Question

ty which will be discussed in chapter 4 2.3 Consider a particle of mass m subject to a one-dimensional potential V(x) that is given by V = 0, x <0; V = 0, 0<x<a; V = Vo, x> Show that bound (E < Vo) states of this system exist only if k cotka = -K where k2 = 2mE/12 and k' = 2m(Vo - E)/h4. 2.4 Show that if Vo = 974/2ma, only one bound state of the system described in problem 2.3 exists. Calculate its energy as a fraction of Vo and sketch its wavefunction, using an iteration similar to that discussed in Section 2.4. 2.5 Show that if V(x) = V(-x), solutions to the time-dependent Schrödinger equation must have definite parity that is, u(x) = fu(-x). Hint: Make the substitution y = -x and show first that u(x) = Au(-x) where A is a constant. 2.6 Consider a particle of mass m subject to the one-dimensional potential V(x) that is given by V = 0 if-a<x<a or if |x| >b · V = Vo if a < x <b where b > a. Write down the form of an even-parity solution to the Schrödinger equation in each region in the case where E < Vo. Note that the particle is not bound in this potential as there is always a probability of quantum-mechanical tunnelling, so solutions exist for all values of E. Show, however, that if *(b - a) » I (where is defined as in problem 2.3) the probability of finding the particle inside the region [x] <a is very small unless its energy is close to that of one of the bound states of a well of side 2a bounded by potential steps of height Vo. In the case where this condition is fulfilled exactly, obtain an expression for the ratio of the amplitudes of the wavefunction in the regions | Sa and x > b. 2.7 The hydrogen atom in a water molecule can vibrate in a direction along the 0-H bond, and
2.5

Answer 1

Tinue dependent sehueduje e is 2uvi) utd- E u 2 w x2 itau=Eu eunyy cpeaalon chanpint 12u(-x)-x) u (-1) - E u(-) New t - . t U 2 м дх2 2 - 2 ul-)V (x) u (-x) E ul-x) 2m degenate fucton th ane eno, ul-1) is vencu nultipu of E . u(-x) ioln E) аи u(-x)A u(x) i e ini La tuu for u(x) ulx) A u(-x) Au(x) A = £1 A u(x) u(-x) (-x) Con mly -kyaute navfwetiom eneryy diHount wartueas Non- dennaey wru ane ae mlans not tno