- 1 By Letting V 2 1 1 2 Show That The Solution To The Schrodinger Equation 6 2 1 At V C 3 6 1 2m Ar Is 1 (64.71 KiB) Viewed 19 times
1. By letting V(2,1) = (1) (2) show that the solution to the Schrodinger equation [6] (2,1) at +V(c)(3,6), (1) 2m ar? is given by V(1,1) = *()e-it/and () satisfies the Time-independent Schrodinger Equation 12 20(1) +V(x)(2) = Ev(). 2m 2.2 = (2) 2. Consider a particle in a one-dimensional infinitely deep potential well (the potential is zero for 0<x<L, but infinite at x = 0 and x = L). Obtain an expression for the energy eigenvalues of this particle by means of a full derivation. [8] 3. Using the fact that equation (2) can be written as V(x) = Ev () calculate (H), (H) and hence AH. Assume the wavefunction is normalized. [5] 4. For free particles (i.e., for 0 <r<L), the time-independent Schrödinger equation be- comes (a) Why is the energy of this particle not quantized? (b) What values may the energy assume? (c) What is signified by the expression x Eut = constant? (d) The general solution for a free wave-particle is '(x) = Acikt + Be-ikt. What is represented by the + and - signs of the argument of e? (e) The wave function of a free particle is not normalizable. What does this imply? (1) Define a wave packet mathematically. (8) The classical particle's speed is double that of the wave speed v = w/k. How do we solve this paradox? =