Quntum Physics I (8.04) Spring 206 Assignment 8 MIT Physics Deprtment Due Fridy, April 22, 206 April 3, 206 2:00 noon Problem Set 8 Reding: Griffiths, pges 73-76, 8-82 (on scttering sttes). Ohnin, Chpter : Scttering nd Resonnces. Sttes of the hrmonic oscilltor [5 points] Consider the stte ψ α defined by ψ α N exp(αâ )ϕ 0, with α C complex number. For the first two questions below it my be helpful to simply expnd the bove exponentil. () Find the constnt N needed for the stte ψ α to be normlized. (b) Show tht the stte ψ α is n eigenstte of the nnihiltion opertor â. Wht is the eigenvlue? (c) Find the expecttion vlue of the Hmiltonin in the stte ψ α. (d) Find the uncertinty in the energy in the stte ψ α. (e) Use the eigenvlue eqution, viewed s differentil eqution to clculte the explicit form of the normlized wvefunction ψ α. 2. Two delt functions- gin [5 points] Consider gin the problem of prticle of mss m moving in one-dimensionl double well potentil V(x) = gδ(x ) gδ(x+), g > 0. You found in the previous set the vlue of the bound stte energy E for the even stte in terms of the energy E 0 = 2 /(2m 2 ). You hd ξ = κ E E 0 = ξ 2 where ξ +e 2ξ = λ, λ mg, 2 with λ unit free, encoding the intensity g of the delt functions, if is constnt, or the seprtion of the delt functions, if g is constnt. We cn thus write λ = 0 0 2, mg
Physics 8.04, Quntum Physics, Spring 206 2 with 0 nturl length scle in the problem once g is fixed. Introduce lso the energy E ssocited with single delt function: mg 2 E. 2 2 Assume now tht this is model for ditomic molecule with intertomic distnce 2. The bound stte electron helps overcome the repulsive energy between the ions. Let the repulsive potentil energy V r (x), with x the distnce between the toms, be given by βg V r (x) =, β > 0, x where β is smll number. The totl potentil energy V tot of the configurtion is the sum of the negtive energy E of the bound stte nd the positive repulsive energy: V tot = E +V r (2). () Write E s E = E f(ξ,λ) where f is function you should determine. Plot E s function of / 0 = λ in order to understnd how the ground stte energy vries s function of the seprtion between the molecules. Wht re the vlues of E for 0 nd for? (b) Write V r in terms of E,β, nd λ. (c) Now consider the totl potentil energy V tot nd plot it s function of / 0 = λ for vrious vlues of β. You should find criticl stble point for the potentil for sufficiently smll β. For β = 0.3 wht is the pproximte vlue of / 0 t the criticl point of the potentil? 3. Finite squre well turning into the infinite squre well [5 points] Consider the stndrd squre well potentil { V 0, for x, V 0 > 0, V(x) = 0 for x >, nd the wvefunction for n even stte coskx, for x, ψ(x) = A e κ x, for x >, () (2) where we included the prefctor to hve consistent units for ψ. We wnt to hve better understnding of the limit s V 0 nd understnd why the discontinuity in ψ in the infinite well does not give trouble. Keeping m nd constnt s we let V 0 grow lrge is the sme s letting z 0 grow lrge. A previous nlysis hs demonstrted tht for the ground stte, in the sitution of lrge z 0, the nstz (2) is ccurtely normlized nd η = k π 2 ( z 0 ), ξ = κ z 0, A π 2z 0 e z 0.
Physics 8.04, Quntum Physics, Spring 206 3 We wnt to see if the expecttion vlue of the Hmiltonin receives singulr contribution from the forbidden region. Since the potentil V(x) vnishes there, we only need to concern ourselves with the contribution from the kinetic energy opertor K = p2 2m. Clculte the contribution to the expecttion of K from the forbidden region x > K x> dxψ (x)kψ(x) The nswer should be in terms of z 0. Interpret your result. 4. Reflection of wvepcket off step potentil [20 points] Consider step potentil with step height V 0 : { V 0, for x > 0 V(x) = 0, for x < 0. () We send in from x = wvepcket ll of whose momentum components hve energies less thn the energy V 0 of the step. For this we need modes with k stisfying We will then write the incident wvepcket s 2 2mV0 k k, k =. (2) 2 Ψ inc (x) = k dkφ(k)e ikx e ie(k)t/, x < 0. (3) 0 Here is the constnt with units of length, uniquely determined by the constnts m,v 0, in this problem, nd Φ(k) is rel, unit-free function peked t k 0 < k 2, Φ(k) = e β 2 (k k 0 ) 2. (4) mv 0 The rel constnt β, to be fixed below, controls the width of the momentum distribution. The units of Ψ inc re L /2 nd tht s why we included the prefctor in (3). Recll tht dk hs units of L. () Write the reflected wvefunction (vlid for x < 0) s n integrl similr to (3). This integrl involves the phse shift δ(e) clculted in clss. Introduce unit free version K of the wvenumber k, unit-free version u of the coordinte x, nd unit-free version τ of the time t s follows K k, x u, t τ. (5) V 0 Nturlly, we will write k 0 = K 0 /. Note tht kx = Ku.
Physics 8.04, Quntum Physics, Spring 206 4 (b) Show tht the group velocity nd the uncertinty reltion for the incoming pcket tke the form du = #K0, u K τ #, d where# represent numericl constnts tht you should fix (different constnts!). Use the pproximtion tht we hve the full gussin Φ(K) 2 to determine the uncertinty K in the incoming pcket in terms of β. Assuming gin tht we hvefullgussin, whtwouldbe(intermsofβ) theminimumpossiblevlueof the uncertinty u for the ssocited coordinte spce probbility distribution? (c) Complete the following equtions by fixing the constnts represented by # E(k) = #V 0 K 2, e 2iδ(E) = #+#K 2 +ik #+#K 2 w(k). (d) Show tht the dely t = 2 δ (E) experienced by the reflected wve implies τ given by # τ = K0, #+#K 2 0 where you must fix the constnts. (e) Prove tht thecomplete wvefunction Ψ(x,t) vlid for x < 0 ndll times, which we now view s Ψ(u,τ) vlid for u < 0 nd ll τ, tkes the form 2 Ψ(u,τ) = nd determine the two missing constnts. 0 ( ) dke β2 (K K 0 ) 2 e i#k2 τ e iku e iku w(k) # (f) Set β = 4 nd K 0 =. Wht re the vlues of K nd u? Wht is the predicted time dely τ? (Not grded: Cn you mke n informed guess if the pcket will chnge shpe quickly?) Now use Mthemtic to clculte nd mke plots of the probbility density 2Ψ(u,τ) 2. Give the plot of the wvefunction for τ = 20, 5, nd 0, nd using u [ 30,0]. Exmine the plot for τ = 20 nd determine the time dely τ by looking t the position of the pek of the pcket. Your nswer should come resonbly close to the nlyticl vlue you determined previously. 5. Scttering off rectngulr brrier. Bsed on Griffiths 2.33. p.83. [0 points] Do only the cses E < V 0 nd E = V 0. Cn you get T = for E < V 0? Find the nswer for E > V 0 in some book (or do it). When does one get T = for E > V 0?
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