5.73 F2018 Lecture 5: Continuum Normalization

5.73 Lecture #5 5-1 Lecture #5: Continuum Normalization Last time: Free Wavepacket encoding of x0, ∆x, p0, ∆p * use of the Gaussian functional form, G(x; x0, ∆x), to avoid calculating integrals * use of stationary phase to encode x0 in |g(k)|eiα(k) * use g(k) because it is automatic to put in e − iE kt ! For moving and spreading free wavepacket: ∆x is time dependent ∆p is not (because free wavepacket is not subject to any force) Today: Normalization of eigenfunctions which belong to continuously (as opposed to discretely) variable eigenvalues. • convenience of ortho-normal basis sets: generalization for continua • we often talk about “density of states”, but in order to do that we need to define what we mean by “state” • computation of absolute probabilities — cannot depend on how we choose to define “state”. 1. Identities for δ-functions. 2. ψ�k, ψ�p,ψ�E for eigenfunctions that correspond to continuously variable eigenvalues. 3. finite box with countable number of discrete states taken to the limit L → ∞. Normalization independent quantity: ⎛ #states ⎞⎛ #particles ⎞ P(x, θ) = ⎜ ⎟⎜ ⎟ ⎝ δθ ⎠⎝ δx ⎠ θ is the argument of the delta-function. So if we integrate over a region of θ and x, we have the absolute probability, ∫∫ d�dx P(x,�). 4. two examples — “predissociation” rate and smoothly varying spectral density. 5.73 Lecture #5 5-2 In Quantum Mechanics, there are two very different classes of systems. * SPATIALLY CONFINED: what is ρE good for? • E is quantized • can count states, easy to compute density of states dn = ρ E dE ∞ • can normalize to 1 = ∫ ψ *E ψ Edx −∞ T: classical period of oscillation 1 * # of encounters/sec: T L/ v (v, classical velocity, is dependent on x) T * SPATIALLY UNCONFINED: • E continuously variable dn • can’t count states, so how to compute ? dE ** * fraction of time in region of length L: • can ask what is the absolute probability of finding the system between E, E + dE and x, x + dx For confined systems, we can express ortho-normalization in terms of Kronecker-δ& δ ij = ∫ ∞ * ψ i ψ j dx −∞ δij = 0 i≠j orthogonal δij = 1 i=j normalized ψ has dimension of L–1/2 δij has dimension of pure number. (Kronecker-δ) For unconfined systems, we are going to ortho-normalize states to Dirac δfunctions In order to do this we need to know better what a δ--function is and what some of its mathematical properties are. One of several equivalent definitions of a δ-function: δ(x − x′) = δ(x, x′) = 1 e− iu( x− x′) du. ∫ 2π What is it good for? ∫ δ(x,x ′)ψ(x)dx=ψ(x ′). δ ( x, x′ ) has dimension of 1/x. shifts a function evaluated at x to the same function evaluated at x′. (Dirac-δ function) 5.73 Lecture #5 5-3 Some useful δ-function identities: We do this so that we will be able to transform between δk, δp, and δE (where E = f(k)) delta-function normalization schemes. 1. δ(ax,ax′) = 1 δ(x,x′) a 1 e.g., δ ( p − p ′ ) = δ ( ! ( k − k ′ )) = δ ( k − k ′ ) ! dimension of p–1 dimension of 1/k nonlecture proof of #1 above δ(ax,ax′) = 1 −iu ( ax −ax ′) e du 2π ∫ change variables v = au dv = a du 1 1 1 − iv x−x′ e ( ) dv = δ ( x, x′ ) ∫ 2π a a but, since δ(ax,ax′) ≡ δ(ax − ax′) = δ(ax′ − ax) = δ ([−a](x − x′) ) δ ( ax,ax′ ) = (δ is an even function), δ(ax,ax′)= 1 δ(x, x′) |a| −1 2. dg(xi ) δ(x, xi ) δ ( g(x) ) = ∑ dx i ! provided that zeros of g(x) dg(xi ) ≠0 dx expand g(x) in the region near each 0 of g(x), i.e., x near x i g(x) ≅ dg dx (x − xi ). x= x i If there is only 1 zero, then identity #1 above gives the required result. It is clear that δ(g(x)) will only be nonzero when g(x) = 0. Otherwise we need to carry out the sum in identity #2. 5.73 Lecture #5 5-4 g(x) = (x – a)(x – b) has zeroes at x = a and x = b. dg d ⎡ 2 = x − x(a + b) + ab ⎤⎦ = 2x − (a + b) dx dx ⎣ dg dx dg dx = a −b x=a δ (g(x)) = ∑ i = dg(x1 ) dx = b−a x=b −1 ( ) δ x,xi (zeroes of g(x)) 1 ⎡δ (x,a) + δ (x,b)⎤ ⎦ a −b ⎣ Other examples: ( δ (x ) δ x2 − a2 = 1/2 ) 1 ⎡δ (x,a) + δ (x − a)⎤ ⎦ 2a ⎣ − a1./2 = 2a1/2δ (x − a) (a > 0) See Merzbacher, Quantum Mechanics, 3rd Edition, pages 630-632. 5.73 Lecture #5 5-5 EXAMPLES A. g(x) = (x–a)(x–b) This has zeroes at x = a, and x = b. You should show that δ(g(x)) = B. 1 [δ (x,a) + δ(x, b) ]. a−b δ ( E 1/2 , E ′1/2 ) g (E ) = E 1/2 − E ′1/2 has one zero at E = E ′, expand g(E) about E = E ′, thus for E near E ′ 1 g (E ) ± E ′ −1/2 (E − E ′ ). 2 you should show that δ ( E 1/2 , E ′1/2 ) = 2 E ′1/2 δ(E , E ′ ) 1/2 This is useful because k ∝ E1/2 or Another property of δ-functions: m ⎛ ⎞ δ(E − E ′ ) = ⎜ 2 [δ(k − k ′ ) + δ(k + k ′ )] for a free particle ⎝ 2h ( E ′ −V0 ) ⎟⎠ ⎛ 2h 2 ⎞ δ ( k E (x ) − k E ′ (x )) = ⎜ ⎝ m ⎟⎠ 1/2 (E ′ −V (x ))1/2 δ(E − E ′ ) d δ(x , x ′ ) dx δ(x , x ′ ) is an even function: d δ(x , x ′ ) ≡ δ′(x , x ′ ) to be an odd function: dx d This is useful because application of δ(x , x ′ ) to f (x ) is capable of picking dx df out evaluated at x′. dx ∴ expect Non-lecture: Use definition of derivative to prove that ∫ ∞ −∞ δ′(x, x ′ ) f (x)dx = − f ′( x ′ ) ∫ ∫ ⎡δ(x + ε, x ′ ) − δ(x, x ′ ) ⎤⎦ d δ(x, x ′ ) = lim ⎣ ε→0 dx ε δ(x + ε, x ′ ) f (x)dx = f ( x ′ − ε) δ(x, x ′ ) f (x)dx = f ( x′) ⎡δ(x + ε, x ′ ) − δ(x, x ′ ) ⎤⎦ f ( x ′ − ε) − f ( x ′ ) ∴ ∫ lim ⎣ f (x)dx = lim = − f ′( x ′) ε ε ε→0 ε→0 5.73 Lecture #5 5-6 There are several useful ways to normalize wavefunctions. Bound states. Particle is confined in space (with tunneling tails outside the box). Space normalized. 1 particle (mostly) in box. Box normalized ∫ ∞ −∞ ψ L,E i (x) (box of length L) * dx ψ L, E i ψ L, E j = δ ij Kronecker - delta OK for bound states, but not continua. dimension of ψ L,E is L-1/2 dimension of δ ij or δ E iE j is 1. Continua. We need some other form of normalization. e.g. ∫ L /2 − L /2 dx (e ikx ) e ik ′x = ∫ * L /2 − L /2 ( ) dxe − i k −k ′ x if k = k ′ we get L if k ≠ k ′ we expect to get 0 (in limit L → ∞) So we can normalize to a delta function in E, p, or k. 5.73 Lecture #5 ∞ * δE : –∞ dxψ δE,E ψ δE,E i j ∫ ( ≡ δ Ei − E j ) 5-7 δ(Ei – Ej) has the useful δ-function property: ∫ dE δ ( E − E ) ψ j δE ,E = ψ δE ,E j This implies that 𝛿(E – Ej) has the dimension of 1/E and that 𝜓𝛿E,E has dimension of L–1/2E–1/2 p E (x) = [2m( E − V(x))] 1/2 δp : ∞ ∫–∞ PE2 2 m = E −V (x ) ( ) dxψ *δp ,p E ψ δp,p E = δ p E i (x) − p E j (x) i ( j ) δ p − p ′ has dimension of 1/p ψ δp,p δk : has dimension of L-1/2 p −1/2 E 1/2 ⎡ 2m ⎤ k E ( x ) = ⎢ 2 ( E −V ( x ) ) ⎥ ⎦ ⎣! ∞ ∫–∞ ! 2 k E2 = E −V ( x ) 2m ( ) dxψ *δk ,k E ψ δk,k E = δ k E i (x) − k E j (x) i ( j ) δ k − k ′ has units of 1/k ψ δk,k has units of L-1/2k −1/2 5.73 Lecture #5 5-8 What are all of these normalization schemes good for? When you make a measurement on a continuum (unbound) system, you ask What is the probability of finding a particle between x, x + dx and 𝛳, 𝛳 + d𝛳 ? 𝛳 can be E, pE(x), or kE(x) The probability is P(x, 𝛳)dxd𝛳 (as shown for ψ Want P(x, 𝛳). Has dimensions L–1 𝛳–1 P(x,θ) = ψ δθ,θ (x)ψ δθ,θ (x) * δ,δE ,ψ δ p ,pE , and ψ δk ,kE ! There is another less abstract way to get this kind of information. “Discretize the continuum” by adding an infinite barrier at x = L and taking the limit L→∞. This way we can use box-normalized states, and actually count the states. The WKB quantization condition (will be derived in Lecture #7) gives dn = dE (2m)1/2 h x + ( E) ∫x ( E) dx( E − V(x)) − −1/ 2 ) 5.73 Lecture #5 5-9 V(x) x x0 L We have a complicated V(x) for x < x0 and constant for x > x0. In the region where V(x) is constant at V(x0) = V0. L ∫x [E − V(x)] −1/ 2 dx = [E − V0 ]−1/2 ( L − x0 ) ∝ L 0 and box normalization causes | ψ |2 ∝ so we get 1 L ⎛ dn ⎞ * P(x, E) = lim ⎜⎝ L ⎟⎠ ψ L,E (x)ψ L,E (x) L→∞ dE dimension dimension L–1E–1 dimension E–1 L–1 2 Schematic Examples * Bound → free transition probabilities * Constant spectral density across a dissociation or ionization limit. 5.73 Lecture #5 5 - 10 Bound-Free Transition (predissociation) V(x) bound (box normalized discrete energy levels) E →L repulsive (continuum of Elevels, can’t really box normalize) xstationary x phase At t = 0, system is prepared in Ψ(x,0) = ψbound(x) Fermi’s Golden Rule: 2 2π free * bound ψ ( E ) Ĥψ dx ρδE (E ) δ E L , E ! ∫ dn (E ) derive this key quantity by box normalizing = δE ⎛ ⎞ dE repulsive state and taking lim ⎜ 1 dn L ⎟ Rate = Γ bound→free = ρδE L→ ∞⎝ L dE ⎠ Then compute the Ĥ integral using two box normalized functions. Constant spectral density on both sides of a bound/free limit II(𝜔) (ω) ω çto bound levels (ω) 𝜔 çto continuum v=0 Intensity(ω) ~ ΔE smooth function of ω, no discontinuity at onset of continuum