Coordinate space

All physical phenomena can be described in different spaces: coordinate, pulsed , phase , etc. Descriptions are mathematically equivalent, but differ in the complexity and intuitiveness of the description. In most cases, the coordinate space is intuitive and the easiest to understand the process that takes place in it, however, in solid state physics it is generally more convenient to use an impulse description.

Content

1 Definition
2 Properties
3 Operators in coordinate space
4 See also
5 notes
6 Literature

Definition

Call ^[1] $n$ ${\ displaystyle n}$ -dimensional vector $n$ ${\ displaystyle n}$ field numbers $P,$ ${\ displaystyle P,}$ these numbers are the coordinates of the vector ${\vec {r}}={\vec {r}}(r_{1},r_{2},\ldots ,r_{n}).$ ${\ displaystyle {\ vec {r}} = {\ vec {r}} (r_ {1}, r_ {2}, \ ldots, r_ {n}).}$ For definiteness, they say that this vector ${\vec {r}}$ ${\ displaystyle {\ vec {r}}}$ is a radius vector , although this is not necessary.

A bunch of $n$ ${\ displaystyle n}$ -dimensional vectors for which operations are defined:

${\vec {a}}={\vec {b}}\;\leftrightarrow \;\left\{{\begin{matrix}a_{1}=b_{1}\\a_{2}=b_{2}\\\ldots \\a_{n}=b_{n}\end{matrix}}\right.$ ${\ displaystyle {\ vec {a}} = {\ vec {b}} \; \ leftrightarrow \; \ left \ {{\ begin {matrix} a_ {1} = b_ {1} \\ a_ {2} = b_ {2} \\\ ldots \\ a_ {n} = b_ {n} \ end {matrix}} \ right.}$
${\vec {a}}+{\vec {b}}=(a_{1}+b_{1},\;a_{2}+b_{2},\ldots ,a_{n}+b_{n})$ ${\ displaystyle {\ vec {a}} + {\ vec {b}} = (a_ {1} + b_ {1}, \; a_ {2} + b_ {2}, \ ldots, a_ {n} + b_ {n})}$
$\lambda \cdot {\vec {a}}=(\lambda \cdot a_{1},\;\lambda \cdot a_{2},\ldots ,\lambda \cdot a_{n})$ ${\ displaystyle \ lambda \ cdot {\ vec {a}} = (\ lambda \ cdot a_ {1}, \; \ lambda \ cdot a_ {2}, \ ldots, \ lambda \ cdot a_ {n})}$

are called $n-$ ${\ displaystyle n-}$ dimensional arithmetic space or $n$ ${\ displaystyle n}$ -dimensional coordinate space $P^{n}$ ${\ displaystyle P ^ {n}}$ .

Properties

Let be $\exists {\vec {0}}=(0,0,\ldots ,0),{\vec {a}}=(a_{1},a_{2},\ldots ,a_{n}),\lambda \in \mathbb {R} ,\mu \in \mathbb {R}$ ${\ displaystyle \ exists {\ vec {0}} = (0,0, \ ldots, 0), {\ vec {a}} = (a_ {1}, a_ {2}, \ ldots, a_ {n} ), \ lambda \ in \ mathbb {R}, \ mu \ in \ mathbb {R}}$

Associativity :

({\vec {a}}+{\vec {b}})+{\vec {c}}={\vec {a}}+({\vec {b}}+{\vec {c}})

{\ displaystyle ({\ vec {a}} + {\ vec {b}}) + {\ vec {c}} = {\ vec {a}} + ({\ vec {b}} + {\ vec { c}})}

Commutativity :

{\vec {a}}+{\vec {b}}={\vec {b}}+{\vec {a}}

{\ displaystyle {\ vec {a}} + {\ vec {b}} = {\ vec {b}} + {\ vec {a}}}

The uniqueness of the solution of the equation :

\forall {\vec {a}},{\vec {b}}\;\exists !\;{\vec {x}}\in P^{n}\;:\;{\vec {a}}+{\vec {x}}={\vec {b}}

{\ displaystyle \ forall {\ vec {a}}, {\ vec {b}} \; \ exists! \; {\ vec {x}} \ in P ^ {n} \;: \; {\ vec { a}} + {\ vec {x}} = {\ vec {b}}}

The existence of a neutral element :

\forall {\vec {a}}\;:\;{\vec {a}}+{\vec {0}}={\vec {a}}

{\ displaystyle \ forall {\ vec {a}} \;: \; {\ vec {a}} + {\ vec {0}} = {\ vec {a}}}

The existence of the opposite vector:

\forall {\vec {a}}\;\exists {\vec {b}}(b_{1}=-a_{1},\;b_{2}=-a_{2},\ldots ,b_{n}=-a_{n})\;:\;{\vec {a}}+{\vec {b}}={\vec {0}}

{\ displaystyle \ forall {\ vec {a}} \; \ exists {\ vec {b}} (b_ {1} = - a_ {1}, \; b_ {2} = - a_ {2}, \ ldots , b_ {n} = - a_ {n}) \;: \; {\ vec {a}} + {\ vec {b}} = {\ vec {0}}}

Scalar multiplication associativity:

\forall \lambda ,\mu \in \mathbb {R} \;:\;\lambda \cdot (\mu \cdot {\vec {a}})=(\lambda \cdot \mu )\cdot {\vec {a}}

{\ displaystyle \ forall \ lambda, \ mu \ in \ mathbb {R} \;: \; \ lambda \ cdot (\ mu \ cdot {\ vec {a}}) = (\ lambda \ cdot \ mu) \ cdot {\ vec {a}}}

The distributivity of multiplication relative to the addition of scalars:

(\lambda +\mu )\cdot {\vec {a}}=\lambda \cdot {\vec {a}}+\mu \cdot {\vec {a}}

{\ displaystyle (\ lambda + \ mu) \ cdot {\ vec {a}} = \ lambda \ cdot {\ vec {a}} + \ mu \ cdot {\ vec {a}}}

The distributivity of multiplication relative to the addition of vectors:

\lambda ({\vec {a}}+{\vec {b}})=\lambda \cdot {\vec {a}}+\lambda \cdot {\vec {b}}

{\ displaystyle \ lambda ({\ vec {a}} + {\ vec {b}}) = \ lambda \ cdot {\ vec {a}} + \ lambda \ cdot {\ vec {b}}}

The existence of basis vectors:

Let be

\left\{{\begin{matrix}{\vec {v_{1}}}={\vec {v_{1}}}(1,0,\ldots ,0)\\{\vec {v_{2}}}={\vec {v_{2}}}(0,1,\ldots ,0)\\\vdots \\{\vec {v_{n}}}={\vec {v_{n}}}(0,0,\ldots ,1)\end{matrix}}\right.

{\ displaystyle \ left \ {{\ begin {matrix} {\ vec {v_ {1}}} = {\ vec {v_ {1}}} (1,0, \ ldots, 0) \\ {\ vec { v_ {2}}} = {\ vec {v_ {2}}} (0,1, \ ldots, 0) \\\ vdots \\ {\ vec {v_ {n}}} = {\ vec {v_ { n}}} (0,0, \ ldots, 1) \ end {matrix}} \ right.}

Then

These vectors are linearly independent
Any vector ${\vec {v}}={\vec {v}}(v_{1},v_{2},\ldots ,v_{n})$ ${\ displaystyle {\ vec {v}} = {\ vec {v}} (v_ {1}, v_ {2}, \ ldots, v_ {n})}$ can be imagined as ${\vec {v}}=v_{1}\cdot {\vec {v_{1}}}+v_{2}\cdot {\vec {v_{2}}}+\ldots +v_{n}\cdot {\vec {v_{n}}}$ ${\ displaystyle {\ vec {v}} = v_ {1} \ cdot {\ vec {v_ {1}}} + v_ {2} \ cdot {\ vec {v_ {2}}} + \ ldots + v_ { n} \ cdot {\ vec {v_ {n}}}}$

Operators in coordinate space

All operators can be generalized to $n-$ ${\ displaystyle n-}$ dimensional case, however, for simplicity, only three-dimensional cases will be considered in this section.

Laplacian :

\Delta ={\partial ^{2} \over \partial x^{2}}+{\partial ^{2} \over \partial y^{2}}+{\partial ^{2} \over \partial z^{2}}

{\ displaystyle \ Delta = {\ partial ^ {2} \ over \ partial x ^ {2}} + {\ partial ^ {2} \ over \ partial y ^ {2}} + {\ partial ^ {2} \ over \ partial z ^ {2}}}

Nabla :

\nabla ={\partial  \over \partial x}{\vec {i}}+{\partial  \over \partial y}{\vec {j}}+{\partial  \over \partial z}{\vec {k}}

{\ displaystyle \ nabla = {\ partial \ over \ partial x} {\ vec {i}} + {\ partial \ over \ partial y} {\ vec {j}} + {\ partial \ over \ partial z} { \ vec {k}}}

Laplace vector operator ^[2] :

{\vec {\Delta }}{\vec {A}}=\nabla (\nabla \cdot {\vec {A}})-\nabla \times (\nabla \times {\vec {A}})

{\ displaystyle {\ vec {\ Delta}} {\ vec {A}} = \ nabla (\ nabla \ cdot {\ vec {A}}) - \ nabla \ times (\ nabla \ times {\ vec {A} })}

Impulse operator :

\mathbf {\hat {p}} =-i\hbar \nabla

{\ displaystyle \ mathbf {\ hat {p}} = -i \ hbar \ nabla}

Notes

↑ Alexandrov P. S. Lectures on analytic geometry. - M .: Nauka, 1968 .-- S. 154-155. - 912 p.
↑ Weisstein, Eric W. Vector Laplacian on the Wolfram MathWorld website.

Literature

[1] Alexandrov P. S. Lectures on analytic geometry. - M .: Nauka, 1968 .-- S. 154-155. - 912 p.

[2] Weisstein, Eric W. Vector Laplacian on the Wolfram MathWorld website.