Question

Problem 1. Wave function An electron is described by a wave function: for x < 0 *(z) = { ce Ce-s/1(1 – e-3/4) for x > 0 : where I is a constant length, and C is the normalization constant. 1. Find C. 2. Where an electron is most likely to be found; that is, for what value of x is the prob: bility for finding electron largest? 3. What is the average coordinate 7 of the electron? 4. What is the standard deviation Ax = V22 – ā?of the electron position? 5. What is the average momentum of the electron? 6. What is the standard deviation Ap= vp2 – of the electron momentum?

Answer 1

The wave function is given by 0 foro & 40 -x/1 (1-é/1) fore x 70 17 the 7 - 1 - 2 ware hination is normalized. 4 (x) 4(a) da See & 22/1 (1-0-2/4) dx = 1 0 (2 te da = 1 ♡ s [e-ald ( 1 - 20 / + - 24/4) s[ + € 42/4] dx = 1 8 - 22/1 -30-32) 7 2. I eld = 1 2 + 1/3 + 4x 1 Cursing gama gamma function) 1 / de [ 2 - 27 + d e 2 x 51 1 & 2 = td 1 ㅕ - 1 ㅕ 6 54 C = 6 51

probability of Hoding the electron af position x is, P(x)= 147212 foro xco се extreme For manimin dP(a) da - 28/ (1 - 02/12 fore x 70 proobability (2 d de ce sa al (1-20 21 '+e //] It] =) de 2011 – 2e - 30/1 te =0 7 e-42 [ - 2 2 - 22/1 + 6 e 30/1 -4e-42/] =0 [1-30 2/1 7 ㅋ + 2e-) = 20 22/7 _ & 2/2 pala ke 7.) - (22-2-1) = (1-2/x-1) (2 6²1-1)=0 -202/ +1=0 ㅋ e-ald 1 on e-2/ 2

deplo) How, 20/1 - 32/1 -42/2 12 A [- 3 e + 6 T e - 4 c2 18 -32)* 16 & -42/1 2 How, foro -xla 1=1, e Hop dep daz C² [ 12 - 11 2 + 16 / 202 12 >0 SO2 e-2/0=, t conroesponds to minima. Foro e-2/ Hod 2 dep daz = 02 [ 18 der 8 + 16 der 16 hem to (² [ 1 / 2 9 412 co 412 e als compespench to maxima. e ald 7 e-2/2 = -2 = 2 - xli ha 7 - cha

So the electrom is most likely to be found x= ama at 3) The is, average coordinate 7 of the electron Sa 14001 - da W = (2 xe 22/1 (1 - 1-2/4) 7 dx (2 S xe -20/+ (1-28 2/1 + 0 24/3) dx = 02 Sale 22 - 2x - 32/1 - 42/4) dx 0 = C2 s [ae 28/1 -32% -axe t xe - 42 / 2] de 6 0² 2 T3) +() TO (E-2) 134 och = 21² + 2 16 x 1312 ((3) = 1) 144 1312 149 ( since eV 51 131 120

So) 134 120 4 How, २२ J22-111/-da c-x-243/(1 - 0-/) O -e-23/4 (1 - २९ * + e-२२/4) dx . xe - २२ २४-९-3x/- + xe-43/in D २ 3 % 3D २(1) (3) + (4) २०-३ (13 = २१ = 2 -2) २८२ ।।53 43 213 27 + 64 11२४ 11513 864 50 2342 144

so the standard deviation is, Ax= 2 22 2312 144 13112 120 13 120/ 144 0.3847 l So the standorrad deviation of electrom position is 0.3867 d Ax =