Question

h2 4. In a region of the x-axis, a particle has a wave function given by y(x) = Ae-*4722° and energy where L is some length. (a) Find the potential energy as a function of x, and sketch V (x) versus x. (b) What is the classical potential (or corresponding force function) that has this dependence? (c) Find the kinetic energy as a function of x. (d) Show that x = L is the classical turning point (i.e. the place where V(x) = E). (e) The potential energy of a simple harmonic oscillator in terms of its angular frequency is given by V 2x2. Compare this with your answer to part (a), and show that the total energy for this wave function can be written E = ħw/2.

Answer 1

wave fonction of a borticle is 4(x) = he- 22/2 22 – (4 Lis some length. Energy E = 7/2mL² – (2) (a) Schrodinger's time independent ear Vm) → potential energy om helt +V() 4x) =€ 4 (2) Now, from (2) = A 2-2,2(-22) - (me*%222) er en -A/c {e-242" - 2.224e-242"} det - & - + 122² { , _ 22,27 (4 Selbstituting eq" (9) in éqnls), we obtain, + m. lap şi-azzy+ vaD4 (2) Y (). O, VODE everyone C1 - 2424) - + the 2 2 mi2 2 222 ^/22 2014 72 2014 x2 for the required answer. - Vaa) FIA: VC) vs x (this is a focorabola) (b) Corresponding force function, f(a) f(n) = – dv(x) = - t? fo(a)= MLч

(6) kinetic energy, ist Ek4= Phant - h d24 im dar of Ek Y = -7 (4/12 & 22 { 1 - 2 422 } ] [ from of Ex 4 = + 2 + 1 - 22 23 FR 22. 31-74922} the required answer. (d) In classically, at tuming point, V(x)=E. Using eqn (2) and egn(5), cwe have, ov, Z=12 or, 1 x zL To is the classical turning point, proved) © The potential energy of a simple hamonic oscillatora in terms of its angular frequency is given by V(n) = 4 mwber from fartas, VOX) = 12 *? - Lex? comparing this with (n) = 1 mober, we get mw12 түЧ Miu The total energy for this wave function is E E = Ek + V (2 E = 1 to [from eano L proved)