VIII Mathematical interlude: Hilbert spaces and linear operators

The most common form of a vector is a collection of components ${\psi }_n$, $n=1,\mathrm{\dots },𝒩$, where $𝒩$ denotes the dimension of the vector space. A function $\psi (x)$ can be consider as a vector in which the discrete index $n$ has been replaced by a continuous index $x$. It is useful to introduce a formalism in which this analogy can be exploited without direct reference to the specific forms of the components ${\psi }_n$ or $\psi (x)$.

In Dirac notation, vectors are denoted as $|\psi ⟩$. These vectors form a complex linear vector space, which entails the following properties: Any vector $|\psi ⟩$ can be scaled by any complex number $\alpha$, i.e., we can form new vectors $|\alpha \psi ⟩=\alpha |\psi ⟩$. Furthermore, any two vectors $|\psi ⟩$, $|\chi ⟩$ can be combined into new vectors by a forming a superposition $|\psi +\chi ⟩=|\psi ⟩+|\chi ⟩$. These operations obey the distributive law $(\alpha +\beta )(|\psi ⟩+|\chi ⟩)=|\alpha \psi ⟩+|\beta \psi ⟩+|\alpha \chi ⟩+|\beta \chi ⟩$. In addition, a vector space possesses a null vector $|\mathbb{𝟘}⟩$ such that $|\mathbb{𝟘}⟩+|\psi ⟩=|\psi ⟩$, and to each vector $|\psi ⟩$ there is an inverse vector $|-\psi ⟩=-|\psi ⟩$ such that $|\psi ⟩-|\psi ⟩=|\mathbb{𝟘}⟩$

These properties are all nicely fulfilled for functions. In particular, if $\psi (x)$ and $\chi (x)$ are functions and $\alpha$, $\beta$ are constants, then $\alpha \psi (x)+\beta \chi (x)$ is also a function.

The scalar product is a generalised versions of the dot product, which associates a complex number $⟨\psi |\chi ⟩$ to any pair of vectors $|\psi ⟩$, $|\chi ⟩$. The scalar product fulfils the important property

$$⟨\psi |\chi ⟩={⟨\chi |\psi ⟩}^{*}.$$

(116)

Consistently with this, the scalar product is linear in the second argument, but conjugate linear in the first argument, i.e., $⟨\psi |\alpha \chi ⟩=\alpha ⟨\psi |\chi ⟩$, $⟨\alpha \psi |\chi ⟩={\alpha }^{*}⟨\psi |\chi ⟩$, $⟨\psi +\phi |\chi ⟩=⟨\psi |\chi ⟩+⟨\phi |\chi ⟩$, $⟨\psi |\phi +\chi ⟩=⟨\psi |\phi ⟩+⟨\psi |\chi ⟩$.

In general, a scalar product must be positive definite, $⟨\psi |\psi ⟩>0$ for $|\psi ⟩\ne |\mathbb{𝟘}⟩$. We call $||\psi ||=\sqrt{⟨\psi |\psi ⟩}$ the norm of the vector $|\psi ⟩$ (this generalises the notion of length of an ordinary vector). A vector with $⟨\psi |\psi ⟩=1$ is called normalised (this generalises the notion of a unit vector). The procedure of passing from a vector $|\psi ⟩$ to the normalised vector $(1/||\psi ||)|\psi ⟩$ is called normalisation. Again in analogy to the case of ordinary vectors, two vectors $|\psi ⟩$, $|\chi ⟩$ fulfilling $⟨\psi |\chi ⟩=0$ are said to be orthogonal to each other.

In conjunction with a certain completeness condition which is always fulfilled in quantum mechanics, a vector space equipped with a scalar product is called a Hilbert space.

Formally, the scalar product can be interpreted as a product $⟨\psi |\cdot |\chi ⟩$ between the vectors $|\chi ⟩$ and the entities $⟨\psi |$, which form the dual vector space. They represent the left entries in the scalar product and therefore are also conjugate linear: $⟨\alpha \psi +\beta \chi |={\alpha }^{*}⟨\psi |+{\beta }^{*}⟨\chi |$. A dual vector is also called a bra, and an ordinary vector is called a ket, alluding to the fact that in the scalar product $⟨\psi |\chi ⟩$ they form a bracket (bra-ket). The introduction of these dual vectors is an important step in the Dirac notation; its usefulness will become clear when we discuss operators (generalised matrices).

A basis is a collection of vectors $|n⟩,n=1,2,3,\mathrm{\dots },𝒩$ such that any vector can be written as a superposition $|\psi ⟩={\sum }_{n=1}^{𝒩}{\psi }_n|n⟩$, where the complex coefficients ${\psi }_n$ are unique. The coefficients ${\psi }_n$ give a representation of the vector, and can be written as a column vector

$$\psi =\left(\begin{array}{c}\hfill {\psi }_1\hfill \\ \hfill {\psi }_2\hfill \\ \hfill \mathrm{⋮}\hfill \\ \hfill {\psi }_{𝒩}\hfill \end{array}\right).$$

(117)

The corresponding dual vector is written as a row vector ${\psi }^{†}=({\psi }_1^{*},{\psi }_2^{*},\mathrm{\dots },{\psi }_{𝒩}^{*})$. While there are many possible bases, in which the same vector is represented by different coefficients, the number $𝒩$ of basis vectors required to obtain all vectors is always the same, and is called the dimension of the vector space ($𝒩$ may be $\mathrm{\infty }$).

An orthogonal basis fulfills $⟨n|m⟩=0$ for any $n\ne m$. If furthermore $⟨n|n⟩=1$ for all $n$ one speaks of an orthonormal basis. In such a basis, the coefficients representing a vector are given by ${\psi }_n=⟨n|\psi ⟩$, and the scalar product takes the explicit form

$$⟨\psi |\chi ⟩=\sum _n{\psi }_n^{*}{\chi }_n={\psi }^{†}\chi .$$

(118)

Thus, a vector $|\psi ⟩$ is normalised if its coefficients in an orthonormal basis obey

$$\sum _n{|{\psi }_n|}^2=1.$$

(119)

For functions $\psi (x)$, the summation over the discrete index is replaced by an integration, $|\psi ⟩=\int \psi (x)|x⟩𝑑x$. In this case the dimension of the vector space is infinite. The orthonormality of a basis can be stated with help of the Dirac delta function, $⟨x|x^{\prime }⟩=\delta (x-x^{\prime })$. In the scalar product, the summation over the discrete index $n=1,2,\mathrm{\dots }$ is again replaced by an integration over the continuous index $x$,

$$⟨\psi |\phi ⟩=\int {\psi }^{*}(x)\phi (x)𝑑x.$$

(120)

This type of integral is called an overlap integral. The expression for the expansion coefficients takes the form $\psi (x)=⟨x|\psi ⟩$, and the normalisation condition translates into

$$\int {|\psi (x)|}^2𝑑x=1.$$

An operator $\widehat{A}$ converts any vector $|\psi ⟩$ into another vector $|\widehat{A}\psi ⟩=\widehat{A}|\psi ⟩$. Linear operators fulfill $\widehat{A}(\alpha |\psi ⟩+\beta |\chi ⟩)=\alpha \widehat{A}|\psi ⟩+\beta \widehat{A}|\chi ⟩$, where $\alpha$, $\beta$ are complex numbers. Operators can be added according to the rule $(\widehat{A}+\widehat{B})|\psi ⟩=\widehat{A}|\psi ⟩+\widehat{B}|\psi ⟩$, and multiplied according to the rule $(\widehat{B}\widehat{A})|\psi ⟩=\widehat{B}(\widehat{A}|\psi ⟩)$.

In Dirac notation, operators are written as $\widehat{A}={\sum }_{nm}{A}_{nm}|n⟩⟨m|$, and the action of an operator is obtained from the multiplication rule $⟨m|\cdot |\psi ⟩=⟨m|\psi ⟩$. Thus,

$$\widehat{A}|\psi ⟩=\sum _{nm}{A}_{nm}|n⟩⟨m|\psi ⟩=\sum _n\left(\sum _m{A}_{nm}⟨m|\psi ⟩\right)|n⟩.$$

(121)

Assuming that the states $|n⟩$, $|m⟩$ in the definition of $\widehat{A}$ form an orthonormal basis, the operator can be represented by $N\times N$-dimensional square matrices

(122)

where, the matrix elements $A_{nm}$ are obtained from $A_{nm}=⟨n|\widehat{A}m⟩\equiv ⟨n|\widehat{A}|m⟩$. Since in an orthonormal basis $⟨m|\psi ⟩={\psi }_m$, the operator then acts on a vector according to the standard rules of matrix multiplication, i.e., $|\phi ⟩=\widehat{A}|\psi ⟩$ is represented by a vector with coefficients ${\phi }_n={\sum }_m{A}_{nm}{\psi }_m$. Furthermore, the operator addition and multiplication rules then translate to the usual prescriptions of matrix addition and multiplication.

The action of an operator is particularly simple in its eigenrepresentation, defined by a basis fulfilling the eigenvalue equation $\widehat{A}|n⟩=a_n|n⟩$. The numbers $a_n$ are called eigenvalues, and the associated vectors $|n⟩$ are called eigenvectors. When appropriate, these eigenvectors are also called eigenfunctions.

If the eigenvectors form an orthonormal basis (as is the case for the hermitian and unitary operators considered below), the eigenrepresentation results in a diagonal matrix, with $A_{nm}=0$ if $n\ne m$ and $A_{nn}=a_n$. In Dirac notation, the operator can then be written as $\widehat{A}={\sum }_n{a}_n|n⟩⟨n|$.

A particularly simple operator is the identity operator $\widehat{I}$ which leaves all states unchanged, $\widehat{I}|\psi ⟩=|\psi ⟩$. Every state is therefore an eigenstate of $\widehat{I}$, with eigenvalue 1. Consequently, in any orthonormal basis this operator takes the same form $\widehat{I}={\sum }_n|n⟩⟨n|$. Representations are simply obtained by multiplying out the identities $|\psi ⟩=\widehat{I}|\psi ⟩$ and $\widehat{A}=\widehat{I}\widehat{A}\widehat{I}$. In a given orthonormal basis, it is useful to decompose the identity $\widehat{I}=\sum {\widehat{P}}_n$ as the sum of projection operators ${\widehat{P}}_n=|n⟩⟨n|$, which fulfill ${\widehat{P}}_n{\widehat{P}}_m=0$ if $n\ne m$, and ${\widehat{P}}_n^2={\widehat{P}}_n$.

For each operator $\widehat{A}$ we can define an adjoint operator ${\widehat{A}}^{†}$ by setting $⟨\psi |{\widehat{A}}^{†}\chi ⟩=⟨\widehat{A}\psi |\chi ⟩$. In an orthonormal basis we then have $A_{nm}^{†}=A_{mn}^{*}$. For many operators, we can also define an inverse operator ${\widehat{A}}^{-1}$ which fulfils $\widehat{A}{\widehat{A}}^{-1}=\widehat{I}$.

Two important types of operators are hermitian operators $\widehat{H}$ and unitary operators $\widehat{U}$. For any two states $|\psi ⟩$, $|\chi ⟩$, hermitian operator fulfill $⟨\psi |\widehat{H}\chi ⟩=⟨\widehat{H}\psi |\chi ⟩$, while unitary operators fulfill $⟨\widehat{U}\psi |\widehat{U}\chi ⟩=⟨\psi |\chi ⟩$. This entails $\widehat{H}={\widehat{H}}^{†}$ and ${\widehat{U}}^{†}={\widehat{U}}^{-1}$. In an orthonormal basis, the matrix elements of a hermitian operator fulfill $H_{nm}=H_{mn}^{*}$, while those of a unitary operator fulfill ${\sum }_l{U}_{nl}{U}_{ml}^{*}={\delta }_{nl}$.

Both classes of operators have the nice property that their sets of normalised eigenvectors form an orthonormal basis. For hermitian operators, the eigenvalues $a_n$ are real, while for unitary operators they fulfill $|a_n|=1$.

Unitary operators are analogous to orthogonal matrices which rotate a coordinate system. In particular, any basis change from one orthonormal basis $|n⟩$ to another orthonormal basis $|\tilde{n}⟩$ can be written as $|\tilde{n}⟩=\widehat{U}|n⟩$, where $\widehat{U}$ is a suitable unitary operator. A common form of unitary operators relates them to a hermitian operator $\widehat{H}$ via $\widehat{U}=\exp (i\tau \widehat{H})$, where $\tau$ is a real constant and the exponential of an operator is defined via its Taylor expansion, $\exp (\widehat{A})={\sum }_{n=0}^{\mathrm{\infty }}{\widehat{A}}^n/n!$. In this case, ${\widehat{U}}^{-1}={\widehat{U}}^{†}=\exp (-i\tau \widehat{H})$. Furthermore, the operators $\widehat{H}$ and $\widehat{U}$ then share the same eigenvectors: If $\widehat{H}={\sum }_n{h}_n|h_n⟩⟨h_n|$, then $\widehat{U}={\sum }_n{u}_n|h_n⟩⟨h_n|$ with eigenvalues $u_n=\exp (i\tau h_n)$.