Linear Spaces

Definition

A linear (vector) space is a set of elements (vectors) with an operation of vector addition and scalar multiplication.

Let $\psi,\phi,\chi$ be vectors in $\mathcal{V}$. And $c,d$ are scalars.

Addition Rules

Their sum is a vector in $\mathcal{V}$, $(\psi+\phi)\in \mathcal{V}$

Commutativity

$\psi+\phi=\phi+\psi$

Associativity

$(\psi+\phi)+\chi=\psi+(\phi+\chi)$

Zero vector

$0 + \psi = \psi$

Inverse vector

$\psi + (-\psi) = 0$

Multiplication Rules

$c\psi\in \mathcal{V}$, $(c\psi+d\phi)\in \mathcal{V}$

Distributivity over addition

$c(\psi+\phi)=c\psi+c\phi$

Compatibility

$c(d\psi)=(cd)\psi$

Identity

$I\psi=\psi I=\psi$, $0\psi=\psi 0=0$

Example of Linear Spaces

$\psi=(x_1,x_2,\cdots,x_n)$, $\phi=(y_1,y_2,\cdots,y_n)$, $\psi+\phi=(x_1+y_1,\cdots,x_n+y_n)$, $c\psi=(cx_1,\cdots,cx_n)$.
(x_1,x_2,⋯), $\sum_i^\infty |x_i|^2$ is finite. $ℓ²$-space
Set of continuous functions
Set of functions with $\int|\psi(x)|^2dx$ is finite. $L²$-spaces
Hilbert Space

Quantum States

State Vector

$|\psi\rangle \in \mathcal{E}$, $\mathcal{E}$ is the state space which is a subset of $L^2$ which is a subset of a Hilbert space.

Scalar Product

$(\varphi_i,\varphi_j)=\delta_{ij}$ implies that the vectors are orthonormal.

The expansion coefficients of $\psi$ can be found with the scalar product of the base vector with $\psi$.

Linear Independence Examples

Example 1

Independent.

$f(x) = x, g(x) = x^2, h(x) = \exp(2x)$

Example 2

Dependent.

$f(x) = x, g(x) = 5x, h(x) = x^2$

Example 3

Dependent.

$f(x) = \cos x, g(x) = \exp ix, h(x) = \sin x$

Wronskian Functional

The Wronskian can be used to determine linear independence: $\begin{vmatrix}f(x) & g(x) & h(x) & \cdots \\f'(x) & g'(x) & h'(x) & \cdots \\ f''(x) & g''(x) & h''(x) & \cdots\\\cdots & \cdots &\cdots & \cdots\end{vmatrix}$. Note that the determinant entries are $N\times N$, where $N$ is the number of functions you are comparing for linear independence. Hence you would only go up to $g'$ if you only had 2 functions.