Question

2.2 Two-level system A particle in the box is described by the following wavefunction 1 1 V(x, t) + V2 V2 = Um(x)e -i(Em/h) In other words, this state is a superposition of two modes: n-th, and m-th. A superposition that involves only two modes (not necessarily particle in the box modes, but any two modes) is called a "two-level system”. A more modern name for such a superposition is a "qubit”. a) Come up with an expression for the probability of finding a particle at position x and time t. It will have four terms. b) Check that this probability distribution is normalized, i.e. that particle will be found somewhere. The following integrals would come handy: Só sin? (x) dx = 1 and sin (9.4 x) sin ( m2 x) dx = 0. c) Your probability distribution is a periodic function of time. What is the frequency of this periodic variation? What is the period T? d) Let n = 1 and m = 2. Plot the probability distribution at the following times: t = 0, t=T/4, t =T/2, t = 3T/4. You can let L = 1. e) Where is the particle most likely to be found at T/4 (i.e. at what point is the proba- bility density largest at this instant of time)? Where is the particle most likely to be found at 3T/4?

Answer 1

P(«,+) *(n,t = 2thca) e 'Ent/5 +th maje Entlik 4*6, 2) 4 (x, t) do 120 m (2773 * Ź [ 14 h (n) eti Entln ti Emily + 4m (a) ttm cx) -i Embla [ tha) ei Enth { [ 14, 6)2 + 14m (O))? + 4h) ) e ti (Em Enita + t (x) In (?) (En-Emitta Ane b P(x, t) da L 2 2 moix sin nix dx sin ² l da + 2 2 2 0 - (Em-Ently mite da + Ź e 0 5sinn x sinmaz sin min sin ni da - il En-Emt/h +! 2 e $ Ž + =1 Proved 2 (normalised)

© "En Emt/h PEX, A) = [ tmbottlanbey then ca) Yunla) e & Tea (a) th (2) ei (5 Emtln In Let, En-Em w ħ :- P(0,7) = { sin? 16x + sin? 297 + sin contra cei mai te iry] 2772 [ a + b Coswt] a sinh tsin 2me b = 6: zsin ta sin a and b not being function of time explicitely, frequency of wo|En-Eml P(x, t) has time period, T = 2 IT w 2t th En-Eml En-Eml

a 272 sin the sin int - i wt Lihat ei P(x, 4) =" 1/4 [sin T2 + sin 27+ + + 2simme sin anime coswt] t=0, P13,0) = { [sin moeten sind 242 + 2 sim me te sin 272 = [sinte atsine 10. En-Em ] = 1/2 [sinthet sin22]? t= "T/A 5 Т-2 Ple, 7/4) txan... t=712 P(x,7/2) { [sin e + sin 27 = Ź [sin te sine ² Ź [sin that sinar t = 3T14 Pla, 3T4) = Ź [sin that P(2,0) . 1.) MA t = 0 1 PETAL/P(2, 31/) t = 1/4, 37/4: mm LMM. ↑ P (2,5/2) T/2 -1