Hilbert spaces

8. Hilbert spaces#

Recommended reference: Griffiths [Gri95], in particular Section 3.1.

So far, we have been considering vectors in an \(n\)-dimensional real or complex space. In quantum mechanics, one encounters infinite-dimensional vector spaces as well. In particular, the state of a quantum system is represented mathematically by a (unit) vector in a Hilbert space.

8.1. Vector spaces#

As is customary in quantum physics and quantum algorithms, we use bra-ket notation for vectors in Hilbert spaces.

Definition 8.1 (Vector space)

A (real or complex) vector space consists of

a set of vectors, denoted by symbols like \(\ket\alpha\)
a distinguished vector \(\mathbf{0}\) called the zero vector[1]
a way of adding vectors, i.e. given two vectors \(\ket\alpha\) and \(\ket\beta\) we can obtain a third vector \(\ket\alpha+\ket\beta\)
a way of multiplying a vector by a scalar: given a scalar \(c\) (a real or complex number, depending on whether we consider a real or complex vector space) and a vector \(\ket\alpha\) we can obtain a vector \(c\ket\alpha\)

such that a number of properties hold:

\(0\ket\alpha=\mathbf{0}\)
\(1\ket\alpha=\ket\alpha\)
\(\mathbf{0}+\ket\alpha=\ket\alpha\)
\(\ket\alpha+\ket\beta=\ket\beta+\ket\alpha\)
\((\ket\alpha+\ket\beta)+\ket\gamma=\ket\alpha+(\ket\beta+\ket\gamma)\)
\(c(\ket\alpha+\ket\beta)=c\ket\alpha+c\ket\beta\)
\((c+d)\ket\alpha = c\ket\alpha + d\ket\alpha\)
\(c(d\ket\alpha)=(cd)\ket\alpha\)

Definition 8.2 (Basis, dimension)

A basis for a vector space is a collection of vectors \((\ket{e_i})_i\) such that every vector can be written in exactly one way as a linear combination of the \(\ket{e_i}\). The number of vectors in a basis is called the dimension of the space.

As in Basis transformations, any two bases contain the same number of vectors, so the dimension does not depend on the choice of basis.

Example 8.1

The simplest examples are the \(n\)-dimensional vector spaces \(\RR^n\) and \(\CC^n\) with coordinatewise addition and scalar multiplication.

An important special case is the two-dimensional space \(\CC^2\). In bra-ket notation, various symbols are used for the standard basis vectors \(\binom{1}{0}\) and \(\binom{0}{1}\) in this space, such as

\(\ket+\) and \(\ket-\);
\(\ket0\) and \(\ket1\) (in the context of quantum computing);
\(\ket\uparrow\) and \(\ket\downarrow\) (in the context of spins).

8.2. Hermitian inner products and Hilbert spaces#

Definition 8.3 (Hermitian inner product)

A Hermitian inner product assigns to vectors \(\ket\alpha\) and \(\ket\beta\) a complex number \(\braket{\alpha|\beta}\) such that the following properties hold:

\(\braket{\alpha|\beta}=\overline{\braket{\beta|\alpha}}\);
\(\braket{\alpha|\alpha}\ge0\), and \(\braket{\alpha|\alpha}=0\) holds precisely when \(\ket\alpha=\mathbf{0}\) (note that \(\braket{\alpha|\alpha}\) is a real number because of the first property);
\(\bra\alpha(\ket\beta+\ket\gamma) = \braket{\alpha|\beta} + \braket{\alpha|\gamma}\);
\(\bra\alpha(c\ket\beta)=c(\braket{\alpha|\beta})\);

Definition 8.4 (Norm of a vector)

The norm or length of a vector \(\ket\alpha\) in a Hilbert space is defined as the real number

\[ \|\alpha\|=\sqrt{\braket{\alpha|\alpha}}. \]

Note

The properties above imply that for any scalar \(c\), the inner product of \(c\ket{\alpha}\) with \(\ket\beta\) equals \(\bar c\braket{\alpha|\beta}\). In other words, while the inner product is (complex) linear in its second argument, it is only conjugate linear in its first argument. This is the usual convention in physics. It differs from the convention in mathematics, where inner products are usually linear in the first argument and conjugate linear in the second argument.

Definition 8.5 (Hilbert space)

A Hilbert space is a complex vector space \(\HH\) equipped with a Hermitian inner product \(\braket{\enspace|\enspace}\) and satisfying a condition called completeness.

We will not make the notion of completeness precise in these notes. Roughly speaking, it means that we do not have to worry too much about convergence of sequences or series in our Hilbert space, and there is a notion of infinite linear combinations.

Example 8.2

We take \(\HH=\CC^n\) and define an inner product by

\[ \braket{\alpha|\beta} = \sum_{i=1}^n\bar\alpha_i\beta_i. \]

Check for yourself that this satisfies all the properties of a Hermitian inner product.

Example 8.3

Consider the space \(L^2(\RR)\) of square-integrable functions on the real line, i.e. functions \(f\colon\RR\to\CC\) such that the integral \(\int_{-\infty}^\infty|f(x)|^2 dx\) exists and is finite. This is a Hilbert space with inner product

\[ \braket{f|g} = \int_{-\infty}^\infty \bar f(x)g(x)dx. \]

Example 8.4

Consider the space \(\ell^2\) of sequences \(z=(z_0,z_1,\ldots)\) of complex numbers such that

\[ \sum_{n=0}^\infty |z_n|^2<\infty. \]

Then \(\ell^2\) is a Hilbert space with inner product

\[ \braket{w|z}=\sum_{n=0}^\infty \bar w_n z_n. \]

8.3. The dual Hilbert space#

With any Hilbert space \(\HH\), we can associate a dual Hilbert space \(\HH^*\). In the bra-ket formalism used in quantum mechanics, for every “ket vector” \(\ket\alpha\) in \(\HH\) we have a dual “bra vector” \(\bra\alpha\) in \(\HH^*\). This correspondence has the following properties:

vector in \(\HH\)	corresponding vector in \(\HH^*\)
ket vector \(\ket\alpha\)	bra vector \(\bra\alpha\)
\(\ket\alpha+\ket\beta\)	\(\bra\alpha+\bra\beta\)
\(c\ket\alpha\)	\(\bar c\bra\alpha\)

In mathematical language, \(\HH^*\) is the space of continuous linear maps \(\HH\to\CC\). The bra vector \(\bra\alpha\) then represents the linear map that sends \(\ket\beta\) to \(\braket{\alpha|\beta}\).

Example 8.5 (The dual of \(\CC^n\))

In the case \(\HH=\CC^n\) we can identify \(\HH^*\) with \(\CC^n\) as well; for a vector \(\ket\alpha\) in \(\CC^n\), the dual vector \(\bra\alpha\) is then simply the same vector. However, it can also be convenient to distinguish \(\HH\) from \(\HH^*\) by denoting a ket vector \(\ket\alpha\) as a column vector and the corresponding bra vector \(\bra\alpha\) by the conjugate transpose of \(\ket\alpha\), so \(\bra\alpha\) is the row vector whose entries are the complex conjugates of those of \(\alpha\). In \(\CC^2\), for example, we have

\[\begin{split} \ket\alpha=\begin{pmatrix}1\\ i\end{pmatrix}\Longrightarrow \bra\alpha=\begin{pmatrix}1& -i\end{pmatrix}. \end{split}\]

This allows us to view \(\bra\alpha\) as a linear map sending \(\ket\beta\) to the scalar obtained by multiplying the row vector \(\bra\alpha\) by the column vector \(\ket\beta\).

Note

If you know about continuous linear maps of Hilbert spaces, here is a useful connection between the mathematical definition of \(\HH^*\) and the bra-ket formalism: the Riesz representation theorem states that every continuous linear map \(\HH\to\CC\) is of the form \(\braket{\alpha|\enspace}\) for some \(\alpha\) in \(\HH\).

8.4. Orthonormal bases#

When working with vector spaces, it is frequently useful to make a choice of basis adapted to the problem at hand. In the context of a Hilbert space, it turns out that a “good” basis is often one that satisfies the following property with respect to the inner product.

Definition 8.6 (Orthonormal system)

An orthonormal system of vectors in a Hilbert space \(\HH\) is a collection of vectors \((\ket{e_i})_{i\in I}\), where \(I\) is some index set, satisfying

\[\begin{split} \braket{e_i|e_j} = \begin{cases} 1&\quad\text{if }i=j,\\ 0&\quad\text{if }i\ne j. \end{cases} \end{split}\]

Definition 8.7 (Orthonormal basis)

An orthonormal basis of a Hilbert space \(\HH\) is an orthonormal system \((\ket{e_i})_{i\in I}\) in \(\HH\) that spans \(\HH\), i.e. every vector in \(\HH\) can be written as a (possibly infinite) linear combination of the \(\ket{e_i}\).

Example 8.6

In the Hilbert space \(\CC^n\) with the standard inner product, the standard basis \((e_1,\ldots,e_n)\) is an orthonormal basis.

Example 8.7

Consider the space \(L^2([0,1])\) of square-integrable functions on the unit interval. An orthonormal basis for this space consists of the functions \(e_n(x)=\exp(2\pi i n x)\) for all integers \(n\). This reflects the fact that every “nice” function on \([0,1]\) can be expressed as a Fourier series.

8.5. Linear subspaces#

Definition 8.8 (Linear subspace)

A linear subspace of a vector space \(V\) is a set of vectors \(W\) contained in \(V\) such that the following properties hold:

The zero vector of \(V\) lies in \(W\).
For all vectors \(w\) and \(w'\) in \(W\), the sum \(w+w'\) is also in \(W\).
For every vector \(w\) in \(W\) and every scalar \(\lambda\), the scalar multiple \(\lambda w\) is also in \(W\).

In quantum physics, linear subspaces are important because they encode quantum states that a physical state can collapse to when a measurement is performed. To make this more precise, we need the notions of orthogonal complement of a subspace and (in the next chapter) of orthogonal projection onto a subspace.

Definition 8.9 (Orthogonal complement)

Consider a Hilbert space \(\HH\) and a linear subspace \(W\) of \(\HH\). The orthogonal complement of \(W\) in \(\HH\), denoted by \(W^\perp\), is the set of all vectors \(\ket{v}\) in \(\HH\) that are orthogonal to all of \(W\), i.e.

\[ W^\perp = \{\ket{v}\in\HH\mid\braket{v|w}=0\quad\text{for all }w\in W\}. \]

It is not hard to verify that the orthogonal complement of a linear subspace of \(\HH\) is again a linear subspace; see Exercise 8.5.

Within a Hilbert space \(\HH\), we will only consider linear subspaces \(W\) that satisfy the completeness condition and are therefore themselves Hilbert spaces.

8.6. Example of a function space: spherical harmonics#

In physics, Hilbert spaces often occur as spaces of functions on some space. We have already seen the examples of \(L^2(\RR)\) and \(L^2([0,1])\), the Hilbert spaces of square-integrable functions on the real line and the unit interval, respectively.

Another useful example is the Hilbert space of square-integrable functions on the unit sphere

\[ S^2 = \{(x,y,z)\in\RR^3\mid x^2+y^2+z^2=1\}. \]

It is useful to work with spherical coordinates \(\theta,\phi\) with \(0\le\theta\le\pi\) and \(0\le\phi\le 2\pi\) such that

\[ x=\sin\theta\cos\phi,\quad y=\sin\theta\sin\phi,\quad z=\cos\theta. \]

Then the inner product of two functions \(f,g:S^2\to\CC\) is defined via the standard area form \(\sin\theta\,d\theta\,d\phi\) as

\[ \langle f,g\rangle_{S^2} = \int_{\phi=0}^{2\pi} \int_{\theta=0}^\pi \overline{f(\theta,\phi)}g(\theta,\phi) \sin\theta\,d\theta\,d\phi. \]

Much like the functions \(\exp(2\pi i m x)\) form an orthonormal basis for \(L^2([0,1])\), one can write down a set of functions forming an orthonormal basis for \(L^2(S^2)\). The standard functions are the spherical harmonics. These are defined by

\[ Y_\ell^m(\theta,\phi) = N_\ell^m \exp(im\phi) P_\ell^m(\cos\theta), \]

where \(\ell\) and \(m\) are integers with \(-\ell\le m\le \ell\). Here \(P_\ell^m\) is an associated Legendre polynomial and \(N_\ell^m\) is an appropriate normalisation factor.

The Spherical Harmonics webpage by Manuel Joffre (École Polytechnique) allows you to make 3-dimensional plots of spherical harmonics. For more background and definitions, see also Griffiths [Gri95] or the Wikipedia and MathWorld pages on spherical harmonics.

8.7. Exercises#

Exercise 8.1

Check that the inner product defined in Example 8.2 is indeed a Hermitian inner product on \(\CC^n\).

Exercise 8.2

The Cauchy–Schwarz inequality states that for vectors \(\ket\alpha,\ket\beta\) in a Hilbert space we have

\[ \bigl|\braket{\alpha|\beta}\bigr|^2\le \braket{\alpha|\alpha}\braket{\beta|\beta}. \]

Show this using the basic properties of the inner product.

Exercise 8.3

The triangle inequality states that for vectors \(\ket\alpha,\ket\beta\) in a Hilbert space we have

\[ \|\alpha+\beta\|\le\|\alpha\|+\|\beta\|. \]

Derive the triangle inequality from the Cauchy–Schwarz inequality.

Exercise 8.4

Consider a Hilbert space \(\HH\) and an orthonormal system of vectors \((\ket{e_i})_{i\in I}\) of \(\HH\). Show that the system is linearly independent, i.e. the only linear relation

\[ \sum_{i\in I} c_i\ket{e_i}=\mathbf{0} \]

is the one where all \(c_i\) are equal to \(0\).

Exercise 8.5

Consider a linear subspace \(W\) of a Hilbert space \(\HH\). Show that the orthogonal complement \(W^\perp\) is again a linear subspace.