4.4 Inner products

对照

An inner product on a vector space $V$ is an additional structure that provides a notion of length for vectors, of angle between vectors, and identifies $V^{\ast }\simeq V$. In the real case:

Definition · Inner product, real case

Definition (Inner product, real case). An inner product on a real vector space $V$ is a symmetric $((v,w)=(w,v))$ map

$$(\cdot ,\cdot ):V\times V\to \mathbf{R}$$

that is non-degenerate and linear in both variables.

Our real inner products will usually be positive-definite ($(v,v)\ge 0$ and $(v,v)=0⟹v=0$) , with indefinite inner products only appearing in the context of special relativity, where an indefinite inner product on four dimensional space-time is used.

In the complex case:

Definition · Inner product, complex case

Definition (Inner product, complex case). A Hermitian inner product on a complex vector space $V$ is a map

$$⟨\cdot ,\cdot ⟩:V\times V\to \mathbf{C}$$

that is conjugate symmetric

$$⟨v,w⟩=\overline{⟨w,v⟩}$$

non-degenerate in both variables, linear in the second variable, and antilinear in the first variable: for $\alpha \in \mathbf{C}$ and $u,v,w\in V$.

$$⟨u+v,w⟩=⟨u,w⟩+⟨v,w⟩,⟨\alpha u,v⟩=\overline{\alpha }⟨u,v⟩$$

An inner product gives a notion of length-squared $||\cdot |{|}^2$ for vectors, with

$$||v|{|}^2=⟨v,v⟩$$

Note that whether to specify antilinearity in the first or second variable is a matter of convention. The choice we are making is universal among physicists, with the opposite choice common among mathematicians. Our Hermitian inner products will be positive definite ($\Vert v{\Vert }^2>0$ for $v\ne 0$) unless specifically noted otherwise (i.e., characterized explicitly as an indefinite Hermitian inner product).

An inner product also provides an isomorphism $V\simeq V^{\ast }$ by the map

$$v\in V\to l_v\in V^{\ast }\text{\hspace{1em}(4.3)}$$

where $l_v$ is defined by

$$l_v(w)=(v,w)$$

in the real case, and

$$l_v(w)=⟨v,w⟩$$

in the complex case (where this is a complex antilinear rather than linear isomorphism).

Physicists have a useful notation due to Dirac for elements of a vector space and its dual, for the case when $V$ is a complex vector space with a Hermitian inner product (such as the state space $H$ for a quantum theory). An element of such a vector space $V$ is written as a “ket vector”

$$|\alpha ⟩$$

where $\alpha$ is a label for a vector in $V$. Sometimes the vectors in question will be eigenvectors for some observable operator, with the label $\alpha$ the eigenvalue.

An element of the dual vector space $V^{\ast }$ is written as a “bra vector”

$$⟨\alpha |$$

with the labeling in terms of $\alpha$ determined by the isomorphism 4.3, i.e.,

$$⟨\alpha |=l_{|\alpha ⟩}$$

Evaluating $⟨\alpha |\in V^{\ast }$ on $|\beta ⟩\in V$ gives an element of $\mathbf{C}$, written

$$⟨\alpha |(|\beta ⟩)=⟨\alpha |\beta ⟩$$

Note that in the inner product the angle bracket notation means something different than in the bra-ket notation. The similarity is intentional though since $⟨\alpha |\beta ⟩$ is the inner product of a vector labeled by $\alpha$ and a vector labeled by $\beta$ (with “bra-ket” a play on words based on this relation to the inner product bracket notation). Recalling what happens when one interchanges vectors in a Hermitian inner product, one has

$$⟨\beta |\alpha ⟩=\overline{⟨\alpha |\beta ⟩}$$

For a choice of orthonormal basis $\{{\mathbf{e}}_j\}$ , i.e., satisfying

$$⟨{\mathbf{e}}_j,{\mathbf{e}}_k⟩={\delta }_{jk}$$

a useful choice of label is the index $j,$ so

$$|j⟩={\mathbf{e}}_j$$

Because of orthonormality, coefficients of vectors $|\alpha ⟩$ with respect to the basis $\{{\mathbf{e}}_j\}$ are

$$⟨j|\alpha ⟩$$

and the expansion of a vector in terms of the basis is written

$$|\alpha ⟩=\sum _{j=1}^n|j⟩⟨j|\alpha ⟩\text{\hspace{1em}(4.4)}$$

Similarly, for elements $⟨\alpha |\in V^{\ast }$,

$$⟨\alpha |=\sum _{j=1}^n⟨\alpha |j⟩⟨j|$$

The column vector expression for $|\alpha ⟩$ is thus

$$\left(\begin{array}{c}⟨1|\alpha ⟩\\ ⟨2|\alpha ⟩\\ ⋮\\ ⟨n|\alpha ⟩\end{array}\right)$$

and the row vector form of $⟨\alpha |$ is

$$\left(⟨\alpha |1⟩⟨\alpha |2⟩\dots ⟨\alpha |n⟩\right)=\left(\overline{⟨1|\alpha ⟩}\overline{⟨2|\alpha ⟩}\dots \overline{⟨n|\alpha ⟩}\right)$$

The inner product is the usual matrix product

$$⟨\alpha |\beta ⟩=\left(\begin{array}{cccc}⟨\alpha |1⟩& ⟨\alpha |2⟩& \dots & ⟨\alpha |n⟩\end{array}\right)\left(\begin{array}{c}⟨1|\beta ⟩\\ ⟨2|\beta ⟩\\ ⋮\\ ⟨n|\beta ⟩\end{array}\right)$$

If $L$ is a linear operator $L:V\to V$ , then with respect to the basis $\{{\mathbf{e}}_j\}$ it becomes a matrix with matrix elements

$$L_{kj}=⟨k|L|j⟩$$

The expansion 4.4 of a vector $|\alpha ⟩$ in terms of the basis can be interpreted as multiplication by the identity operator

$$1=\sum _{j=1}^n|j⟩⟨j|$$

and this kind of expression is referred to by physicists as a “completeness relation”, since it requires that the set of $|j⟩$ be a basis with no missing elements. The operator

$$P_j=|j⟩⟨j|$$

is the projection operator onto the $j$’th basis vector.

Digression. In this book, all of our indices will be lower indices. One way to keep straight the difference between vectors and dual vectors is to use upper indices for components of vectors, lower indices for components of dual vectors. This is quite useful in Riemannian geometry and general relativity, where the inner product is given by a metric that can vary from point to point, causing the isomorphism between vectors and dual vectors to also vary. For quantum mechanical state spaces, we will be using a single, standard, fixed inner product, so there will be a single isomorphism between vectors and dual vectors. In this case the bra-ket notation can be used to provide a notational distinction between vectors and dual vectors.

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上一节：4.3 Change of basis · 下一节：4.5 Adjoint operators

附录：约定与记号

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正文：英文 · 原著转录

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原书 PDF · 印刷页 38、39、40、41

来源版本：2025-10-20

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