Notions from Linear Algebra and Bra-Ket Notation

Abstract

The Schrödinger equation (1.14) is linear in the wave function $$\psi (\boldsymbol{x},t)$$. This implies that for any set of solutions $$\psi _{1}(\boldsymbol{x},t)$$, $$\psi _{2}(\boldsymbol{x},t),\ldots$$, any linear combination $$\psi (\boldsymbol{x},t) = C_{1}\psi _{1}(\boldsymbol{x},t) + C_{2}\psi _{2}(\boldsymbol{x},t)+\ldots$$with complex coefficients Cnis 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 $$\psi (\boldsymbol{x},t)$$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.