Inner-product spaces

Alexei Gilchrist

3 States
Inner-products and norms are defined. Equipping a vector space with an inner-product gives an inner-product-space, and let’s us define bases.
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Inner-product spaces

1 Inner products

An Inner-product, or scalar-product, is a map from a pair of vectors to a complex number. It can be represented by a simple tuple: \((\bullet,\bullet)\), and is defined for the entire vector space \(\mathbb{V}\):

\[\forall\; |\psi\rangle,|\phi\rangle\in \mathbb{V}\quad (|\psi\rangle,|\phi\rangle)\rightarrow \mathbb{C}\]

The inner product helps capture the concepts of angle and length from spatial vectors, and will allow us to define orthogonality and then orthonormal bases.

An inner product has the following three properties:

Linearity
The inner-product is linear in the second component:
\[(|x\rangle,a|y\rangle+b|z\rangle) = a(|x\rangle, |y\rangle)+b(|x\rangle,|z\rangle)\]
Symmetry
Swapping the order of the arguments produces the complex-conjugate of the result
\[(|\psi\rangle, |\phi\rangle) = (|\phi\rangle,|\psi\rangle)^*\]
Positivity
An inner product of a vector with itself is always a non-negative real number.
\[ (|\psi\rangle,|\psi\rangle) \ge 0 \]

Note that the linearity and symmetry properties imply the innner-product will be anti-linear, or conjugate-linear in the first component: \((a|x\rangle+b|y\rangle,|z\rangle) = a^*(|x\rangle, |z\rangle)+b^*(|y\rangle,|z\rangle)\). Be aware that some books use the convetion that the inner-product is linear in the first component but the reason for using the second component will become clear when we introduce the Dirac notation.

Example

Say \( |a\rangle,|b\rangle \in \mathbb{C}^n \), i.e. the vector space is a stack of \(n\) complex numbers like \((a_1,a_2,\ldots,a_n)\). Then the inner product is defined as

\[(|a\rangle,|b\rangle) = \sum_j a_j^* b_j\]
Where \(a_j\) and \(b_j\) are the components of the vectors \(|a\rangle\) and \(|b\rangle\).

Example

Square integrable functions are also vectors. These are functions \(f(x)\) such that \(\int_{-\infty}^{\infty} |f(x)|^2 dx\) is finite. The inner-product for these functions is defined as

\[(\psi(x),\phi(x)) = \int_{-\infty}^{\infty} \psi(x)^*\phi(x) dx\]

It’s a good exercise to verify that both examples above satisfy linearity, symmetry, and positivity.

Two vectors \(|a\rangle\), and \(|b\rangle\) are said to be orthogonal if their inner product is zero \((|a\rangle,|b\rangle)=0\).

2 Norm

A norm, denoted with double bars \(\lVert \bullet\rVert\), associates a positive real number to every vector

\[\forall |a\rangle \in \mathbb{V}\quad \lVert |a\rangle\rVert \in \mathbb{R}\]
and captures the notion of the length of a vector. To do so it obeys three properties:
Positivity
\[\begin{gather*}\forall |x\rangle\in\mathbb{V}\quad \lVert x\rVert\ge 0, \\ \lVert x\rVert = 0 \iff |x\rangle = |\mathbf{0}\rangle.\end{gather*}\]
Where \(|\mathbf{0}\rangle\) is the null vector.
Linearity
\[ \forall |x\rangle\in\mathbb{V}, a\in\mathbb{C}\quad \lVert a|x\rangle\rVert = |a|\cdot\lVert |x\rangle\rVert \]
Triangle Inequality
\[\forall |x\rangle,|y\rangle\in\mathbb{V}\quad \lVert |x\rangle+|y\rangle\rVert \le \lVert |x\rangle\rVert+\lVert |y\rangle\rVert\]
 That last requirement is an embodiment of the maxim ‘‘the shortest distance between two points is a straight line’’.

Given an inner-product, we can define a “natural” norm as

\[\lVert |x\rangle\rVert = \sqrt{(|x\rangle,|x\rangle)}.\]
This is not necessarily possible in the other direction—it’s possible to have a norm without having an inner-product.

3 Basis

One of the key tools that we will use in quantum mechanics is that of a basis, and in particular orthonormal bases. So let’s start with some definitions.

3.1 Definitions

Definition: Span
If \(s\) is a set of vectors in \(\mathbb{V}\), then the span of \(s\) is the set of all vectors formed by linear combinations of the vectors in \(s\).
Definition: Basis

A basis is a set of vectors in \(\mathbb{V}\) that are linearly independent, and the span of the vectors is the whole vector space \(\mathbb{V}\).

They are useful because any vector \(|y\rangle\) can be expanded interms of the basis.

Definition: Orthogonal basis
An orthogonal basis is a basis of vectors that are pair-wise orthogonal:
\[ (|x_j\rangle,|x_k\rangle) = 0 \quad\text{if}\; j\ne k \]
Definition: Orthonormal basis
An orthonormal basis is an orthogonal basis where every vector has norm 1:
\[ (|x_j\rangle,|x_k\rangle) = \begin{cases} 0\; j\ne k \\ 1\; j= k \end{cases}\]

3.2 Basis expansion

3.3 Components and representations

© Copyright 2022 Alexei Gilchrist