Question

Hello, I need help with a problem for my Quantum Mechanics class. Please explain as if I am learning for the first time. I want to be able to understand and do problems like this on my own. Thank you in advance for your help! The infinite square well has solutions that are very familiar to us from previous physics classes. However, in this class we learn that a quantum state of the system can be in a superposition state - a linear combination of a set of eigenstates. While a quantum system that is purely an eigenstate does not change over time, a quantum state that is in a superposition state will change its value as a function of position. a) Create a normalized square wave pulse between 0 and -a/2 in a square well that is bounded by -a and a. b) Follow the procedure to create the time dependent version of this wavefunction. Given a Hamiltonian H and an initial state ), what is the probability that the eigenvalue a; of the observable A is measured at time t? 1. Diagonalize H (find the eigenvalues E, and eigenvectors En)). 2. Write ((0)) in terms of the energy eigenstates En). 3. Multiply each eigenstate coefficient by e-iEnt/ħ to get y(t)). 4. Calculate the probability Pa; = [(a}\¥(t))/?.

Answer 1

tabo - a copy af (a, f=0) = 1 akaso other wise for normal izatia ja tie du = 5 105 da = 198 199 = 1 4 (0,t=0) = = 5/0 a £x40 =o otherwise. formal solun for peorticle in a bex Ethe - dx = 2ME TO ret 20 e 2ME – k2 = 4 = A elka Beit no due = - Bry

ก => 4 Caza)=0 A eika, Belka _o ...) yon=-a) = 0 Э лека - дека – о. - 2) D A = - Berika peettin în egur as - B ezika, Beikao » BEO e qika - 1 12nt n integer > ka= nt 123.0 » k = na # 8.4 (az = - Bela Belka +B ( einn elke zika) =-Beira (eikinta) ei k(n+a)) = - Belka ni sin k(a+a) =-gi Belka sin konta) c sink (nta) ก ก rika Scanned with CamScanner

it any wave fn follow the form (5), then one can get time dependet wave for as follows s For any eigen state in its time 2 depadace pare e Ht it HE H(L) » Ince>= ē itt(e) M(8) Whe I thin) = (ain) =ē l'Ente ind 4 (ast) = I cm ē Ent sin (max) previestsly we te D me checos (24) /Styro show that Lokant For sin mill a n=2 m En= n²7 42 smar - 2² m² 72 72 8 mar = marr amar cos(2m+ Rx) = ame) &n= (2met 22 smar For