Appel W. Mathematics for Physics and Physicists

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Chapter

Bras, kets,

and all that sort of thing

14.1

Reminders about ﬁnite dimension

In this section, we consider a vector space E, real or comple x, with ﬁnite di-

mension n, equipped with a scalar product (·|·) and with a given orth onormal

basis (e

, . . . , e

). Some elementary properties concerning the scalar product,

orthonormal bases, and the adjoint of an endomorphis m will be recalled;

these results should be compared with those holding in inﬁnite dimensions,

which are quite different.

14.1.a Scalar product and representation theorem

THEOREM 14.1 (Representation theorem) The space E is isomor phic to its dual

space E

∗

. More precisely, for any linear form ϕ on E, there exists a unique a ∈ E

such that

∀x ∈ E ϕ(x) = (a|x) .

Proof

Existence. Using the orthonormal basis (e

, . . . , e

), deﬁne a =

j=1

ϕ(e

) e

For any x =

i=1

∈ E, we have ϕ(x) =

i=1

ϕ(e

), hence ϕ(x) = (a|x).

Uniqueness. Assume a and a

′

∈ E satisfy

(a|x) = (a

′

|x) ∀x ∈ E.

378 Bras, kets, and all that sort of thing

Then (a − a

′

|x) = 0 for any x ∈ E, and in particular (a −a

′

|a −a

′

) = 0, which

implies that (a − a

′

) = 0 and so a = a

′

14.1.b Adjoint

DEFINITION 14.2 (Adjoint) Let u be an endomorphism of E. The adjoint

of u, if it exists, is the unique v ∈ L (E) such that , for any x, y ∈ E, we have



v(x)









u( y)



. I t is denoted u

∗

, and so by deﬁnition we have

∀x, y ∈ E



∗

(x)









u( y)



Proof of uniqueness. Let v, w both be adjoints of u. Then we have

∀x, y ∈ E





u( y )





v(x)







w(x)





which shows that v(x) − w(x) ∈ E

⊥

= {0} for any x ∈ E and he nce v = w.

THEOREM 14.3 (Existence of the adjoint) In a euclidean or hermitian spac e E

(hence of ﬁnite dimension), any endomorphism admits an adjoint. Moreover, if u ∈

L (E) and if B is an orthonormal basis of E, the matrix representing u

∗

in the

basis B is the conjugate transpose of that of u:

mat

∗

) =

mat

(u).

Proof. For any x ∈ E, the map ϕ

: y 7→





u( y )



is a linear form; according to

Theorem 14 .1, there ex ists a unique vector, which we denote u

∗

(x), such that ϕ

is the

map y 7→



∗

(x)





It is easy to see that the map x 7→ u

∗

(x) is linear. So it is the adjoint of u.

Another viewpoint, less elegant but more constructi ve, is the following.

Choose an or thonormal basis B. Notice ﬁrst that if x, y ∈ E are represented by

matrices X and Y in this basis, we have (x|y) =

X Y . Notice also that if

X AY =

X BY for any X , Y ∈ M

(C), then in fact A = B.

Let now u, v ∈ L (E), and deﬁne U = mat

(u) and V = mat

(v). Thanks to the

preceding remark,

v is an adjoint of u ⇐⇒ ∀x, y ∈ E



v(x)









u( y )



⇐⇒ ∀X , Y ∈ M

(R)

(V X )Y =

X (U Y )

⇐⇒

V = U .

Hence it sufﬁces to deﬁne u

∗

as the endomorphism represented by the matrix

U in

the basis B.

COROLLARY 14.4 (Properties of the adjoint) Let u, v ∈ L (E) and let λ ∈ C :

(λu + v)

∗

= λu

∗

+ g

∗

, Id

∗

= Id

, (u ◦ v)

∗

= v

∗

◦u

∗

)

∗

= u, rank(u

∗

) = rank(u), tr (u

∗

) = tr (u),

det u

∗

= det u, λ ∈ Sp(u) ⇐⇒ λ ∈ Sp(u

∗

Moreover, if u is invertible, then u

∗

is invertible and (u

∗

)

−1

= (u

−1

)

∗

Kets and bras 379

14.1.c Symmetric and hermitian e ndomorphisms

DEFINITION 14.5 (Hermitian/symmetric endomorphisms) Let u ∈ L (E)

be an endomorphism of E. Then u is called hermitian, symmetric, or s elf-

adjoint if

∀x, y ∈ E



u(x)









u( y)



or equivalently if u

∗

= u. In other words, u is hermitian if and only if th e

matrix representing u in an orthonormal b asis sat isﬁes A =

A (hermitian

matrix).

In the euclidean case (for a real vector space), the terminology used is

symmetric or self-adjoint, and amounts to saying that the matrix representing

u in an orthonormal bas is is symmetric.

THEOREM 14.6 (Spectral theorem) Let u be a normal endomorphism of E, that

is, such that u u

∗

= u

∗

u. Then there exists an orthonormal basis E = (e

, . . . , e

)

such that the matrix representing u in E is diagonal.

Similarly, if A ∈ M

A, then there exist a unitary matrix U

and a diagonal matrix D such that A = U D

U .

Since a self-adjoint endomorphism obviously commutes with its adjoint,

the following corollary follows:

COROLLARY 14.7 (Reduction of a self-adjoint endomorphism) Let u be a self-

adjoint endomorphism (i.e., hermitian in the complex case, or symmetric in the real

case). Th en u can be diagonalized in an orthonormal basis.

Also the reader should be able to prove the following result as a n exercise:

THEOREM 14.8 (Simultaneous reduction) If u are v two commuting self-adjoint

endomorphisms of E, then they can be diagonalized simultaneously in the same or-

thonormal basis.

14.2

Kets and bras

In t he remainder of th is chapter, H is a separable Hilbert s pace (i.e., ad-

mitting a countable Hilbert basis).

14.2.a Kets |ψ〉 ∈ H

DEFINITION 14.9 (Kets) Following Dirac, vectors in the space H will b e

generically denoted |ψ〉. They are also called the kets of the space H.

A function with values in H will be denoted t 7→



ψ(t)



380 Bras, kets, and all that sort of thing

An important Hilbert space is the space L

(R). A function ψ ∈ L

(R)

will be denoted |ψ〉. If th is function is furthermore a function of time t, the

function x 7→ ψ(x, t) will be denoted



ψ(t)



Remark 14.10 (Spaces of quantum mechanics) In the setting of quantum mechanics, it is usual

to work at the same time wi th two or even three Hilbert spaces:

• an “abstract” Hilbert space H, the vectors of which are denoted |ψ〉;

• the space L

, in which the vectors are denoted ψ (meaning a function x 7→ ψ(x)); th is

space w ill be denoted L

(RR

RR, d x) for added precision;

• the space L

again, but in which the vectors are denoted Ψ : p 7→ Ψ(p); this sp ace will

be denoted L

(RR

RR, d p).

The square integrable functions ψ correspond to Schrödinger’s formalism of wave mechanics, in

position representation in the case of the space L

(R, dx), and in m omentum representation

in the case of the space L

(R, dp). The function p 7→ Ψ(p) is the Fourier transform of the

function x 7→ ψ(x). We know that the Fourier transform gives an isometry between L

(R, dx)

and L

(R, dp).

When working at the same time with the abstract space H and t he space L

(R), the

distinction can be be made explicit, for instance, by writing ϕ a function in L

(R) and |ϕ〉 the

corresponding abstract vector. However, we w ill not make thi s distinction he re.

In the case of a particle conﬁned in an interval [0, a] (because of an inﬁnite potential

sink), th e space used is L

[0, a] for the position representatio n and ℓ

for the momentum

representation; going from one to the other is then done by means of Fourier series.

Moreover, one should notice that all that is done here in the space L

(R) may be easily

generalized to L

) or L

We will denote 〈·, ·〉

the scalar product in H. The norm of a vector |ψ〉

is therefore



|ψ〉



def



|ψ〉, |ψ〉



14.2.b Br as 〈ψ| ∈ H

′

DEFINITION 14.11 (Topolog ic a l dual space) The (topological) dual space

of H, denoted H

′

, is the s pace of continuous linear forms on H. Often,

we will simply write “dual” for “topological dual.”

DEFINITION 14.12 (Bras) Elements of H

′

are called bras and are denoted

generically 〈ϕ|. The result of the application of a bra 〈ϕ| on a ket |ψ〉 is

denoted 〈ϕ|ψ〉, which is therefore a bracket.

Example 14.13 Consider the Hilbert space L

(R). The map

ω: L

(R) −→ C

ψ 7−→

+∞

−∞

ψ(x) e

−x

Dirac did not simply make a pun when inventing the terminology “bra” and “ket”: this

convenient notation became quickly indispensable for most physicists. Note th at Dirac also

coined the words ferm ion and boson, and introduced the terminology of c-number, the δ-

function, and the useful notation }h = h/2π.

Kets and bras 381

is li near. De ﬁning g : x 7→ e

−x

, we have ω(ψ) = 〈g, ψ〉

for any ψ ∈ L

; by the Cau chy-

Schwarz inequality, we have

∀ψ ∈ L



ω(ψ)



〈g, ψ〉



¶ kgk

kψk

and hence the linear form ω is continuous and is an element of th e topological dual of L

(R).

More generally, if we ch oos e a ket ϕ ∈ H, a linear form

: |ψ〉 7−→



|ϕ〉, |ψ〉



is associated to ϕ using the scalar product. The Cauchy-Schwarz inequality

shows that this linear form ω

is continuous (see Theorem A.51 on page 583).

It corresponds therefore to a bra, wh ich is denoted 〈ϕ|:

〈ϕ|ψ〉

def



|ϕ〉, |ψ〉



In other words:

To each ket |ϕ〉 ∈ H is associated a bra 〈ϕ| ∈ H

′

The converse is also true, as proved by the Riesz representa tion theorem:

THEOREM 14.14 (Riesz representation theorem) The map |ϕ〉 7→ 〈ϕ|is a semi-

linear isomorphism of the Hilbert space H with its topological dual H

′

. In other words,

for any continuous linear form 〈ϕ|, there exists a unique vector |ϕ〉 such that

∀|ψ〉 ∈ H 〈ϕ|ψ〉 =



|ϕ〉, |ψ〉



Hence there is a one-to-one correspondance between bras and kets:

To each bra 〈ϕ| ∈ H

′

one can associate a ket |ϕ〉 ∈ H.

Because of this, it could be conceivable to work only with H and to forget

about H

′

, stating that 〈ϕ|ψ〉 is nothing more nor less than a notation for



|ϕ〉, |ψ〉



. However, th is result will not remain true when we consider

“generalized bras. ”

When we deﬁne on H

′

the operator norm



〈ϕ|



′

def

= sup

k|ψ〉k



〈ϕ|ψ〉



the isomorphism above,

H −→ H

′

|ϕ〉 7−→ 〈ϕ|,

becomes a semilinear isometry:



〈ϕ|



′



|ϕ〉



In particular it is permissible to write kϕk without ambiguity whether ϕ

corresponds to a bra or a ket.

382 Bras, kets, and all t hat sort of thing

Remark 14.15 The semilinearity of the map above means that 〈λϕ| = λ 〈ϕ| for any λ ∈ C

and ϕ ∈ H. This is a result of the convention chosen; if we associate to a ket |ϕ〉 the linear

map

|ψ〉 7−→



|ψ〉, |ϕ〉



the result map H → H

′

is a linear isomorphism.

Counterexample 14.16 Consider again the Hilbert space H = L

(R). Let x

∈ R. Then the

map δ

: ψ 7→ ψ(x

) is a linear form on the subspace D of continuous functions

in H.

However, it does not take much effort to discover that it is not representable as a scalar

product with a square integrable function. W h at’s the problem? Certainly, it must be the case

that δ

is continuous?

Well, in fact, it is not! It is continuous on the space D wit h its topology (of uniform

convergence), but not on L

with the hermitian topology (based on integ rals, so deﬁned “up

to equality almost everywhere”).

To see this, it sufﬁces to deﬁne a sequence of functions ψ

: x 7→ e

−nx

. Then, with

= 0 and δ = δ

(which is no loss of generality), we have

kψ

+∞

−∞

−nx

dx =

−−→

n→∞

whereas δ(ψ

) = 1 for all n ∈ N. So there does not exist a constant α ∈ R

such that



δ(ψ)



¶ α kψk

for all ψ ∈ H, whic h means that δ is not continuous.

14.2.c Ge neralized bras

We will denote also with the generic notation 〈ψ| the generalized bras, namely

linear forms which are not necessarily in H

′

. Such a form may not be contin-

uous, or it may not be deﬁned on the whole of H (as seen in the case of δ,

in counterexample 14.16); still it will be asked that it be deﬁned at least on a

dense subspace of H.

One of the most important example is the following:

DEFINITION 14.17 (The bra 〈x

|) Let x

∈ R. We denote by 〈x

| the discon-

tinuous linear form deﬁned on the dense subspace of continuous functions

〈x

|ψ〉 = ψ(x

The bra 〈x

| is thus another name for the Dirac d istribution δ(x − x

Consider now a broader approach to the notion of generalized bra. For

this, let T be a tempered distr ibution. Since S is a dense subspace of L

(R),

the dual S

′

of S contains the dual of L

(R), which is L

(R) itself because

of the Riesz representation theorem.

Hence we have the following situation:

S ⊂ L

(R) ⊂ S

′

Since functions in L

are deﬁned up to equality almost everywhere, δ

is not well deﬁned

on the whole of H. We therefore restrict it to the dense subspace of continuous functions.

In fact, the dual of L

(R) is isomorphic to L

(R), but via this isomorphism one can

identify any continuous linear form on L

(R) to a function in L

(R).

Kets and bras 383

This is called a Gelfand triple. It is tempting to deﬁne the bra 〈T | by

〈T |ϕ〉 = 〈T , ϕ〉; but this last expression is linear in T , in the sens e that

〈αT , ϕ〉 = α 〈T , ϕ〉, whereas the hermitian product is semilinear on the left.

To dea l with this minor inconvenience, the following deﬁnition is therefore

used:

DEFINITION 14.18 (Generalized br a ) L et T be a tempered distribution. The

generalized bra 〈T | is deﬁned on t he Schwartz space S by its action on any

|ϕ〉 ∈ S given by

〈T |ϕ〉

def

= 〈T , ϕ〉.

Remark 14.19 It is immediate to check that, if T

is a regular distribution, associ ated to a

function ϕ ∈ S , then we have

∀ψ ∈ H







+∞

−∞

ϕ(x) ψ(x) dx,

which is equal to the hermitian scalar product 〈ϕ|ψ〉 or (ϕ|ψ) in L

(R).

If T is a “real” distribution (such as δ), we do, however, have 〈T |ϕ〉 = (T |ϕ).

14.2.d Ge neralized kets

In the case where H = L

(R) or L

[a, b], it is possible to associate a general-

ized ket to some generalized bras. If 〈ψ| is a linear form deﬁned by a regular

distribution

〈ψ|ϕ〉 =

+∞

−∞

ψ ϕ,

where ψ is a locally integrable function, we deﬁne the associated generalized

ket |ψ〉, namely, simply the function ψ. This does not necessarily belong

to L

and is therefore not necessarily an element of H. All that is a sked is

that the integral be deﬁned.

DEFINITION 14.20 (Ket | p〉 a nd bra 〈 p|) Let p be a real number. The f unc-

tion

: x 7→

2π }h

i p x/ }h

is not square integrable. It still d eﬁnes a noncontinuous linear form on the

dense s pace L

∩ L

Ω

: L

(R) ∩ L

(R) −→ C

ϕ 7−→

+∞

−∞

(x) ϕ(x) dx = eϕ(p),

where eϕ is the Fourier transform of ϕ, with the conventions usual in quan-

tum mechanics. This generalized bra is denoted 〈p| = 〈Ω

|. The ket that

corresponds to the function ω

is denoted |p〉 = |ω

〉 /∈ H.

384 Bras, kets, and all t hat sort of thing

Moreover, th e notion of d istribution can be seen as a generalization of

that of function. Hence, th e generalized bra 〈p| not only acts on (a subspace

of) L

(R), associating to a function its Fourier transform evaluated at p, but

can be extended to tempered distributions. If |T 〉 is a ket formally associated

to a tempered distribution T ∈ S

′

, then we have 〈p|T 〉 =

T (ω

THEOREM 14.21 The family of kets |p〉 is orthonormal, in the sense that for all

p, p

′

∈ R, we have 〈p



′

〉 = δ(p − p

′

Proof. Apply the deﬁnition 〈p| p

′

〉 = Ýω

(ω

′

), and remember that the Fourier

transform of ω

′

in the sense of distributions is δ

′

The boundary between “bra” and “ket” is not clear. A distribution is a

linear form, thus technically a generalized “bra.” But as a generalization of

a function, it can — formally — be denoted as a ket. Thus we deﬁne the

〉 associated to the “generalized function” δ(x − x

). In particular, for any

function ϕ ∈ S , we have

〈ϕ|x

〉 = 〈x

|ϕ〉 = ϕ(x

Also, the kets |x〉 satisfy th e or thonormality relation:

〈x|x

〉 = δ(x − x

) = δ

valid in t he sense of tempered distributions — that is, which makes sense only

when applied to a function in the Schwart z s pace.

14.2.e Id =

|ϕ

〉〈ϕ

Let (ϕ

)

n∈N

be an orthonormal Hilbert basis of H. We know that this prop-

erty is characterized by the fact that

∀x ∈ H x =

∞

n=0

(ϕ

|x) ϕ

Using Dirac’s notation, this equation is written as follows:

∀|ψ〉 ∈ H |ψ〉 =

∞

n=0

〈ϕ

|ψ〉 |ϕ

〉,

or also, putting the scalar 〈ϕ

|ψ〉 after the vector |ϕ

〉

∀|ψ〉 ∈ H |ψ〉 =

∞

n=0

|ϕ

〉 〈ϕ

|ψ〉.

Or still, formally, the operator

∞

n=0

|ϕ

〉〈ϕ

| is the identity operator. This is

what physicists call a closure relation.

For easy remembering, this will be stated as

Kets and bras 385

THEOREM 14.22 (Closure rela tion) A family (ϕ

)

n∈N

is a Hilbert basis of H if

and only if

Id =

∞

n=0

|ϕ

〉〈ϕ

This point of vie w allows us to recover easily another characterization of a

Hilbert basis, namely th e Parseval identity. Indeed, for any |ψ〉 ∈ H, we have

kψk

= 〈ψ|ψ〉 = 〈ψ|



∞

n=0

|ϕ

〉〈ϕ



|ψ〉 =

∞

n=0

〈ψ|ϕ

〉〈ϕ

|ψ〉

∞

n=0



〈ϕ

|ψ〉



which can be compared with the formula d) in Theorem 9.21 on page 259.

14.2.f Generalized basis

We now extend the notion of basis of the space L

(R), to include distribu-

tions, seen a s generalizations of functions. Let (T

)

λ∈Λ

be a family of tempered

distributions, indexed by a discrete or continuous set

Λ. To generalize the

idea of a Hilbert basis, the ﬁrst idea would be to impose t he condition

Id =

〉〈T

| dλ, t hat is, |ϕ〉 =

〉〈T

|ϕ〉 dλ,

where the notation

dλ may represent an integral, a sum, or a mixture of

both. Still, there would be the problem of deﬁning correctly the integral on

the right: the integ rand is not a sca lar but a vector in an inﬁnite-dimensional

space.

It is easier to consider another characterization of a Hilbert basis : the

Parseval formula, which only involves s ca lar quantities.

DEFINITION 14.23 (Generalized basis) The family (T

)

λ∈Λ

is a generalized

basis, or a total orthonormal system, if and only if we have

∀|ϕ〉 ∈ S kϕk

= 〈ϕ|ϕ〉 =



〈T

|ϕ〉



dλ.

The family (T

)

λ∈Λ

is a total system (or a complete system) if, for all ϕ ∈ S ,

we have



〈T

, ϕ〉 = 0 for any λ ∈ Λ



⇐⇒



ϕ = 0



The choice of index is not without meaning. A generalized basis can be useful to “diago-

nalize” a self-adjoint operator which h as, in addition to eigenvalues, a continuous spectrum.

386 Bras, kets, and all that sort of thing

As in the case of the usual Hilbert bas is (see Theorem 9.21), a generalized

basis is a total system. However, the normalization of such a family (the

elements of which are not a priori in L

) is d elicate.

Note that a generalized basis is not in general a Hilbert basis of L

, for

instance, because any Hilbert b asis is countable.

THEOREM 14.24 (The basis {〈x| ; x ∈ RR

RR}) Denote by 〈x| = 〈δ

| the generalized

bra associated to the distribution δ

. Then the family of distributions



〈x| ; x ∈ R



is a generalized basis of L

(R).

Proof. Indeed, for any ϕ ∈ S , by the very deﬁnition of the L

norm, we have

kϕk

+∞

−∞



ϕ(x)



dx =

+∞

−∞



〈x|ϕ〉



dx.

THEOREM 14.25 (The basis {〈 p| ; p ∈ RR

RR}) Denote by 〈p| = 〈Ω

| the general-

ized bra deﬁned by the regular distributions Ω

associated to the function

: x 7−→

iπx/ }h

2π }h

Then the family



〈p| ; p ∈ R



is a generalized basis of L

(R).

Proof. Let ϕ ∈ S . Then 〈p|ϕ〉 = eϕ(p); the Parseval-Plancherel identity (Theo-

rem 10.36), which is a translation of the fact that the Fourier transform is an isometry

of L

(R), can be written as

kϕk

= keϕk

+∞

−∞



eϕ(p)



dp =

+∞

−∞



〈p|ϕ〉



dp.

Finally, it can be shown that the closure relation is still valid for two gen-

eralized bases



〈x| ; x ∈ R



and



〈p| ; p ∈ R



. Indeed, if ϕ, ψ ∈ L

(R),

the relation 〈ϕ|ψ〉 =

〈ϕ|p〉〈p|ψ〉 dp holds: this is (once more) the Parseval-

Plancherel identity! Similarly, we have

〈ϕ|ψ〉 =

+∞

−∞

ϕ(x) ψ(x) dx =

+∞

−∞

〈x|ϕ〉 〈x|ψ〉 dx

+∞

−∞

〈ϕ|x〉〈x|ψ〉 dx.

It w ill be convenient to state this as follows:

THEOREM 14.26 (Generalized closure relation)

Id =

+∞

−∞

|x〉〈x| dx and Id =

+∞

−∞

|p〉〈p| dp.

Remark 14.27 This shows that it is possible to write, formally, identities like

|ϕ〉 =

+∞

−∞

|x〉〈x|ϕ〉 dx, that is, ϕ(·) =

+∞

−∞

(·) ϕ(x) dx.