Hilbert Space | Quantum Mechanics

Hilbert space is a mathematical framework that provides a way to describe the state of a quantum system using wave functions.

It allows us to calculate probabilities, measure the similarity between states, and represents the full range of possible states for the system.

A wave function is a mathematical function that assigns a complex number to each point in the Hilbert space.

These complex numbers contain information about the probabilities of different outcomes when we make measurements on the system.

Mathematical Explanation

In a vector space, the set of unit vector $ê_{1}ê_{2}ê_{3}…..$ from the orthonormal basis

i.e.

we can express any vector in this space as a linear combination of $ê_{1}ê_{2}ê_{3}…..$

Similarly, a space can be defined in which a set of functions $\phi_{1}(x),\phi_{2}(x),\phi_{3}(x)……$ from the orthonormal basis of the coordinate system. The corresponding infinite-dimensional linear vector space is called Hibert Space.

Properties of Hilbert Space:

(i) The inner product or scalar product of two functions $\psi_{i}\left(x\right)$ and $\psi_{j}\left(x\right)$ defined in the interval $a\leq x\leq b$ is defined as

$≺\psi_{i}|\psi_{j}≻=\intop_{a}^{b}\psi_{i}^{*}(x)\psi_{j}(x)dx$

(ii) Two functions $\psi_{i}\left(x\right)and \psi_{j}\left(x\right)$ are said to be orthogonal if their inner product is zero i.e.

$≺\psi_{i}|\psi_{j}≻=\intop_{a}^{b}\psi_{i}^{*}(x)\psi_{j}(x)dx=0$

This is known as the orthogonality of two wave functions.

(iii) The norm of a function $\psi_{i}\left(x\right)$ is defined as

$N=\sqrt{\prec\psi_{i}|\psi_{j}\succ}=\left[\intop_{a}^{b}\psi_{i}^{*}(x)\psi_{j}(x)dx\right]^{1/2}$

(iv) A function is said to be normalized if the norm of the function is unity i.e.

$N=\sqrt{\prec\psi_{i}|\psi_{j}\succ}=\left[\intop_{a}^{b}\psi_{i}^{*}(x)\psi_{j}(x)dx\right]^{1/2}$ = 1

This is known as the normalization condition of a particular wave function.

(v) Functions that are orthogonal and normalized are called orthonormal functions and they will satisfy the condition:

$≺\psi_{i}|\psi_{j}≻=\intop_{a}^{b}\psi_{i}^{*}(x)\psi_{j}(x)dx=\delta_{ij}$

(vi) A set of functions $\psi_{1}\left(x\right),\psi_{2}\left(x\right),\psi_{3}\left(x\right)…..$ is linearly independent if there exist a relation like $c_{1}\psi_{1}\left(x\right)+c_{2}\psi_{2}\left(x\right)+c_{3}\psi_{3}\left(x\right)…….=0,$

where all $c_{1},c_{2},c_{3}…..$ are zero.

Otherewise, they are said to be linearly dependent. A set of linearly dependent.

A set of linearly independent functions is complete.