3. The wave function [Last revised: Saturday 28th November, 2020, 12:53] 115 Continuous spectrum So far we have assumed observables with a discrete spectrum of possible values. • This is in particular the case of Sz. B But there are observables, like the position or the momentum of a free particle, that have a continuum of possible values. As a consequence of postulate III, we will need an operator with a continuous set of eigenvalues. Then the operator will be acting on a Hilbert space of infinite dimension. B As we will see in this chapter, most of the results that we have already obtained can be generalized to a continuous spectrum. 116 Continuous spectrum Let us revise first the concept of orthonormal basis. • B If A is an operator with a (non-degenerate) discrete spectrum, A a = a a . | i | i Then a is an orthonormal basis, {| i} 1,a = a0 a a0 = d (Krönecker delta), d = aa0 aa0 8 0,a = a0 ⌦ ↵ < 6 B Consider now a generic operator X with a continuous: spectrum, X x = x x . | i | i Then x is an orthonormal basis in the following sense, {| i} • , x = x0 (!!) x x = d(x x ) (Dirac delta), d(x x )= 0 0 0 8 − − 0,x = x0 ⌦ ↵ < 6 : 117 Continuous spectrum B The Dirac delta is strictly not a function, it is a distribution that assigns a complex number to each smooth function in x = x0 constrained to satisfy • x0+e dx d(x x0) f (x)= dx d(x x0) f (x)= f (x0), ˆ • − ˆx e − − 0− equivalently, a distribution that fulfills • dx d(x x0)=1, d(x)=0, x = 0. ˆ • − 8 6 − From now on, definite integrals extend from • to • unless otherwise stated. − 118 Continuous spectrum The Dirac delta can be otained as the limit when L • or e 0+ of the B ! ! functions: L 1 ikx sin Lx (a) dL(x)= dk e = 2p ˆ L px − 1 x2/(2e2) (b) de(x)= e− (Gaussian) (2pe2)1/2 e/p (c) de(x)= (Breit-Wigner) x2 + e2 e e q(x + 2 ) q(x 2 ) dq(x) (d) de(x)= − − e ! dx q(x)=Heaviside step function 119 Continuous spectrum For continuous spectra the completeness relation is generalized as follows • A :  a a = I, X : dx x x = I. a | ih | ˆ | ih | B Any vector in the Hilbert space of an operator A can be expressed in the basis of eigenvectors, A : y =  a a y  a ca | i a | ih | i ⌘ a | i where c is the component of y along a . a | i | i B In the case of X with a continuous spectrum, X : y = dx x x y dx x y(x) | i ˆ | ih | i ⌘ ˆ | i where y(x)= x y is the wave function of y , analogous to the component of h | i | i y along x . | i | i 120 Continuous spectrum Notice that the vectors in the Hilbert space of A must be normalized, • 2 y y =  y a a y =  c⇤a ca =  ca = 1. h | i a h | ih | i a a | | Likewise, 2 y y = dx y x x y = dx y⇤(x)y(x)= dx y(x) = 1. h | i ˆ h | ih | i ˆ ˆ | | This requires y(x) L2(R),asquare-integrable function in R. 2 121 Continuous spectrum B The probability to obtain an eigenvalue a of the observable A on the state y is | i p = y P y = y a a y a h | A,a | i h |  | iih i| | i ai=a ! 2 =  ai y i | h | i | for the general case of a degenerate eigenvalue. B Analogously, the probability to obtain a value x of the observable X in the interval [x , x ] on the state y is 0 1 | i x1 p = y P y = y dx x x y [x0,x1] h | X,[x0,x1] | i h | ˆ | ih | | i ✓ x0 ◆ x1 = dx y(x) 2. ˆ x0 | | Hence, y(x) 2 is the probability density to obtain a value x of X that gives the B | | probability to obtain x in the interval [x, x + dx]. 122 Continuous spectrum Finally, the braket of two states • f y = dx f x x y = dx f⇤(x)y(x) h | i ˆ h | ih | i ˆ is the overlap between their wave functions. B And for an arbitrary function F = F(X), F(X) x = F(x) x , | i | i the matrix element f F(X) y = dx f F(X) x x y = dxF(x) f x x y h | | i ˆ h | | ih | i ˆ h | ih | i = dxF(x)f (x)y(x). ˆ ⇤ 123 Position representation The position of a particle will be given by the eigenvalue of the position operator, • ~X =(X1, X2, X3)=(X, Y, Z). The (infinite-dimensional) orthonormal basis of eigenvectors of X~ is ~x where {| i} ~x labels every point in the 3-dimensional space, 3 ~x ~x0 = d (~x ~x0)=d(x x0)d(y y0)d(z z0) (orthonormality) − − − − 3 ⌦ ↵ d x ~x ~x = I (closure). ˆ | ih | B The wave function in this basis is called the position representation, ~x y y(~x)=y(x, y, z). h | i ⌘ The normalization of the physical states is expressed as 3 3 3 2 1 = y y = d x y ~x ~x y = d x y⇤(~x)y(~x)= d x y(~x) . h | i ˆ h | ih | i ˆ ˆ | | 124 Position representation The Hilbert space of physical states is formed by the square-integrable functions • in R3, L2(R3). It is remarkable that, in particular, the wave function of ~x , | 0i 3 ~x ~x0 = d (~x ~x0) − ⌦ ↵ is not a square-integrable functionand hence does not belong to = L2(R3). H In order to incorporate it, one must generalize to include distributions. B H This is the so called rigged or equipped Hilbert space, introduced to account for the continuous spectrum, as was done implicitly in the previous section. B Furthermore, we should not be concerned that a state described by a delta function cannot be normalized, because it is not physical (but a limiting case), since one cannot measure the position ~x with infinite precision. Let us discuss this below. 125 Position representation B In practice, when we measure the position X of the state y , it collapses to a state | i that is actually a superposition of a continuum of eigenstates of X in the interval [x D/2, x + D/2] where D is a narrow (but non zero) range around x that our − detector cannot resolve, x+D/2 y dx0 x0 x0 y . | i ! ˆx D/2 − ↵⌦ ↵ These can be considered as eigenvectors of a common eigenvalue x, analogous to the case of a degenerate eigenvalue a of an operator A with a discrete spectrum, y a a y p = a y 2, | i !  | iih i | i ) a  | h i | i | ai i assuming that x y does not appreciably change within an infinitesimal D h | i x+D/2 2 2 p(x)= dx0 x0 y = D x y . ) ˆx D/2 | | | h | i | − ⌦ ↵ y(~x) is the amplitude and y(~x) 2 is the probability density to find a particle at ~x. B | | 126 Position representation Let us now introduce the operator that produces a space displacement • (translation) from the position ~x to ~x + x~0: T(x~ ) ~x = ~x + x~ . 0 | i | 0i There is a continuous set of translations, ~x R3, that can be composed 0 2 (multiplied), T(~x1)T(~x2)=T(~x1 + ~x2). They have the mathematical structure of a Lie group called T3. 127 Position representation Translations are represented by unitary transformations acting on , B H ~x T(~x ) ~x , ~x0 T(~x ) ~x0 | i ! 0 | i ! 0 † ↵ ↵ ~x T (~x0)T(~x0) ~x0 = ~x + ~x0 ~x0 + ~x0 h | 3 3 ↵ = ⌦d (~x + ~x ~x0 ↵ ~x )=d (~x ~x0)= ~x ~x0 0 − − 0 − † 1 T (~x )=T− (~x )=T( ~x ). ⌦ ↵ ) 0 0 − 0 B The elements of a Lie group can be written in terms of the generators. Consider first the generator of translations in R, T(x ) x = x + x . 0 | i | 0i We define the generator K as the operator such that T(dx) I idxK ⌘ − for an infinitesimal translation dx. 128 Position representation B K is self-adjoint, 1 † † † T− (dx)=I + idxK, T (dx)=I + idxK K = K, ) and satisfies the differential equation dT T(x + dx)=T(x)+ dx dx T(x + dx)=T(x)T(dx)=T(x) (I idxK) = T(x) idxT(x)K − − dT = iTK ) dx − with the boundary condition T(0)=I. Hence, iKx T(x) = e− 1 = I iKx + ( iKx)2 + ... − 2! − x N = lim I iK . N • − N ! ⇣ ⌘ 129 Position representation A finite translation x is the composition of N • infinitesimal translations x/N B ! generated by K. The displacement x is the parameter of this one-dimensional Lie group. Next we will investigate how K commutes with the position operator in one • dimension: XT(dx) x = X x + dx =(x + dx) x + dx , | i | i | i T(dx) X x = T(dx)x x = x x + dx , | i | i | i [X, T(dx)] x = dx x + dx = dxT(dx) x x ) | i | i | i 8 | i [X, T(dx)] = dxT(dx). ) 130 Position representation B Writing this expression in terms of the generator, to leading order in dx, [X, T(dx)] = [X, I idxK]= idx [X, K] − − dxT(dx)=dx (I idxK) = dxI, − we find that [X, K]=iI or [X,¯hK]=i¯hI.