A Dirac’s delta Function The 1-d delta function, δ(x), is defined through a limiting procedure so that δ(x)=0 for x = 0 (A.1a) and δ(0) = ∞. The meaning of this last relation, taking into account (A.1a), is that b dxf(x)δ(x)=f(0) (A.1b) a for any well-behaved function f(x) and for any pair a, b such that a<0 <b. Some functions producing in the limit the delta function, as defined by (A.1a, A.1b), are given below: ⎧ ⎪ 1 ε ⎪ lim , Lorentzian, (A.2a) ⎪ → + 2 2 ⎪ ε 0 π x + ε ⎪ 2 ⎪ 1 x ⎨⎪ lim √ exp − , Gaussian, (A.2b) σ→0+ σ 2π 2σ2 δ(x)= ⎪ sin(x/ε) ⎪ lim , Dirichlet, (A.2c) ⎪ ε→0+ πx ⎪ ⎪ |L| ⎪ 1 ⎩⎪ lim eikxdk, Fourier. (A.2d) →∞ L 2π −|L| From the definition of the delta function it follows that δ (x − x ) δ(g(x)) = n , (A.3) |g (x )| n n where g(x) is a well-behaved function of x, the sum runs over the real roots, { } ≡ xn ,ofg(x) = 0, and g (xn) (dg/dx)x=xn =0forallxn. The derivative of the delta function is properly defined as δ (x)=0,x= 0 (A.4a) 338 A Dirac’s delta Function and b dxf(x)δ(x)=−f (0) (A.4b) a for any well-behaved function f(x) and for any pair a, b such that a<0 <b. The integral of the delta function is the so-called theta function dθ(x) δ(x)= , (A.5a) dx where 1 , for x>0 , θ(x) ≡ 0 , for x<0 . (A.5b) The definition of the delta function is generalized to more than one inde- pendent cartesian variable: δ (r)=δ (x1) δ (x2) ...δ (xn)(A.6) in an n-dimensional space. More explicitly we have 1 δ (r − r)=δ (x − x) δ (y − y)= δ (r − r) δ (φ − φ) , 2-d , (A.7) r δ (r − r)=δ (x − x) δ (y − y) δ (z − z) 1 = δ (r − r) δ (cos θ − cos θ) δ (φ − φ) r2 1 = δ (r − r) δ (θ − θ) δ (φ − φ) , 3-d . (A.8) r2 sin θ The completeness relation (1.4) of Chap. 1 allows us to obtain many series or integral representations of the delta function. For example, the Fourier representation of the d-dimensional delta function is, according to (1.4) and the footnote on p. 3, ∗ 1 ik · (r−r) δ (r − r )= φk (r) φk (r )= e −→ Ω Ω→∞ k k 1 Ω · r−r 1 k · r−r = dkeik ( ) = dkei ( ) . (A.9) Ω (2π)d (2π)d Equation (A.9) generalizes in d-dimensions (A.2d). Similarly, the delta function over the spherical angles θ and φ can be expressed as a sum over spherical harmonics, Ym(θ, φ), according to (1.4): 1 δ (cos θ − cos θ) δ (φ − φ)= δ (θ − θ) δ (φ − φ) sin θ ∞ ∗ = Ym(θ, φ)Ym (θ ,φ) , (A.10) =0 m=− A Dirac’s delta Function 339 where the spherical harmonics are given in terms of the associated Legendre |m| polynomials, P , by the following expression: 0 (|m|+m) (2 +1)( −|m|)! | | Y (θ, φ)=i(−1) /2 P m (cos θ)eimφ . (A.11) m 4π( + |m|)! Equation (A.10) is valid because the set Ym(θ, φ) is orthonormal π 2π ∗ δ δ sin θdθ dφYm(θ, φ)Y m (θ, φ)= mm . 0 0 B Dirac’s bra and ket Notation In quantum mechanics, the states of a particle are described by eigenfunctions, φn (r), of any hermitian operator, L, corresponding to a physical quantity such as its momentum, its Hamiltonian, etc. The same physical states can equally well be described by the Fourier transforms, φn (k), of φn (r)orbyanyother representation of φn (r) such as the set {an,} of the expansion φn (r)= an,ψ (r)(B.1) in terms of a complete orthonormal set of functions {ψ}. It is very convenient to introduce for a physical state a single symbol that will be independent of any particular representation and such that the various representations, φn (r), φn (k), and {an,}, can be reproduced easily whenever they are needed. This symbol is denoted by |n or by |φn and is called a ket. Thus State of a particle characterized by quantum number(s) n ⇐⇒ | φn or |n To each ket |φn corresponds one and only one conjugate entity denoted by φn| and called a bra. We define also the inner product of two kets, |φ and |ψ, (in that order) as a complex quantity denoted by φ | ψ and such that ∗ φ | ψ = ψ | φ . (B.2) In particular, for an orthonormal set we have φn | φm = δnm . (B.3) Physically, the inner product φ | ψ gives the probability amplitude for a par- ticle being in state |ψ to be observed in state |φ. A complete set of kets, {|φn}, defines an abstract Hilbert space. Some usual complete sets of kets for a spinless particle are the following: (a) The position kets {|r}, where r runs over all positions in the domain Ω; |r corresponds to the state where the particle is completely localized at the point r. 342 B Dirac’s bra and ket Notation (b) The momentum kets {|k}; |k corresponds to the state where the particle has momentum k. (c) The Bloch kets {|k,n}; |k,n corresponds to the state where the particle has crystal momentum k and belongs to the nth band. (d) The kets {|φ}; |φ is an eigenstate of the Hamiltonian. The indexes fully characterize this eigenstate. According to the above, the inner product r | ψ gives the probability ampli- tude for a particle being in the state |ψ to be observed at the point r; but this probability amplitude is nothing other than ψ (r). Hence ψ (r)=r | ψ . (B.4) Similarly, ψ(k)=k | ψ . (B.5) If in (B.4) we choose |ψ = |k, then we obtain 1 · r r | k = √ eik . (B.6) Ω The rhs is the r-representation of a normalized plane wave; if in (B.4) we choose |ψ = |r,thenwehave r | r = δ (r − r) . (B.7) Equation (B.7) follows from the orthonormality of the set {|r}. From (B.4) and (B.7) we obtain ψ (r)=r | ψ = drδ (r − r) r | ψ = dr r | rr | ψ = r| dr |rr | ψ . (B.8) Since (B.8) is valid for every r| and every |ψ, we end up with the complete- ness relation for the set {|r}: dr |rr| =1. (B.9) Similarly, we obtain the relation Ω 1= |kk|−→ dk |kk| . (B.10) Ω→∞ d k (2π) Using (B.9) and (B.10) we can easily transform from ψ(r)toψ(k)andvice versa 1 · r ψ(k)=k | ψ = dr k | rr | ψ = dr √ e−ik ψ (r) Ω 1 · r = √ dre−ik ψ (r) , (B.11) Ω B Dirac’s bra and ket Notation 343 1 · r ψ (r)=r | ψ = r | kk | ψ = √ eik ψ(k) k Ω k √ Ω · r = dkeik ψ(k) . (B.12) (2π)d Notice that the definition of the inner product coincides with the usual defi- nition in the r-representation: φ | ψ = dr φ | rr | ψ = drφ∗ (r) ψ (r) . (B.13) The form of any operator in the Dirac notation follows by calculating its matrix elements. For example, taking into account that the position opera- tor, rˆ, satisfies by definition the relation r| rˆ = r| r and that the momentum operator, pˆ, satisfies the relation k| pˆ = k| k,wehave r | rˆ | ψ = r r | ψ = rψ (r) , (B.14) k | pˆ | ψ = k k | ψ = kψ (k) . (B.15) Using (B.10) and (B.15) we obtain the matrix element r | pˆ | ψ: 1 · r r | pˆ | ψ = r | kk | pˆ | ψ = √ eik kψ(k) k k Ω ∂ · r 1 ∂ 1 k · r = −i eik √ ψ (k)=−i √ ei ψ(k) k ∂r Ω ∂r k Ω ∂ = −i ψ (r) . (B.16) ∂r The last relation follows from (B.12). In a similar way one can show that ∂ k | rˆ | ψ =i ψ(k) . (B.17) ∂k The relations (B.14)–(B.17) can be generalized for operators that are func- tions of rˆ or pˆ. Since by definition the function f Aˆ of any operator Aˆ is an operator with the same eigenstates as Aˆ and eigenvalues f (Ai), where Ai are the eigenvalues of Aˆ,wehave r | f (rˆ) | ψ = f ∗ (r) ψ (r) , (B.18) k | g (pˆ) | ψ = g∗ (k) ψ(k) , (B.19) ∂ r | g (pˆ) | ψ = g∗ −i ψ (r) , (B.20) ∂r ∂ k | f (rˆ) | ψ = f ∗ i ψ (k) . (B.21) ∂k 344 B Dirac’s bra and ket Notation For hermitian operators f ∗ (r)=f (r)andg∗ (k)=g (k). As an application of (B.18) and (B.20) we have 8 9 pˆ2 2 r + V (rˆ) ψ = − ∇2 + V (r) ψ (r) . (B.22) 2m 2m r For particles with spin 1/2 we have to specify whether the spin is up (↑)or down (↓). Thus a complete set is {|r, ↑ , |r, ↓} or {|k, ↑ , |k, ↓}. Assuming that the spin quantization axis is the z-axis, we have for the spin operator s = x0sx + y0sy + z0sz: 1 σ = +1 for spin up , s |r,σ = σ |r,σ , (B.23) z 2 σ = −1forspindown, r, ↑|s | ψ 1 r, ↑|ψ = σ , (B.24) r, ↓|s | ψ 2 r, ↓|ψ where σ is the vector defined by the three Pauli matrices 01 0 −i 10 σ = x + y + z .