Question

4. Infinite well The quantin-mechanically allowed bound states in an infinite well have energies E =an with n = 1,2,3,... and a n constant. (a) Determine the energy difference AE between neighbouring states + 1 and n. (b) Simplify this expression for when n gets very large (do not just give the limiting valuel). (c) Bohr's correspondence principle states that for large quantum numbers , quantum theory must approach classical behaviour. Does it look like AE(n) found in (1) is in accordance with this principle? (Explain in a few words.) (c) For large n, calculate the fractional energy difference (d) Does this offer a reconciliation with the correspondence principle? (Words only, no calcula- tion.)

Answer 1

Girew, quantum mechanically allowed bound states at sin an infinite se well fiave evergies fw=an, n=1,2,3, train wil Q = Compto eslo W To determine energy difference. beth hti and in het, wo write, En = ant & Enti = a (nu) 2 it in Energy difference between {n+th and WH state is 4E = Enti - Ew = a (n+1) - an² O no loroon od -a [ m+ 2n+1 -12] ab voque opala a Qntu. oor THE = a (anti) © - for e TE"becomes Can't dare 4E = aw (2+ ) use to Wyo

© Bohrso correspondo principle principle states that for large quantum number w , quantum therory approach classical behaviour. from expression of TE (m) is is looking like a CE = a (2011) will increase as w is going. to a large value? so, consecutive energy dexed reparation increases. Energy - ei o tota continuum never approaches. But Bohry Correspondence uprinciple, s still is in accordance if you think from probability distribution for large W. N613 - Us I Fig. 27 a haw n =23 for, W-l probability of finding the partiele at midde is high but as w increases say for n=23 case

The probability of finding the particle through out the box e uniform. In classical pieture , the probability of finding the particle in the box in uniform through all prorog the box: Peloriced This classical pictare his matched in the limit of large , for example if the particle in a highly excited! cay = 23 (fig. 2) , the probability density is characterized by rapid fluctuations. Then probability of finding the quantum particle in the interval ax does not on where it taken drerage value of Pelagical and quantam merges some what. algori So, Bohrs correspondence principle e not violated. ; erable para la © calculation of TE for nora • TE = 2(2741) anr P!) C - MA page - 3

sa roho g Yes this i ut = offer a seconciliation with correspondence principle of Bohre kort to , in this way if if we can hoplot, we can say it is going to m ated with classical limit for n 700 1 01 111? gid Way That means LTE rice Ef- fi te o meine Ef = Ep tow that means stimo od 14 Ep (nt oth stades test and his with states prwid. dosa over large o f Ef 1194 it means consecutive energy levels are game hence energy continuum inn and sand hence reconciliates Bohre principle of correspondenee. Page - 4