Question

A particle in an infinite well of width L is in its ground state. a) If L is 30 cm, what is the ground state energy? (3 marks) Where is the particle most likely to be found? Use sketching to further explain. (4 marks)

Answer 1

linian V The potential for a pot peorticle infinite pot well of pot well of width tis for ocach fase 820,X-L. As the potential is time, independent so, the time independent schrodinger region ORX<Lis +2. dig 2 EY equation for for the reg oon, 2mE Where dart, knezo K²2 h² dy O .. mis mass of partide; The solution of ® is Y(a) z À Strik& Bloska. Here the boundary conditions are at aro (11) Y la at x=L from boundary condition. (n)=0 at to we get ♡ X (x)=0 co B = 0 Hence. Pley.. A sinkre Again from bound condition @ cony A sink L=0 or, sinkl zo get E AS A to

Normalization condition or, sinke 2: sinna . n 21; 2, 3, .. So, KLz 2. NI orok El so Hence the wave fanction, is in casa A sin ma سبه fisme) pa da JA sin sina na dna L. 2nTIR ganai 6 091 no L So corrente inTRA da L sin 2 nna or WN 207 no 20 so, Yning E sin YT MTX squaring 3 we get K? 2. N? m2 L2 nam 2mE or ta L2 or . En ne til ħ 2 2m 22 This is the expression for with energy sos state!

so, a For g round state . n=1 is ground state energy T2 h 2 : Ei 2 m L² 1301.42?x7846 28x28 ? 72 on 492 2 mL N h2 2 8m2² 2 2 (6.626x10-341 8. mx (30x102)? [AS L = 30cm = 30x102m 67 6,0 98 x 10 N im b The particle: most likely found for the maximum value of 14h (2112 | Im (2) 12 = Now, sine max MAR

YIURI fi 1/2 -- Palan -- lla 3414 4₂ in .85418 248 From the the expression ofan (In ceste for nas kilag 1 z 2 strax powe get: at Xo ! 4., 1 Pon 2 sin 1 x 1 / 2 2 than so, we get maxima at az 12. In general manima of (in (asp have to put sin nix ཀ༣ 's sin pm * for we 9712 = 1 or, sin nta 2 Ogi, non a PL 2 n 2 an where þ- 1, 2, 3, -

Az 2 n=2. so, so Hence for for n21 we get PL 1 Ž [butting p=1] This is manima for ground state For 1st encited pl PA Hence maxima area= which we & get & from of graph. similarly can cal culate manoma's for higher value of n be Az 2 2 x 2 32 4 st we no