Hilbert Space

Describes in an infinite-dimensional vector space the wave function of a particle. In the discrete case where a particle can only take three positions, the Hilbert Space would be ${\mathbb{C}}^3$ with each basis vector corresponding to the probability of the particle taking on that position. Also Cauchy Complete
Visual explanation

Topics

• Square Integrable Vector Space
  • Physicists use $L^2$ to describe wave functions so that they may have a finite dot product no matter if the wave function lives in an infinite-dimensional space
• Hilbert Space Bases

$⟨{\Psi }_n|{\Psi }_m⟩={⟨{\Psi }_m|{\Psi }_n⟩}^{\ast }$

• Anti commutative in a way

$⟨\Psi |\Psi ⟩=1\iff \Psi \text{ is normalized}$

$⟨{\Psi }_n|{\Psi }_m⟩=0\text{ for }n\ne m\iff {\Psi }_n\text{ and }{\Psi }_m\text{ are orthogonal}$

$⟨{\Psi }_n|{\Psi }_m⟩={\delta }_{nm}\iff \{{\Psi }_n\}\text{ is orthonormal}$

• The basis $\{{\Psi }_n\}$ is orthonormal

$\Psi (x)=\sum _{n=1}^{\infty }{c}_n{\Psi }_n(x)\iff \{{\Psi }_n\}\text{ is complete}$

$c_n=⟨{\Psi }_n|\Psi ⟩$

• Fourier analysis trick
• Equivalent to $c_n=e_i\cdot \vec{v}$