Question

solu 4. The principle of superposition (in quantum mechanics) states that if two or more Solutions are each solution to the Schrodinger cquation, then their linear combinations are also solution to the Schrodinger equation. Each of y(x) = A sin ax, V(x) = B cos ax, V(X) Celax, and y(x) =D elox is a solution to the particle in the box problem. Show that linear combinations v(x) = A sin ax + B cos ax and y(x) = Celax + Celax are solutions. Determine the eigenvalue in each case? (Hint: Hv=Ev).

Answer 1

Here Hy EY Fon Simplicity we Conrid, H-d Poteution eauns V D, Like H- Partiele in bn Now, Y b= ASin ox, Y(a=B Cas ax Jy) -A cosaxd d Ya) BEsindn).d A Sindx.d d B-Casda.d d -aAsind) deasd 2m 2m The eigen Punctim The eigen funetion 2m 2m Linear Combination) Y ASindxt B esd = A Co3ax.a +B(-sind). d d4C A.-sindn).ai B eax d a(Asinda Beo dn) -l The eigen furetion Sume aftr linear embinattn 2m Linear combnotion alsa he ealution f Sehodingr eanation

Simigarly -dx Ya) De (-id) De DE ian ia-i) (Dea c.e (a) ) -e) c) Hy (a) =to The eigen Punetiom = ar ti 1 The eigen Punetion -d Tm ..The linaan emb iuation ceiax Deax e eion(ia)Dea- i) da C (eDeian) 2m Lm : Aptr einear embinorin te eigenm ralue ane Sane So The Pour wane Funetin aud are the Solutian of Sehrodiugen eauatin. (Aroved) Hne eigen ralue =a'Rhm Hue luean esmbian And