Adjoints and inner products

Adjoints and inner products#

Dual vector spaces#

Definition#

Let \(V\) be a vector space over \(\CC\). the dual vector space \(V^*\) is the space of all linear maps \(f:V \to \CC\).

Properties and notation#

  1. \(V^*\) is a vector space.

Consider linear maps \(f_{1,2}\), and \(a,b \in \CC\). Then we can define a linear map \(a f_1 * b f_2\) by their action on a vector \(\ket{v}\).

(175)#\[(a f_1 + b f_2)(\ket{v}) = a f_1(\ket{v}) + b f_2(\ket{v})\]

One can show that this defines a linear map: for any \(c,d\in \CC\) and \(\ket{v_{1,2}} \in V\),

\[(a f_1 + b f_2)(c \ket{v_1} + d \ket{v_2}) = c (a f_1 + b f_2)\ket{v_1} + d (a f_1 + b f_2)(\ket{v_2})\]

which follows from \(f_{1,2}\) being linear maps/

  1. \(\text{dim}(V^*) = \text{dim}(V)\). It is instructive to show this. Consider a basis \(\ket{i}\) of \(V\), \(i = 1,\ldots d = \text{dim}(V)\). A general vector \(\ket{v}\) can be expressed as

\[\ket{v} = \sum_{i = 1}^d c_i \ket{i}\]

for a unique set of coefficients \(c_i \in \CC\). Now

\[f(\ket{v}) = \sum_{i = 1}^d c_c f(\ket{i})\]

Thus, the map \(f\) is completely specified by \(d\) complex numbers \(f(\ket{i}) = c_i \in CC\). Thus, if we define \(f_i\) by \(f_i(\ket{j}) = \delta_{ij}\), we can show that each \(f_i\) is a linear map. Furthermore, \(f_i\) is linearly independent of \(f_{j \neq i}\) (you should convince yourself of this).

Any function \(f\) can always be written as \(f = \sum_{i = 1}^d c_i f_i\), so this basis is maximal, and \(f_i\) form a complete basis for \(V^*\). There are \(d\) such independent basis functions, so \(\text{dim}(V^*) = d\).

  1. Notation. We can express \(f \in V^*\) as a “bra vector” \(\bra{f}\). We then call elements \(\ket{v} \in V\) “ket vectors”. We can then write

\[\brket{f}{v} \equiv f(\ket{v})\]

as a “bra(c)ket”. I didn’t do this, please blame Dirac. Anyhow the notation is unfortunately standard. With this notation we can define the linear structure of \(V^*\) as

\[\bra{a f_1 + b f_2} = a \bra{f_1} + b \bra{f_2}\]

  1. Finally, The dual vector space for \(V^*\) is \(V\), or \((V^*)^* = V\): for any \(\ket{v}\) the map from \(V^* \to \CC\) is just \(\ket{v}: f \to \brket{f}{v}\).

Example#

If \(V = \CC^3\) is represented as the space of column vectors. we can represent \(V^*\) as the set of row vectors. That is, consider a basis

\[\begin{split}\ket{1} = \begin{pmatrix}1 \\ 0 \\ 0 \end{pmatrix}\ ; \ket{2} = \begin{pmatrix}0 \\ 1 \\ 0 \end{pmatrix}\ ; \ket{3} = \begin{pmatrix}0 \\ 0 \\ 1 \end{pmatrix}\end{split}\]

We can define any linear map \(f\) by \(f(\ket{i} = c_i\). Then if \(\ket{v} = \sum_i a_i \ket{i}\),

\[\begin{split}f(\ket{v}) = \sum_i c_i a_i = \begin{pmatrix} c_1 & c_2 & c_3 \end{pmatrix} \begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix}\end{split}\]

Adjoint maps#

Since \(\dim V = \dim V^*\), we expect that there is an isomorphism (a map that is one-to-one and onto) between them. Choosing such a map leads to a choice of “inner product” on \(V\) itself: a way of assigning to \(\ket{v}\) a number corresponding to some notion of its length.

Definition#

Let \(V\) be a vector space over \(\CC\). An adjoint map is a map \(\cal{A}: V \to V^*\), which we denote by \(A\ket{v} \equiv \bra{f_v}\) with the properties

  1. Skew symmetry: \(\brket{f_w}{v} = \brket{f_v}{w}^*\).

  2. Positive semi-definiteness:

\[\brket{f_v}{v} \equiv ||v||^2 \geq 0\ ; ||v|| = 0\ \text{iff}\ \ket{v} = 0\]

In general we write \(\bra{f_v} = \bra{v}\).

Properties#

  1. Antilinearity. Using skew-symmetry you can show that for \(\ket{v_{1,2}} \in V\), \(a, b \in \CC\),

\[\bra{a v_1 + b v_2} = a^* \bra{v_1} + b^* \bra{v_2}\]

  1. Schwarz inequality

\[|\brket{v}{w}| \leq ||v||\ ||w||\]

  1. Triangle inequality

Examples#

  1. \(V = \CC^3\).

\[\begin{split}A\begin{pmatrix} c_1 \\ c_2 \\ c_3 \end{pmatrix} = \begin{pmatrix} c_1^* & c_2^* & c_2^* \end{pmatrix}\end{split}\]

If

\[\begin{split}\ket{v} = \begin{pmatrix} c_1 \\ c_2 \\ c_3 \end{pmatrix}\ , \ket{w} = \begin{pmatrix} d_1 \\ d_2 \\ d_3 \end{pmatrix}\end{split}\]

then

\[\brket{v}{w} = c_1^* d_1 + c_2^* d_2 + c_3^* d_3\]

  1. \(V = M_2(\CC)\).

\[\brket{M_1}{M_2} = \text{tr} (M_1^{*})^T M_2) = \sum_{i,j} (M_1)^*_{ij} (M_2)_{ij}\]

  1. \(V = L^2(\CR)\), the space of complex square-integrable functions on the real line where \(\ket{\psi}\) is represented by the function \(\psi(x)\). A good inner product, which defines an adjoint map, is

(176)#\[\brket{\chi}{\psi} = \int_{-\infty}^{\infty} dx \chi(x)^* \psi(x)\]

Additional definitions and a comment#

  1. \(\brket{v}{w}\) is the inner product of \(\ket{v}\), \(\ket{w}\).

  2. \(||v||^2 = \brket{v}{v}\) is called the norm of \(\ket{v}\).

  3. \(V\) with an adjoint map is called an inner product space.

  4. An inner product space (over \(\CC\)) is called a Hilbert space if either:

  • \(\text{dim}(V) < \infty\), or

  • Cauchy sequences in \(V\) are complete.

To explain the last possibility, note that \(\ket{v_i}\), \(i = 1,\ldots,\infty\) is a Cuachy sequence if for any \(\eps > 0\), there exists some integer \(N\) such that

\[|| v_n - v_m || < \eps\ \forall n, m \geq N\]

Such a sequence is complete if it converges to a vector in \(V\).

  1. There is no unique adjoint map.

Actions of operators#

Goven a linear operator \(A\) and \(\ket{v} \in V\), \(A\ket{v}\) is a vector and \(\bra{w} A \ket{v}\) is a complex number. We can therefore define \(\bra{A w} \equiv \bra{w} A\) such that \(\brket{A w}{v} = \bra{w} A \ket{v}\).

Orthonormal bases#

Definitions#

Let \(V\) be a vector space over \(\CC\).

  1. \(\ket{v} \in V\) is a normal vector if \(||v||^2 = \brket{v}{v} = 1\).

  2. \(\ket{v},\ket{w} \in V\) are orthogonal if \(\brket{v}{w} = 0\).

  3. An orthonormal basis is a basis \(\ket{i} \in V\), \(i = 1,\ldots,d = \text{dim} V\) such that for \(\cal{A}: \ket{i} \to \bra{i}\), \brket{i}{j} = \delta_{ij}$.

Examples#

  1. We can write \(\ket{v} = \sum_i v_i \ket{i}\); the antilienarity of the adjoint map means that \(\bra{v} = \sum_i \bra{i} v^*_i\). This means that

(177)#\[\brket{v}{v} = \sum_{i,j} v^*_i \brket{i}{j} v_j = \sum_i |v_i|^2\]

Similarly, for \(\ket{w} = \sum_i w_i \ket{i}\),

(178)#\[\brket{w}{v} = \sum_i w^*_i v_i\]

This works if we idenfity

(179)#\[\begin{split}\ket{v} \to \begin{pmatrix} v_1 \\ v_2 \\ \vdots \\ v_n \end{pmatrix}\end{split}\]

and thus

(180)#\[\bra{v} \to \begin{pmatrix} v_1^* & v_2^* & \ldots & v_n^* \end{pmatrix}\]

The basis element \(\ket{i}\) is a column vector with all zeros except a \(1\) in the \(i\)th row.

  1. If \(V = M_2(\CC)\), the space of \(2\times 2\) complex matrices, a natural inner product is

\[\brket{m}{n} = \text{tr} (m^T)^* n\]

where \(m,n\) are \(2\times 2\) matrices. This clearly defines an adjoint map from \(\ket{n}\) to a linear map. An orthonormal basis is:

(181)#\[\begin{split}\ket{1} \to \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} \ ; \ \ \ket{2} \to \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}\ ; \ \ \ket{3} \to \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}\ ; \ \ \ket{4} \to \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}\end{split}\]

  1. Consider the vector space of complex functions on the interval \(0,L\) with Dirichlet boundary condittions. You can convince yourself that the basis

\[\ket{n} \to \psi_n(x) = \sqrt{\frac{2}{L}} \sin \frac{n\pi x}{L}\]

is orthonormal with respect to the inner product (176)

The Gram-Schmidt machine#

Theorem: every finite-dimensional vector space or infinite dimensional vector space with a countable basis has an orthonormal basis.

Proof (partial): Given a basis \(\ket{v_1},\ket{v_2},\ldots,\ket{v_d}\), we can construct a basis iteratively. Define

(182)#\[\begin{split}\begin{align} \ket{1} & = \frac{\ket{v_1}}{||v_1||}\\ \ket{2} & = \frac{\ket{v_2} - \brket{1}{v_2}\ket{1}}{\sqrt{||v_2||^2 - |\brket{1}{v_2}|^2}}\\ \ket{k} & = \frac{\ket{v_k} - \sum_{n = 0}^{k-1} \ket{n}\brket{n}{v_k}}{\sqrt{||v_k||^2 = \sum_{n = 1}^{k-1} |\brket{v_k}{n}|^2}} \end{align}\end{split}\]

Matrix elements of operators#

Since \(\ket{i}\) is a basis, we can write the action of operators in this basis: \(A\ket{j} = A_{ij}\ket{i}\). As notation, we will sometimes write

(183)#\[A = \bra{i} A_{ij} \ket{j}\]

We understand this to mean

\[A\ket{v} = \sum_{i,j} \ket{i} A_{ij} \brket{j}{v} = \sum_{i,j}A_{ij} v_j \ket{i}\]

where \(\ket{v} = \sum_i v_i \ket{i}\), and for dual vectors \(\bra{v} = \sum_i \bra{i} v_i^*\),

\[\bra{v} A = \sum_{i,k} \bra{i} v^*_k A_{ki}\]

Thus

\[\bra{v} A \ket{w} = \sum_{i,j} v^*_i A_{ij} w_j\]

A particularly important example is the identity operator \(\bf{1}\) for which \(\bf{1}_{ij} = \delta_{ij}\). This can be represented as above by:

(184)#\[{\bf 1} = \sum_i \ket{i}\bra{i}\]

for any orthonormal basis. This is called a resolution of the identity, associated to a given basis.

In this basis, an important operator on \(A\) is the transpose. That is given a linear operator \(A\), we can define the transpose \(A^T\) via its matrix elements

(185)#\[(A^T)_{ij} = A_{ji}\]

In particular, we can write

\[\bra{v}A = \bra{i} v^*_k A_{ki} = \bra{i} (A^T)_{ik} v^*_k\]

Adjoints of operators#

The vector \(A\ket{v} \equiv \ket{Av} = A_{lk} v_k \ket{l}\) has a natural adjoint

\[{\cal A} : A\ket{v} \to \bra{l} v_k^* A_{lk}^* = \bra{l} v_k^* (A^T)^*_{kl} \equiv \bra{v} A^{\dagger}\]

which defines the Hermitian conjugate \(A^{\dagger}\). We can either define it as \({\cal A}: A\ket{v} \to \bra{a} A^{\dagger}\) or via its matrix elements in an orthonormal basis,

(186)#\[A^{\dagger}_{ij} = (A^T)^*_{ij} = A^*_{ji}\]

Hermitian and unitary operators#

  1. Definition. A Hermitian operator is an operator \(A = A^{\dagger}\).

Note that this does not mean the operator has real matrix elements. The following operator on \(\CC^2\) is Hermitian:

\[\begin{split}\sigma_y = \begin{pmatrix} 0 & i \\ -i & 0 \end{pmatrix}\end{split}\]

  1. Definition. A Unitary operator is an operator \(U\) such that \(U^{\dagger} = U^{-1}\).

An important property of this operator is that it is norm-preserving:

\[|| U\ket{v}||^2 = \bra{v} U^{\dagger} U \ket{v} = \bra{v} U^{-1} U \ket{v} = \brket{v}{v} = ||v||^2\]

  1. An example of a unitary operator acting on \(\CC^2\):

\[\begin{split}U = \begin{pmatrix} \cos\theta & \sin\theta e^{i\phi} \\ - \sin\theta e^{-i\phi} & \cos\theta \end{pmatrix}\end{split}\]

  1. Two nontrivial examples for \(L^2(\CR)\):

  • The position operator \({\hat x}: \psi(x) \to x \psi(x)\). Since

\[\begin{split}\begin{align} \bra{\chi} \hat{x} ket{\psi} & = \int dx \chi^* (x \psi(x)) = \int dx (x\chi)^* \psi \\ & = \brket{\chi}{x\psi} = \brket{x\chi}{\psi} \end{align}\end{split}\]

as expected for a Hermitian operator.

  • the operator \(\hat{p} = - i\hbar \frac{\del}{\del x}\), when acting on \(\psi(x)\).

\[\begin{split}\begin{align} \bra{\chi} {\hat p}\ket{\psi} & = \int_{-\infty}^{\infty} \chi^* (-i\hbar) \frac{\del \psi}{\del x}\\ & = (-i \hbar) \int_{-\infty}^{\infty} dx \frac{\del}{\del x} (\chi^* \psi) + i \hbar \int dx \frac{\del \chi^*}{\del x} \psi \\ & = - i \hbar \chi^* \psi \Big|_{-\infty}^{\infty} + \int_{-\infty}^{\infty} dx\left(-i\hbar \frac{\del \chi}{\del x}\right)^*\psi\\ & = \bra{\chi}{\hat p}^{\dagger} \ket{\psi} \end{align}\end{split}\]

The second line follows from integration by larts, and the boundary terms vanish because \(\psi\) is sequare integrable. In other words for every \(\ket{\psi},\ket{\chi}\), \(\bra{\chi} {\hat p} \ket{\psi} = \bra{\chi} {\hat p}^{\dagger} \ket{\psi}\). From this we can deduce that \({\hat p} = {\hat p}^{\dagger}\).

The same argument follows for the case of complex functions with periodic boundary conditions. For Dirichlet boundary conditions, \({\hat p}\) fails to be an operator on teh Hilbert space, as the derivative of a function with Dirichlet boundary conditions does not in general satisfy Dirichlet boundary conditions. (Similarly for Neumann boundary conditions).

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