Question

1. The eigenfunctions of a particle in a square two-dimensional box with side lengths a = b = L are non, (x, y) = { sin ("T") sin (9,7%) = xn, (x)}n, (y) where n. (c) and on, (y) are one-dimensional particle-in-a-box wave functions in the x and y directions. a. Suppose we prepare the particle in such a way that it has a wave function V (2,y) given by 26,0) = Võru (s. 1) + Vedra . 2 what is the probability that a measurement of energy yields a value 51?r? E = 2m2 b. For the prepared state in part a, what is the probability that a measurement of the particle's position yields a value in the interval x € 0, L/4 and y E (0, L/2)?

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Answer 1

Waue fonction for 2-D box 4 na my (13) = { sm (1927) som (ny ny) - Anala? Angla) - 0 Given Ware finction Cory) - T14,(1,0) +412(213) ' sum of two-a/D Iinction me=1, my=1 dne-1, My = 2 " Energy or 2.D box when length are equal - a=b=l (given) Energy = he ? (nz?ny?) Energy of egh ② becomes . E= 4272 (U21 113?) her?/?+22) m 2ml ami? ami? Probality Energy will be = sari? ami? then the function I - 4,1 (x,y) must be zero die Rasom () (1972)

Yo [+] = [• ] : [94][j ] = Leclashineser ] [er(s) ] : [ure) * sr][ure) * 7] - [5]as so s]: 0|