Question

answer part 1 and 2 together. please dont answer 2 if you dont answer question 1 first they go together.

1. Consider an electron in a 1-D box of length L = 1 nm that has energy eigenstates given by ºn(x) = 2sin(n7x) when x has units of nanometers. Now consider that an atomically-precise hammer is used to deform the bottom of the box such that the electron feels an effective potential given by (2) V(x) = V(x) = -2(x – 0.5)² +0.5. a. Compute the energy exxpectation value of an electron described by 41(x) after the box is deformed. b. Explain if you think the original energy eigenstates, yn(x), are stationary states in the deformed box. 2. Considering the modified box in Question 1, write down the Hamiltonian operator after the box has been modified. Using this Hamiltonian, use the linear variational method and a bsis of the first 3 energy eigenstates of the ordinary particle in a box to estimate the ground state energy of the modified system.

Answer 1

[Q.1 (as) Given ID box of length h= lnm and Pole) = Vesin (252) Now when deform the bottom of the box the perturbation is VID= –2 (x- 0.5 +0.5 Now the correction of the energy of Ist of Perturbation is 44 4 4 42 then ist Order ! Ր Ա Ր Ր Ո Ր Ր Ր Հ 4 * vm) 4 dx . 4, (n) = √2 Sin(x) dy - iz Sin (Aa) (-2(X-0.53 +0.5] V2 sin (an) de --2682.59-0.5) GK (+8) de - ... ait) de + leaked 2 35 (loss=o) ao +5 **) as 21-0.5) > [(xorij tete dasən 2 de de-C - S6 13 Iy Now Calculate I., I₂ I3 & Ty we get I cos de when a=0 Pusat Put n-o.s=t & o-0.5=t da= at when x=1 -0.5-t 0.5

0.5 (2) tat = 0 [.. This function is odd / ANN Now 12. - 119-0.5) Costa de Put 1-0.5=t de=dt 0.5 + 0.5 CO24 (+408) # jas or isnt 41) d apie savo 1.- for sucer 0.5 t sin( 2att) 3th Sincent til at -0.5 an Add function +4 (0.6) Sin(*+1) – (-0.6)*cm ( -*-+1)] - 0.125 x0.0421 +0.125x4000312) be 6. Mo 0.0006] - Dobele = 0.000100 Now Iz= [du= 1 we can do 19. (woo ) - S 27 Now put the value of I. I. Is Jy in ea 0.

8.16 when all observables is are are independent of time then independent of an this called the stationary state. After deformation of box, the eigen state remain will be the stationary state, because when we find the correction of steltec in pertar bation theory then these states are stationasy as well as this perturbed potential is also the time independent. Therefore the correction of state & are time independent then the original state are stationary after the deformation.