Linear Spaces

Definition

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

Let $ψ, ϕ, χ$ be vectors in $V$ . And $c, d$ are scalars.

Addition Rules

Their sum is a vector in $V$ , $(ψ + ϕ) \in V$

Commutativity

$ψ + ϕ = ϕ + ψ$

Associativity

$(ψ + ϕ) + χ = ψ + (ϕ + χ)$

Zero vector

$0 + ψ = ψ$

Inverse vector

$ψ + (- ψ) = 0$

Multiplication Rules

$c ψ \in V$ , $(c ψ + d ϕ) \in V$

Distributivity over addition

$c (ψ + ϕ) = c ψ + c ϕ$

Compatibility

$c (d ψ) = (c d) ψ$

Identity

$I ψ = ψ I = ψ$ , $0 ψ = ψ 0 = 0$

Example of Linear Spaces

$ψ = (x_{1}, x_{2}, \dots, x_{n})$ , $ϕ = (y_{1}, y_{2}, \dots, y_{n})$ , $ψ + ϕ = (x_{1} + y_{1}, \dots, x_{n} + y_{n})$ , $c ψ = (c x_{1}, \dots, c x_{n})$ .
(x_1,x_2,⋯), $\sum_{i}^{\infty} | x_{i} |^{2}$ is finite. $ℓ²$-space
Set of continuous functions
Set of functions with $\int | ψ (x) |^{2} d x$ is finite. $L²$-spaces
Hilbert Space

Quantum States

State Vector

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

Scalar Product

$(φ_{i}, φ_{j}) = δ_{i j}$ implies that the vectors are orthonormal.

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

Linear Independence Examples

Example 1

Independent.

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

Example 2

Dependent.

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

Example 3

Dependent.

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

Wronskian Functional

The Wronskian can be used to determine linear independence: $| \begin{matrix} f (x) & g (x) & h (x) & \dots \\ f^{'} (x) & g^{'} (x) & h^{'} (x) & \dots \\ f^{″} (x) & g^{″} (x) & h^{″} (x) & \dots \\ \dots & \dots & \dots & \dots \end{matrix} |$ . 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.