Question

1) A particle with mass m moves under the influence of a potential field
V (x
. The particle wave function is stated by:

for $-\infty< x< \infty$ where and are constants.
(a) Show that is not time dependent.
(b) Determine as the normalization constant.
(c) Calculate the energy and momentum of the particle.
(d) Show that

Answer 1

Pese Az ont fait le ten] I let us calculate the ik 2 - не it rey] Y at it 24 ət sit AX exp[-* om te ] Esłax ] * NE nzuflete ] + it st 3 ih 24 =-ħk Y c) = HY - Now, since time independent schrodinger egnation. ьthе от м. - HY=64 Above equation can be compared to this with E= - the Hence, potential is time independent since wave fuction satisfies time independent Schrodinger equation. 2m

probability of finding To determine A we need to use particle bielwaren (-00,00) to be 1. | 146, 1² dx=1 (14248 248-] *1 imaginary part will cancel out since eilas (ei)* = eit eig=1 À 2ef[-22 19 Jan 1 tabe 22kat pada per 242 ! cpct), 4-1 4 (9) fte* dt = at 1 + 2 A Taal 282 24

② Since in part we calculated H4 =E4 hence energy of particle is is momentum will be plona - it 2400 plans = . (4.2 ex°[-2 Vente 3-ni kheap die box exp[-2*] @ yana sih cep [416][ 046 *VE] -2** c*++*+&]] fotos=-1reffen ) 2. caff-71 (1-2) fro --it ko (4.2.10 pas - ik [+- 24] <E). Į y t r de @ since 44 =E4 Keys from part @ we have 20 of 21

- 8 | 4* Huda = 1 4 H. (H4) dx = 1 4 4. E4 de - E l 4* H4 de TE FT* Ex du -'co It E [l 'eruda a lo[rude]: SE ZEP>- E) = 0