Notions from Linear Algebra and Bra-Ket Notation

Abstract

The Schrödinger equation (1.14) is linear in the wave function ψ(x, t). This implies that for any set of solutions ψ₁(x, t), ${\psi }_{2}(x,t),\ldots$, any linear combination $\psi (x,t) = {C}_{1}{\psi }_{1}(x,t) + {C}_{2}{\psi }_{2}(x,t) + \ldots$ with complex coefficients C _n is also a solution. The set of solutions of equation (1.14) for fixed potential V will therefore have the structure of a complex vector space, and we can think of the wave function ψ(x, t) as a particular vector in this vector space. Furthermore, we can map this vector bijectively into different, but equivalent representations where the wave function depends on different variables. An example of this is Fourier transformation (2.5) into a wave function which depends on a wave vector k,

$$\psi (k,t) = \frac{1} {{\sqrt{2\pi }}^{3}} \int \nolimits \nolimits \!{d}^{3}x\,\exp \!\left (-\mathrm{i}k \cdot x\right )\psi (x,t).$$