Quantum Mechanics

4

Mathematical Preliminaries

4.1 LINEAR VECTOR SPACE

The concept of linear vector space is the backbone of quantum mechanics. The important feature of quantum mechanics is the principle of superposition. If ψ₁ and ψ₂ are two states of a system, then there exists a state corresponding to ψ = c₁ψ₁ + c₂ψ₂, where c₁ and c₂ are complex numbers. It is the concept of linear vector space which expresses the idea of superposition. The simple concept of linear vector space is not sufficient in quantum mechanics. The structure of quantum mechanics requires imposition of additional properties on linear vector space. More particularly, quantum mechanics requires the linear vector space to be Hilbert space. The simplest way of expressing the Hilbert space ...