Question

3. [Total: 24 pts] a) (8 pts) Calculate the probability of finding a particle in the classically forbidden regime for the ground state of the 1D harmonic oscillator. Simplify the integral expression for the probability as much as possible - the integral can only be solved numerically. b) (8 pts) For the 1D harmonic oscillator, the energy eigenstates are either even or odd. This is indeed a special case of a more general statement: If V(x) is an even function (that is, V(-2) = V(x)), then the stationary state wavefunctions x() are always either even or odd. Prove this general statement. c) (8 pts) Consider the following one-dimensional potential: V(x) = + for a 50 and V(x) = 5mW222 for a > 0 (3) Find the energy levels. (Hint: This actually requires very little calculation.)

Answer 1

win) 4.1) Hum 0 ground state wave fenchon Poln = 1m w]y e-mx E = V(no) = Įmwne Yo = 40 - Bobability of finding purbele in classical forbi delen regsen Pa s 14 cm)? de+ 74.03pdae = protein S*, - mwil det senast den I doint software integral home any to solve there please do your 12 - 6 chrodinger wave enn - +2 dz 4 (n) + veny (h) = E4 (2) in die box -x 4(H) + V ulpen) = E41-21) gm der given Vlan)= ven lenen -h2 d241-1) + V/24 (-1) = E4(-4) gm dyr this from ean 6 & 2 either ey in of 4 4 ) are the soln roped for even potential ean 0 [tid 2 eur/m] P10 = E412) 4(n) in eano 1 - 22 k vm)] +-) = E41-4) - Y(n) soin Er du? im ax? M son

-ARUN'S- PAGE NO. DATE / ci som vans Here we have the burning point & en unow that at hunning point Toled Enagy sequent to potenbel energo E = x 2 = 1 / 1 fx = 0 or c= 1 / 2 this is the broblem like one wall WVB approximation. Shpan = (nr. 2) ash (43) ** = ") 3m { -v). do - me ft-camyon let y = - dys su dve H=1 y 12 & unit of yotol du = 1 n dy = H2 xl dy da = 1 1 2 2 4 Yajt dy (hoge) at some l' toy magt dy = sumex na s'y**(1-336

-ARUN'S in the SE JE S 454 (1-3) = buy resistono 12 n ;) : 42 x 83 ;) / DATE (3nt 3) ath = 12m EXK x t ff. (an+3)+ - Dom EXO :=/ F = alon+ 1) FT® E = (4943) h support = fmo then we get vina I mwana ne 0,,3- -- eun no Ten a 13h+2) kw. ) du n=0 EF 3 tw) x 2 n=1 = 7 ho es nog to una live symmetrie y z dy. le there onergy are looking Tag Harmonic of cl Motor (Ho) but odd quantum no. (n=1, 3,5.. so will also will be be ground ground state ster wawe finchion function go of this polerhul Tua (22) equal to get enclad skate wave function af symmetre Ho

win) 4.1) Hum 0 ground state wave fenchon Poln = 1m w]y e-mx E = V(no) = Įmwne Yo = 40 - Bobability of finding purbele in classical forbi delen regsen Pa s 14 cm)? de+ 74.03pdae = protein S*, - mwil det senast den I doint software integral home any to solve there please do your 12 - 6 chrodinger wave enn - +2 dz 4 (n) + veny (h) = E4 (2) in die box -x 4(H) + V ulpen) = E41-21) gm der given Vlan)= ven lenen -h2 d241-1) + V/24 (-1) = E4(-4) gm dyr this from ean 6 & 2 either ey in of 4 4 ) are the soln roped for even potential ean 0 [tid 2 eur/m] P10 = E412) 4(n) in eano 1 - 22 k vm)] +-) = E41-4) - Y(n) soin Er du? im ax? M son