Question

Quantum Mechanics

Please help me to solve this exercise step by step.

I will appreciate it a lot. Write clear

2. An important problem of quantum mechanics is that of the particle subject to a linear res- titutive force (harmonic oscillator). The stationary Schrödinger equation for this problem, in one dimension, has the form h² #20 Ika-6 = EⓇ 2m 8x2 + 2k2O = Eº where k is the oscillator constant. Solutions of the following types are proposed: a) 6 = Ajeaz? + Aze-az?, with a real and positive, b) Q = (B1 + B2x)e-bz”, with b real and positive. Find the values that the constants A1,A2,B1,B2, a and b such that these functions to be physically acceptable solutions and standardized to the unit. Find the value of E in each case in terms of the system parameters.

Answer 1

Solution . Hasimonic Oscillator - tiaden + *x++ = Ep. - am ax² 1 2 () a = A, e ax² + Age-ax? substitute & in above eq? 24 = A, (200) e axe + A, (-29x) € *** are = 2011 ( 2.2axect te ox"> - 20 A2 (2. - 2ax é a earz => 1 x 22 (, (2 az +1) €022-A2 (-20x2+i)=1924) +_ *zº(a,learance Aeran) = f(Aeantech Now, comparing A.. => che (29z++1) + <kx? = € =) - 2 tiase? - tin - E-4kx? 24?tit + tags +(-4 xxy =0.0 - comparing Az. s tě. (1-2234 + 4 km? = € 5 - 2a telesa i things + 4 kw)-E zo. => 20x -' tia +(-1 kx) = o → for physically acceptable ware fu, &. 1A=0 , as $0 when a s Now, Ś(A) ²e-29% = 1 ( Mormalisation coudition m which is not possible 21A2? le Put 2ax²=t - Yarduodt du= dt Ja to ya - Scanned with CamScanner ya JE

a 14,1?. Va = 1 - insteyen Now, 19,1°H1421°=1 0+ 129 1 from eq? ☺ 2.4.22 - The +6=– *x2 =0 G ckx - H2 - x' + 7 h 2m 2m . © 4- (6,+B2x) é be? for physically acceptable function, ¢ has to be zero, as a 50, B2 = 0 = $ = B, e-b2. Using Monnalisation conclition, S 18,12€2bx dx = 1. -baz Also, 18,14+1821=1 -ba? So, th, B, (-26)/-26x² e-bx² + bx 2 + 1 Rx² B, et buth ( 2 )* + Kxee B = EBS " E. B, e-bar. => -2 b?ti”.(2014. c? + 7 m V Scannedy with CamScanner TIETO