2.9 Dirac Notation Vectors ji that are orthonormal hk ji = k,j span a vector space and express the identity operator I of the space as (1.

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1 14 Fourier Series 2.9 Dirac Notation Vectors ji that are orthonormal hk ji k,j san a vector sace and exress the identity oerator I of the sace as (1.132) I NX jihj. (2.99) Multilying from the right by any vector gi in the sace, we get j1 gi I gi NX jihj gi (2.1) which says that every vector gi in the sace has an exansion (1.133) in terms of the N orthonormal basis vectors ji. Thecoe cientshj gi of the exansion are inner roducts of the vector gi with the basis vectors ji. These roerties of finite-dimensional vector saces also are true of infinitedimensional vector saces of functions. We may use as basis vectors the hases ex(inx)/. They are orthonormal with inner roduct (2.1) (m, n) Z e imx j1 e inx Z dx e i(n m)x dx m,n (2.11) which in Dirac notation with hx ni ex(inx)/ and hm xi hx mi is hm ni Z hm xihx ni dx Z e i(n m)x The identity oerator for Fourier s sace of functions is I dx m,n. (2.12) nihn. (2.13) So we have fi I fi nihn f i (2.14) and hx fi hx I fi hx nihn fi hn fi (2.15) which with hn fi f n is the Fourier series (2.2). The coe cients hn fi f n

2 are the inner roducts (2.3) hn fi Z hn xihx fi dx 2.1 Dirac s Delta Function 15 Z hx fi dx Z f(x) dx. (2.16) 2.1 Dirac s Delta Function A Dirac delta function is a (continuous, linear) ma from a sace of functions into the real or comlex numbers. It is a functional that associates a number with each function in the function sace. Thus (x y) associates the number f(y) with the function f(x). We may write this association as Z f(y) f(x) (x y) dx. (2.17) Delta functions o u all over hysics. The inner roduct of two of the kets xi that aear in the Fourier-series formulas (2.15) and (2.16) is a delta function, hx yi (x y). The formula (2.16) for the coe cient hn fi becomes obvious if we write the identity oerator for functions defined on the interval [, ] as for then hn fi hn I fi Z I Z hn xihx fi dx xihx dx (2.18) Z hx fi dx. (2.19) The equation yi I yi with the identity oerator (2.18) gives yi I yi Z xihx yi dx. (2.11) Multilying (2.18) from the right by fi and from the left by hy, we get f(y) hy I fi Z hy xihx fi dx Z hy xif(x) dx. (2.111) These relations (2.11) and (2.111) say that the inner roduct hy xi is a delta function, hy xi hx yi (x y). The Fourier-series formulas (2.15) and (2.16) lead to a statement about the comleteness of the hases ex(inx)/ f(x) f n Z e iny f(y) einx dy. (2.112)

3 16 Fourier Series Interchanging and rearranging, we have Z e f(x) in(x y)! f(y) dy. (2.113) But f(x) and the hases are eriodic with eriod, so we also have Z! ein(x y) f(x +`) f(y) dy. (2.114) Thus we arrive at the Dirac comb ein(x y) ` 1 or more simly " # cos(nx) n1 (x y `) (2.115) ` 1 (x `). (2.116) Examle 2.13 (Dirac s Comb) The sum of the first 1, terms of this cosine series (2.116) for the Dirac comb is lotted for the interval ( 15, 15) in Fig Gibbs overshoots aear at the discontinuities. The integral of the first 1, terms from -15 to 15 is 5.. The stretched Dirac comb is e in(x y)/ ` 1 (x y `). (2.117) Examle 2.14 (Parseval s Identity) Using our formula (2.35) for the Fourier coe cients of a stretched interval, we can relate a sum of roducts fn g n of the Fourier coe cients of the functions f(x) and g(x) to an integral of the roduct f (x) g(x) Z Z fn g n dx einx/ f (x) dy e iny/ g(y). (2.118) This sum contains Dirac s comb (2.117) and so Z Z fn g n dx dy f (x) g(y) 1 Z dx Z dy f (x) g(y) ` 1 e in(x y)/ (x y `). (2.119)

4 2.1 Dirac s Delta Function x 14 Dirac Comb Sum of Series Figure 2.11 The sum of the first 1, terms of the series (2.116) for the Dirac comb is lotted for 15 ale x ale 15. Both Dirac sikes and Gibbs overshoots are visible. x But because only the ` tooth of the comb lies in the interval [,], we have more simly Z Z Z fn g n dx dy f (x) g(y) (x y) dx f (x) g(x). (2.12) In articular, if the two functions are the same, then Z f n 2 dx f(x) 2 (2.121) which is Parseval s identity. Thus if a function is square integrable on an interval, then the sum of the squares of the absolute values of its Fourier coe cients is the integral of the square of its absolute value. Examle 2.15 (Derivatives of Delta Functions) Delta functions and other generalized functions or distributions ma smooth functions that vanish at

5 18 Fourier Series infinity into numbers in ways that are linear and continuous. Derivatives of delta functions are defined so as to allow integrations by arts. Thus the nth derivative of the delta function (n) (x y) mas the function f(x) to ( 1) n times its nth derivative f (n) (y) at y Z Z (n) (x y) f(x) dx (x y)( 1) n f (n) (x) dx ( 1) n f (n) (y) (2.122) with no surface term. Examle 2.16 (The Equation xf(x) a) Dirac s delta function sometimes aears unexectedly. For instance, the general solution to the equation xf(x) a is f(x) a/x+b (x)wherebisan arbitrary constant (Dirac, 1967, sec. 15), (Waxman and Peck, 1998). Similarly, the general solution to the equation x 2 f(x) a is f(x) a/x 2 +b (x)/x+c (x)+d (x) inwhich (x) is the derivative of the delta function, and b, c, and d are arbitrary constants The Harmonic Oscillator The hamiltonian for the harmonic oscillator is H 2 2m m!2 q 2. (2.123) The commutation relation [q, ] q q i~ imlies that the lowering and raising oerators r m! a q + i r m! and a i q (2.124) 2~ m! 2~ m! obey the commutation relation [a, a ] 1. In terms of a and a, which also are called the annihilation and creation oerators, the hamiltonian H has the simle form H ~! a a (2.125) There is a unique state i that is annihilated by the oerator a, as may be seen by solving the di erential equation r m! hq a i 2~ hq q + i i. (2.126) m!

6 Since hq q q hq and 2.11 The Harmonic Oscillator 19 hq i ~ i the resulting di erential equation is dhq i dq (2.127) dhq i dq m! ~ q hq i. (2.128) Its suitably normalized solution is the wave function for the ground state of the harmonic oscillator m! 1/4 m!q hq 2 i ex. (2.129) ~ 2~ For n, 1, 2,...,thenth eigenstate of the hamiltonian H is ni 1 n! a n i (2.13) where n! n(n 1)...1isn-factorial and! 1. Its energy is H ni ~! n ni. (2.131) The identity oerator is I nihn. (2.132) n An arbitrary state i has an exansion in terms of the eigenstates ni i I i nihn i (2.133) and evolves in time like a Fourier series n X 1, ti e iht/~ i e iht/~ nihn i e i!t/2 with wave function n (q, t) hq, ti e i!t/2 1 X n 1 X n e in!t nihn i (2.134) e in!t hq nihn i. (2.135) The wave functions hq ni of the energy eigenstates are related to the Hermite olynomials (examle 8.6) H n (x) ( 1) n e x2 dn dx n e x2 (2.136)

7 11 Fourier Series by a change of variables x m!/~ q sq and a normalization factor se (sq) 2 /2 m! 1/4 e m!q 2 /2~ m! 1/2q hq ni 2 n n! H n(sq) H n. ~ 2 n n! ~ (2.137) The coherent state i i e 2 /2 e a i e 2 /2 n n n! ni (2.138) is an eigenstate a i i of the lowering (or annihilation) oerator a with eigenvalue. Its time evolution is simly, ti e i!t/2 e 2 /2 n e i!t n n! ni e i!t/2 e i!t i. (2.139) 2.12 Nonrelativistic Strings If we clam the ends of a nonrelativistic string at x and x, then the amlitude y(x, t) will obey the boundary conditions and the wave equation as long as y(x, t) remains small. The functions y n (x, t) sin n x y(,t)y(, t) (2.14) v 2 (2.141) f n sin n vt + d n cos n vt (2.142) satisfy this wave equation (2.141) and the boundary conditions (2.14). They reresent waves traveling along the x-axis with seed v. The sace S of functions f(x) that satisfy the boundary condition (2.14) is sanned by the functions sin(n x/). One may use the integral formula Z sin n x to derive for any function f 2 S the Fourier series f(x) f n sin n x m x sin dx 2 nm (2.143) n1 (2.144)