Quantum Hilbert Space

The Quantum Hilbert Space is the Hilbert Space used to represent quantum system states. These states are acted on by Operators.

Position and Momentum basis

• Move the position and momentum basis into their own notes

In classical mechanics we keep track of a Particle’s position and momentum directly. In quantum mechanics Particle’s are described by their Wave function. The Wave function can be expressed in different orthonormal basis. Two commonly used basis are the position basis and the momentum basis. Both of these basis are continuous, represented by the $x$ and $p$ variables respectively, and in both cases taking values in $\mathbb{R}$.

Properties

• Quantum states’ inner product is time independent

Equations

The results below are frequently useful when manipulating algebraic expressions related to quantum spaces

Result	Comments
$|\psi ⟩={\int }_{\infty }^{\infty }|x⟩⟨x\mid \psi ⟩dx$	Representation of a Wave function $\psi$ in the position basis
$|\psi ⟩={\int }_{\infty }^{\infty }|p⟩⟨p\mid \psi ⟩dp$	Representation of a Wave function $\psi$ in the momentum basis
$⟨x\mid p⟩=\frac{1}{\sqrt{2\pi \hslash }}\exp (\frac{i}{\hslash }px)$	Transformation between the position and momentum basis
$X|x⟩=x|x⟩$	$|x⟩$ is an eigenstate of $X$ with eigenvalue $x$
$P|p⟩=p|p⟩$	$|p⟩$ is an eigenstate of $P$ with eigenvalue $p$

• $|p⟩$ is the eigenstate associated with a definite momentum $p$
• $|x⟩$ is the eigenstate associated with a definite momentum $x$
• $|\psi ⟩$ is a generic Quantum state
• $\hslash$ is Planck’s reduced constant
• $X$ is the Position Operator
• $P$ is the Momentum Operator