Question

Consider a plane-wave solution to the free Schrödinger equation (V = 0) in one space dimension, with momentum pi. At time t = 0, the wavefunction takes the form Up (0,0) = P12/h. The lower index pı labels the momentum of this state. 1) What does the wavefunction look like at a later time t? 2) Next, consider another plane-wave state with a different momentum p2, with the wavefunction at t = 0 taking the form Up (2,0) = 1222/h. Suppose experimentally, we are able to prepare a state with wavefunction at t = 0 as the following linear combination (3,0) = (Upi (2,0) + Up (2,0)) and let it evolve according to the free Schrödinger equation. In this case, what is the probability density of finding the electron at position = 0 and time t, i.e., p(0,t) = V*(0,t) 4 (0,t)? 3) During the period from t=0 to t= T, what is the time-averaged value of the probability density of finding the electron at position r=0? (Hint: the time averaging of a continuous function F(t) is usually defined as lo ).) 4) How does the result of 3) look in the classical limit, h→0?

Answer 1

. Page nose Gwen; Here we consider a plane ware solution to the free schrodinger equation in one space dimension, with momentum poi At time, it so the wave function is in the form of Up, (2,0) = e iPix it. ) The objective is to determine how the wave function looks like at a later time, t 1. At time ,t=0; Up, (8,0)=ePix /h. Similarly, ut time, t= t;fpp.cx, t) se i (Pix +6,6] /t 2) Now we consider another plane wave state with momentum P2, and its wave function is, at tao. Up(x,0) = e 'Paxla Experimentally, the wave function at to is; ♡ (x,0) = (YP,(2,0) + 4p, (2,0)). The objective is to determine the probability density of finding the election at position to and time t We know, the probability density Plo, t) = 4* (0,4) Y CO,6) [.Co, t) + P (0, 1)] = [44,6,) +46,47] 5 (14,6,4))*+ 148,40,4) |*+2 Re[year. Cont) 44,6,4)) *{1+1 +2 Re [eiciteitze/27 { [2+ 2 cos

Page no = 2 * ** la cos 2 LETERS] Plo,=2 con 2 cerita de 3) Now , The objective is to determine the time averaged value of probability density of finding the election which is at position 2=0; during the period from t=0 to t:T : We know; The time - average value of probability density Sdt P(t) 7. + SPCC) at Now substitute PCE) value in above equation; + Process Centent de + is cose cerere de =+f( 14 cos t . + [6+ sin lepot () Time average value of = + (1+ree Sin (,) ] I probability density. ' 4) In classical limit , hao looks like as, when we apply the classical timit, then we have find that the time average of probability density is equal to one.