Equivelently, one may say that the norm of a unit vector is equal
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| to $1$, and write $\Vert \bu\Vert=1$, where $\bu$ is the vector in |
to $1$, and write $\Vert \bu\Vert=1$, where $\bu$ is the vector in |
| question. |
question. |
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| Let $\bv$ be a non-zero vector. To \emph{normalize} $\bv$ is to find |
Let $\bv$ be a non-zero vector. To \emph{normalize} $\bv$ is to find |
| the unique unit vector with the same direction as $\bv$. This is done |
the unique unit vector with the same direction as $\bv$. This is done |
| by multiplying $\bv$ by the reciprocal of its length; the |
by multiplying $\bv$ by the reciprocal of its length; the |
| corresponding unit vector is given by $\bu=\frac{\bv}{\Vert |
corresponding unit vector is given by $\bu=\frac{\bv}{\Vert |
| \bv\Vert}$. |
\bv\Vert}$. |
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| \paragraph{Note:} The concept of a unit vector and normalization makes |
\paragraph{Note:} The concept of a unit vector and normalization makes |
| sense in any vector space equipped with a real or complex norm. |
sense in any vector space equipped with a real or complex norm. |
| Thus, in quantum mechanics one represents states as unit vectors |
Thus, in quantum mechanics one represents states as unit vectors |
| belonging to a (possibly) infinite-dimensional Hilbert space. To |
belonging to a (possibly) infinite-dimensional Hilbert space. To |
| obtain an expression for such states one normalizes |
obtain an expression for such states one normalizes |
| the results of a calculation. |
the results of a calculation. |
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| \paragraph{Example:} Consider $\mathbb{R}^3$ and the vector |
\paragraph{Example:} Consider $\mathbb{R}^3$ and the vector |
| $\bv=(1,2,3)$. The norm (length) is $\sqrt{14}$. Normalizing, we obtain |
$\bv=(1,2,3)$. The norm (length) is $\sqrt{14}$. Normalizing, we obtain |
| the unit vector $\bu$ pointing in the same direction, namely |
the unit vector $\bu$ pointing in the same direction, namely |
| $\bu=\left(\frac{1}{\sqrt{14}},\frac{2}{\sqrt{14}},\frac{3}{\sqrt{14}}\right)$. |
$\bu=\left(\frac{1}{\sqrt{14}},\frac{2}{\sqrt{14}},\frac{3}{\sqrt{14}}\right)$. |
