Question

3. (a) A particle of mass m is stuck in a 2D box of length I i. What are the wavefunctions? ii. What are the energies of the ground state and first excited state?

Quatum Mechanics Question

Answer 1

Salution cCrsider the Schvödinger quators pardcle confined /stuck in a box ox Let's for a 2 D and La te Schrädinger eaon So We can for tis case - x2 2 know that V(x,yj outside the uation For His we withuin the box aud Vy = box So E 2 Since left side of tuis equado (Hauciltouian) independent product uave functo separion ofiables appreach express, 2D the productof tua Sum of two terts ow ry cain we So the i'ke in In tu's approach fanchoun in dependent can loe as luave 1» сonponent. e

Sinuilarly for euadibn ) (33 2 Ex X(x) m 4) partic te Hhe total energy of e uehere The diferenhal ez (3) aud (4) can be so lued salution Hue form f and have X y Co (ky in Ky) YCy) app ly bouh day se Alow we are h egh aud (6) and o) -o Condifons ニ0 aud By ニ0 E O oubrue eah 12) can we Now, the wave functo for box of the for, a Cou struct and particle in 2- 2 boundasyy condi tions ane e maing and

CL avd , = So din N and be satisfed Huese both egh can 2 Ameg L) and cau be satisfred ouly if egh 8) and () mEr L aud gdve which 2 2 2- and Eny

caupeleyindepentad be sahsfedy any tuo diff orext Cor nil values of nx aol ny total enogy Because coudious ane and can The elowed values of tHe ane know given by 2- Ehe y state ground for 2- E1,1 excited state, (1,2) /(2,1) for st Ec, 2)/02,1) values of Eny anol En eah Cair be The functibry the ueave substituded i'n be come N Sh (hи) B/ oTMalizahoh constaut uhere be deternuned ustng normalizatiy whwch can conditon

2- 2 N 2 хp (xx), ч8 L N2 w2. 2 fuches - maltzed So canuplete ueawr nor