5 . Consider a particle in a box of length [ = I in a state defined by the wavefunction , $ ( 20 ) = Aac 2 ( 1 - 20 ) ( 1 ) ( 2 ) Apply the

(a) Apply the normalization condition to determine A in φ(x). (b) Compute the expectation value of the energy directly. Express it in the form E = (const)h 2/8m. What is the value of const? (c) Sketch φ(x) inside the box. Based on visual inspection, what two particle-in-a-box eigenfunctions probably contribute most to φ(x)? (d) Expand φ(x) in terms of the particle in a box eigenfunctions. Write the first four terms in the wavefunction with numerical coefficients. If you work this problem by hand, the following integrals will be useful: Z u 2 sin u du = −u 2 cos u + 2 cos u + 2u sin u (2) Z u 3 sin u du = −u 3 cos u + 3u 2 sin u − 6 sin u + 6u cos u (3) Alternatively, you are welcome to use computer software like Mathematica (available free to students at http://mysoftware.ucr.edu), online integral calculators, or your calculus-ready graphing calculator to evaluate the necessary integrals numerically. For example, the following website is fairly easy to use: https://www.zweigmedia.com/RealWorld/integral/integral.html (e) Evaluate the average energy using your approximate wavefunction from part (d), and compare it to the result you got from part (b).

5 . Consider a particle in a box of length [ = I in a state defined by the wavefunction ,$ ( 20 ) = Aac 2 ( 1 - 20 )( 1 )( 2 ) Apply the normalization condition to determine A in $ ( 20 ) .( b) Compute the expectation value of the energy directly . Express it in the form { -( const ) h 2 /8 m . What is the value of const ?!( c ) Sketch $ ( 20 ) inside the box . Based on visual inspection , what two particle-in - a - boxeigenfunctions probably contribute most to $ ( 20 ) ?'( d ) Expand & ( 20 ) in terms of the particle in a box eigenfunctions . Write the first four termsin the wavefunction with numerical coefficients . If you work this problem by hand , thefollowing integrals will be useful :`\ u 2 sinu du = - 4- cosu + 2 cos u + Zu sinu*( 2 )| 43 sinudu = - "' cos u + 3 u* sinu - 6 sinu + bucosu( 3 )Alternatively , you are welcome to use computer software like Mathematica ( available*free to students at http : / / my software . Ucr . edu ) , online integral calculators , or yourcalculus- ready graphing calculator to evaluate the necessary integrals numerically . Forexample , the following website is fairly easy to use :"https : / / www . zweigmedia . com / Real World / integral / integral . htm ]( @ ) Evaluate the average energy using your approximate wavefunction from part ( d ) , andcompare it to the result you got from part ( b ) .