Continuum Normalization. good for cooking or inspecting wiggly functions and for crudely evaluating integrals of wiggly integrands.

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1 5.73 Lecture #5 5-1 Last time: Gaussian Wavepackets How to encode x in gk ( )e ikx dk = ψ(x) Continuum Normalization or k in g( x) e ikx = ψ(k) stationary phase: good for cooking or inspecting wiggly functions and for crudely evaluating integrals of wiggly integrands. v group v phase Today: Normalization of eigenfunctions which belong to continuously (as opposed to discretely) variable eigenvalues. convenience of ortho-normal basis sets we often talk about density of states, but in order to do that we need to define state computation of absolute probabilities cannot depend on how we choose to define state. 1. Identities for δ-functions. 2. ψ δk, ψ δp,ψ δ for eigenfunctions corresponding to continuously variable eigenvalues. 3. finite box with countable discrete states taken to the limit L. Normalization independent quantity: #states # particles δθ δx δθ is the argument of the delta-function. So if we integrate over a region of θ and x, we have the absolute probability. 4. two examples predissociation rate and smoothly varying spectral density.

2 5.73 Lecture #5 5-2 In Quantum Mechanics, there are two very different classes of systems. * SPATIALLY CONFIND: quantized what is ρ can count states, easy to compute density of states dn good for? =ρ T: classical period of oscillation * # of encounters /sec: 1 T * fraction of time in region of length L: can normalize to L/ v T 1 = ψ * ψ * SPATIALLY UNCONFIND: continuously variable dn can t count states, so how to compute? ** can ask what is the absolute probability of finding the system between, + and x, x + For confined systems, we can express ortho-normalization in terms of Kronecker-δ δ ij = ψ * i ψ j δ ij = 0 i j orthogonal δ ij = 1 i = j normalized For unconfined systems, we are going to ortho-normalize states to Dirac δ-functions 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 definition of δ -function: 1 δ( x x ) = δ(x, x ) = 2π e iu(x x ) du. What is it good for? δ(x, x )ψ(x) = ψ(x ). shifts a function evaluated at x to the same function evaluated at x.

3 5.73 Lecture #5 5-3 Prove some useful Identities We do this so that we will be able to transform between δk, δp, and δ (where = f(k)) normalization schemes. 1. δ(ax,ax ) = 1 δ(x,x ) e.g., δ(p p ) = δ(h(k k )) = 1 δ(k k ) a h nonlecture proof δ(ax,ax ) = 2 1 π e iu(ax ax ) du v = au dv = a du change variables 1 1 δ(ax,ax ) =, 2 π a e iv(x x ) dv = 1 δ(x x ) a but, since δ(ax,ax ) =δ(ax ax ) = δ(ax ax) = δ([ a](x x )) 1 (δ is an even function), δ(ax,ax ) = δ( xx ), a dg(x 2. δ ( g(x) ) = i ) 1 δ(x,x i ) i { dg(x i ) 0 zeros of g(x) expand g(x) in the region near each 0 of g(x), provided that i.e., x near x i g(x) dg (x x i ). 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.

4 5.73 Lecture #5 5-4 XAMPLS A. g(x) = (x a)(x b) This has zeroes at x = a, and x = b. 1 You should show that δ( gx ( )) = [δ(x, a) + δ(x, b)]. a b / / B. δ( 12, 12 ) / / g ( ) = has one zero at =, expand g() about =, thus for near / g ( )± ƒ 1 12 ( ). 2 / / / you should show that δ( 12, 12 ) = 2 12 δ(, ) 12 m / This is useful because k 1/2 δ( ) = k k 2 2h ( V 0 ) δ( ) Another property of δ -functions: δ( xx ), is an even function: d x x ) δ(, d expect δ( xx ), δ (x, x ) to be an odd function: d This is useful because δ( xx ), is capable of picking df out evaluated at x. Non-lecture: Use definition of derivative to prove that δ (x, x )f(x) = f(x) d [δ(x + ε,x ) δ(x,x )] δ(x,x ) = lim ε 0 ε δ(x + ε,x )f(x) = f(x ε) δ(x,x )f(x) = f(x ) lim [δ(x + ε, x ) δ(x, x )] fx = lim fx ε) ( f ( ( ) x ) = f ( x ) ε 0 ε ε 0 ε

5 5.73 Lecture #5 5-5 δk δ * Our goal is to create ortho-normalized ψ s that look like e ikx : normalized to a δ-function in k * std. defn. of δ(k,k) = ψ δk,k ψ δk,k 1 e ix(k k ) 2π δ function in k ψ δk,k (2π) 1/2 e ikx for V(x) = constant. ψ δk,k is said to be normalized to δ(k,k ). What is the probability of finding the system, which is described by ψ δk,k, to be located between 0 x L? L * 1 L L 0 ψ δk,k ψ δk,k = 2π 0 = 2π = P δk (L) probability grows without limit as L But, more interestingly, what is the probability of finding a system in a δk-normalized state within a region of length equal to one de Broglie λ? 2π λ 1 λ = h/ p = P k δk (λ) = = 2π k δk normalized states (for V(x) = constant) have: 1/k particle per λ of x 1 or 2 π particle per unit length

6 5.73 Lecture #5 What about ordinary space normalization? ψ k = N k e ikx ψ k = N k e i kx ψ * k ψ k = N k 2 e i( GNRALIZ δ(k, k) 1 2π e iu(k k) du = k k)x 0 if k k * ψ δk,k ψ δk, k 5-6 if k = k THIS IS TH PROBLM! Can t specify N k. where ψ δk,k (2π) 1/2 e ikx 1 ik ( k ) x, thus δ ( kk, ) = e 2 π notation δ(k, k) =δ(k k) =δ(k k, 0) when δ(k,k ) is multiplied onto f(k) and integrated over all k, we get f(k ) δ(k, k)f(k)dk = f( k) δ(k,k ) is zero when k k and is one when k = k ψ ψ δ kk, with eigenvalue k 0 0 δ kk, = k 0 0ψ δ kk, 0 δ xx, 0 is eigenfunction of k ˆ = 1 i x is eigenfunction of x ˆ = x with eigenvalue x 0 : k ˆ ψ

7 5.73 Lecture #5 5-7 Other normalization schemes for free particle δp δ * ψ δp,p = N p e ipx/h what is value of N p? * δ(p,p ) = N p N p exp [ix(p / h p / h)] * = N p N p 2πδ(p/ h,p / h) δ(p,p ) using h identity #1 ψ δp,p = (2πh) 1/2 e ipx/h 0 L ψδp,p ψ δp,p = 2πh 1 * L 1 h particle per λ = p p h particle per unit * length ( ( h 2 ± δ δ * ψ δ = N e ikx ± e ikx ) k = 2m V 0 ) degenerate pair of states 12 / you show that m 1/4 2m 1/2 + ψ δ, = 2π 2 h 2 cos h 2 x m 1/4 2m 1/2 ψ δ, = 2π 2 h 2 sin h 2 x + ψ δ is orthogonal to ψ, δ, = another 1 2 λ +* + ψ δ, ψ δ, from ψ δ, 0 = 1 probability for δ - normalized state per λ * Volume of N-dimensional phase space occupied by a δp normalized state is h N

8 5.73 Lecture # Thus there are particles per λ for a δ - normalized state. 2 m 1/ 2 ) or ( particles per unit length h 2 So we have assembled all the basic stuff we will need, at least for V(x) = constant problems. Now use it to examine a problem we understand perfectly. -L/2 L/2 / / 2 12 sin 2m n 12 ψ n = L h 2 x cos n even n odd 12 2m n / k n = = n h 2 h 2 2 8mL 2 n 1 particle per box of length L 1 L particle per unit length 0 as L λ particle per λ L 4 normalization schemes (δk, δp, δ, box): each gives different #/L or #/λ. Why - because each scheme defines state differently. However, expect that # particles # states δx δθ must be independent of normalization scheme k, p, or box Why? Because the probability of finding a system between x, x + AND θ, θ + dθ is observable. We have completely specified what counts as an observation.

9 5.73 Lecture #5 5-9 Normalization-Independent Quantity for general V(x): lim dn 1 L/2 * ψ ψ δθ = # states # particles L dθ L L/2 δθ δθ δx 123 density of states (# states per unit θ The infallible way to get the invariant reference density is to box normalize (so that one can count states) and then take limit L. Why? Because most realistic potentials become smooth and flat at large enough x. V(x) x_() - inner turning point x = L Procedure: 1. Box normalize ψ n ( is quantized) 2. Compute dn from (n) 3. take limit L dn but 1 dn remains finite L example: 0 L ψ = 2 1/2 sin 2m n 1/2 x L h 2 L * ψ ψ 0 n n = 1 by construction (for box normalization)

10 5.73 Lecture # h 2 2nh 2 n = 8m n L 2 1/ 2 n = n 2 8mL 2 dn = 8mL 2 h 2 dn 2L m ) 1/ 2 ρ () = = h ( 2 RFRNC DNSITY lim dn 1 L ψψ = 2 m / = 2m / * Px,x + δx;, + δ) = L 2 L ( 0 3 h 2 12 h { δx δ # # indep. of L L THIS SCTION TO B RPLACD dn δ for ψ δ, dn δp dp for ψ δp, dn δk dk for ψ δk δ ± 2m h 2 1/2 above derived reference density = lim L dn δ 1 L L ψ * δ ψ δ /2 m 2h 2 derived earlier dn δ = 2 ( ) = 2 2 because each is doubly degenerate δp 1 L L 0 * ψ δp ψδp = 1 h dn 1/ 2 δp 1 dp h = 2m h 2 dn δp dp = 1/ 2 2m = 2m p = p

11 5.73 Lecture # δk 1 L ψ * δk L 0 dn δk dk ψ δk = 1 2π = 2m h 2 k = k dn δθ dθ δ 2/ δp p / k δk / see the pattern? These results are only valid for problem. They illustrate all continuum normalization problems where it is desired to calculate probabilities. 2 Schematic xamples * Bound free transition probabilities * Constant spectral density across a dissociation or ionization limit.

12 5.73 Lecture # Bound-Free Transition (predissociation) V(x) L bound (box normalized discrete energy levels) repulsive (continuum of - levels, can t really box normalize) x stationary phase x at t = 0 system is prepared in Ψ(x,0) = ψ bound (x) Fermi s Golden Rule: Rate = Γ bound free = 2π free* () ˆ bound 2 ψ δ Hψ ρδ () h ρ δ = n δ () repulsive state and taking lim 1 dn L L derive this key quantity by box normalizing ˆ Then compute the Hintegral using two box normalized functions. Constant spectral density on both sides of a bound/free limit I(ω) ω (ω) v = 0 Intensity(ω) ~ smooth function of ω, no discontinuity at onset of continuum