Hilbert Spaces and Dirac Notation

The objects we have just defined obey a set of fundamental properties, always valid whatever the dimension of the Hilbert space, and independently of whether the operators are bounded or not. In the latter case one can still define an adjoint, but it is mathematically more subtle, and we shall return to it in the next theme, devoted to the theory of linear operators. Since the whole planet uses Dirac notation, these properties are in effect the algebraic rules of quantum calculation, and are therefore imperative to master.

This lesson gathers together all the useful formulas seen so far, even at the cost of repeating them, and adds a few more. We first give those that hold in any dimension, then the formulary specific to finite dimension, where Dirac notation reduces to an explicit matrix calculation.

1. Calculation rules in any dimension

Properties of the inner product.

Let $(\ket{\phi}, \ket{\psi}) \in \mathcal{H}^2$ be two vectors of the Hilbert space and $(\lambda, \mu) \in \mathbb{C}^2$ scalars. The inner product satisfies the following properties:

1. Hermitian symmetry:

   $$\boxed{\braket{\phi}{\psi} = \braket{\psi}{\phi}^*.}$$

   (1)

2. Antilinearity on the left: the bra associated with $\lambda \ket{\psi}$ is $\lambda^* \bra{\psi}$, that is:

   $$\boxed{(\lambda \ket{\psi})^\dagger = \lambda^* \bra{\psi} \qquad \textrm{and} \qquad \braket{\lambda \psi}{\phi} = \lambda^* \braket{\psi}{\phi} }$$

   (2)

3. Linearity on the right:

   $$\boxed{\braket{\phi}{\lambda \psi} = \lambda \braket{\phi}{\psi}.}$$

   (3)

4. Linear combinations: the preceding properties allow one to expand in the usual way complicated expressions such as the following: for all $\alpha, \beta, \gamma, \delta \in \C$, one has:

   $$\boxed{ \braket{\alpha \phi + \beta \psi}{\gamma \phi + \delta \psi} = \alpha^* \gamma \braket{\phi}{\phi} + \alpha^* \delta \braket{\phi}{\psi} + \beta^* \gamma \braket{\psi}{\phi} + \beta^* \delta \braket{\psi}{\psi}. }$$

   (4)

5. Norm: by definition,

   $$\boxed{\|\ket{\psi}\|^2 = \braket{\psi}{\psi} \ge 0,}$$

   (5)

   with equality if and only if $\ket{\psi} = 0$ is the null vector (sometimes, but rarely, written $\ket{\varnothing}$).

Properties of the adjoint.

We rewrite here the two formulas that serve as definitions:

and

which also allows one to prove, via Hermitian symmetry, that:

Properties of adjoint operators.

For all continuous linear operators $\hat{A}, \hat{B}$ on $\mathcal{H}$ and every scalar $\lambda \in \mathbb{C}$, one has:

Ket-bra operator.

A bracket is a number, but a ket-bra is an operator, called the outer product (unrelated to the exterior product of differential forms). To two vectors $\ket{\phi}$ and $\ket{\psi}$, one associates the operator $\ket{\phi}\bra{\psi}$ defined by:

Decomposition of the identity operator.

The identity operator on $\Hilb$ is written $\mathbf{1}$, trivially defined by $\mathbf{1} x = x, \forall x \in \Hilb$. In a countable Hilbert basis $\{\ket{e_i}\}_{i \in I}$ where $I$ is either finite or countably infinite, the identity operator $\mathbf{1}$ is written:

This formula is extremely useful in practice. It is also called the resolution of the identity or the closure relation. The right-hand side is a sum in finite dimension, or a convergent series in countably infinite dimension. It is false in the uncountable-dimensional case.

Hilbert decomposition.

The preceding formula allows one, for example, to recover directly the decomposition theorem seen in lesson 1, by defining $\psi_i$ as the component of the ket $\kpsi$ on $\ket{e_i}$:

This is a sum or a series depending on the dimension. The decomposition of the ket has its counterpart for bras:

The norm is then written:

A few particular operators.

The next theme presents in detail the theory of linear operators on a Hilbert space. Nevertheless we note for what follows a few very common cases. Among the operators from $\Hilb \to \Hilb$, we shall have in particular an operator:

1. $\hat A$, Hermitian or self-adjoint if $\hat{A}^\dagger = \hat{A}$,
2. $\hat U$, unitary if $\hat{U}^\dagger = \hat{U}^{-1}$,
3. $\hat P$, a projector if $\hat{P}^2 = \hat{P}$,
4. $\hat P$, an orthogonal projector: a self-adjoint projector.

Unitary operators are bijective, linear, and preserve the inner product. We have therefore already met them above: they are the isometric isomorphisms (in this instance the isometric automorphisms, since we are speaking here only of linear operators from $\Hilb$ into itself).

2. Formulary in finite dimension

One can further specify the rules of quantum calculation when the Hilbert space is of finite dimension $n$. To begin with, all the formulas of the previous section are valid, where the sums encountered there are to be written as finite sums: $\sum_i \longrightarrow \sum_{i=1}^{n}$.

Moreover, once an orthonormal basis $\mathcal{B} = \left(\ket{e_i}\right)_{i=1}^n$ has been chosen, Dirac notation corresponds to an explicit matrix calculation, which we now detail (beware, what follows makes no sense at all in infinite dimension).

Kets as column vectors.

By canonically representing the kets of the basis $\mathcal{B}$ by $(n,1)$ matrices, also called column vectors:

where the $1$ is in $i$-th position, and via the algebraic decomposition:

one obtains the writing of all kets as column vectors: for $\ket{v} \in \mathcal{H}$, one writes:

where, unless necessary, we shall omit the mention of the basis $\mathcal{B}$ in which this expansion is made. Moreover, the components are obtained by orthogonal projections:

Bras as conjugate row vectors.

The linear form $\varphi_u$ associated with the vector $u$ must satisfy, for every vector $v$:

by antilinearity of the Hermitian inner product on the left. To obtain this sum as a matrix product, $\bra{u}$ must be the row vector of the conjugate coordinates of $u$:

because then the inner product $\braket{u}{v}$ is obtained as the usual matrix product:

One then notes that the bra $\bra{u}$ is the conjugate transpose of the ket $\ket{u}$. The dagger operation therefore amounts, in finite dimension, to taking the conjugate transpose:

This is false in infinite dimension, and would not even make sense, since the transpose is not defined¹.

Matrix representation of operators.

In an orthonormal basis $\{\ket{e_i}\}_{i=1}^n$ in finite dimension, every linear operator $\hat{A}$ decomposes as:

which leads, in finite dimension, to the natural identification between the linear operator $\hat{A}$ (written with a hat), and its matrix $A$ (without a hat). The formula above is indeed the counterpart, in the Dirac formalism, of the matrix expansion $A = \sum_{ij} A_{ij} E_{ij}$, where $E_{ij}$ is the elementary matrix with a $1$ in position $(i,j)$ and zeros elsewhere. Here, $E_{ij}$ is therefore the matrix representing the operator $\ket{e_i} \bra{e_j}$. The coefficients $A_{ij}$ are naturally called the matrix elements of the operator $\hat{A}$ in the basis $\{\ket{e_i}\}$. They equal:

but for aesthetic reasons one will more often write them in the form:

The identity operator has of course the identity matrix for its matrix, and its matrix elements are the $\delta_{ij}$ (Kronecker symbol).

Trace of an operator.

We shall often be led in what follows to consider the trace of an operator $\hat{A}$. In Dirac notation it is computed by:

This expression is independent of the choice of orthonormal basis.

Adjoint and conjugate transpose.

In finite dimension, the matrix representing the adjoint $\hat{A}^\dagger$ is the conjugate transpose of the matrix representing the operator $\hat{A}$:

We propose a quick proof here, because it shows how the calculation rules seen above are used in practice. The matrix elements of $\hat{A}^\dagger$ are by definition:

But, by definition of the adjoint, one has:

Using Hermitian symmetry, one has $\braket{\smash{\hat{A}} e_i}{e_j} = \braket{e_j}{\smash{\hat{A}} e_i}^*$, but this is nothing other than the matrix elements $A_{ji}^*$. Therefore:

which proves the property. In finite dimension, one will therefore be able to check explicitly by a matrix calculation whether an operator is Hermitian, or unitary, etc.

3. References