11.17 Show that the wave fun nite square well satisfy th that m(x)*f,(x) dx = FIGURE 11.21 SECTION 11.5 (Transitions; Perturbation Theory (Problem 11.10) 11.11 (a) A classical point charge q of mass m is in a cir- cular orbit of radius r around a fixed charge Q (with q and Q of opposite sign, of course). Starting from Eq. (11.1), derive a formula for the radiated power P in terms of q, m, r, and Q. (b) By what factor is P changed if we double q (leaving m, r, and Q un- changed)? (c) What if we double Q (with m, r, and q unchanged)? 11.12 atom was that they failed to predict the correct fre- quency for the radiation emitted. According to classi- cal electromagnetic theory, the frequency of emitted radiation should equal the frequency of the orbiting electron. (a) Calculate the orbital frequencies, forb (1) and forb (2), of a classical electron in the n= n3 2 Bohr orbits of a hydrogen atom. (b) Now find the frequency f,(2→1) of the actual photon emitted in the 2 1 transition. Show that f,(2 1) is not equal to either forb 2) or forb 1) (or their ayerage or 11.18 (a) The first excited st 2.11 eV above the ground diation can cause transit els? What sort of radiatic (b) Answer the same que in hydrogen, which are 4.5 x 10 eV apart. (TH ting discussed in Sectic questions for the lowest which are 0.48 MeV apa One of the difficulties with classical models of the 11.19 The atoms of a certain gy levels: E = 0, E2 Es = 12.4, all measure frared light with wavele 3200 nm shines through cause? If the gas was se ground state, would yo %3D 1 and portions. The transition zone between the near and far fields necessarily con- at lial field. When portions of the field lines are offset from near langed position moves outward portiransverse component, as shown in Fig. 11.1(b) and (c).* While the ra- dial component of the electric field falls like 1/r it can be shown that the transverse component falls like 1/r. Consequently, at large distances it is the transverse component that dominates and carries radiated energy away from (b) the charge. The total power P radiated by any single charge q (moving nonrelativis- tically) can be shown to be ms ns 2kq a P = (c) (11.1) 3c FIGURE 11.1 ol- (a) Electric field lines from a static where a is the charge's acceleration. This formula accurately describes the charge are radial. (b) When the ply so, power radiated by any macroscopic system of moving charges. For example, in charge is given an abrupt kick to the right, changes in its electric field propagate outward at speed c distant portions of the field still point outward from the original TV or radio broadcasting, electric charges are made to oscillate inside the rods of an antenna, and the resulting radiated power is given by (11.1). (See Prob- n- lem 11.2.) Notice that the power (11.1) depends on the acceleration a. Thus a charge moving at constant velocity does not radiate. We should also mention that with an assembly of many accelerating charges, the fields produced by the uferent charges can sometimes interfere destructively, with no net radiated power. For example, consider a uniform ring of charge rotating at a constant (open circle) position. (c) The transverse disturbance linking near and far fields continues to move radially outward as the charge ats #. a- coasts forward. by er. steady current loop and does not radiate any power.

11.17 Show that the wave fun nite square well satisfy th that m(x)f,(x) dx = FIGURE 11.21 SECTION 11.5 (Transitions; Perturbation Theory (Problem 11.10) 11.11 (a) A classical point charge q of mass m is in a cir- cular orbit of radius r around a fixed charge Q (with q and Q of opposite sign, of course). Starting from Eq. (11.1), derive a formula for the radiated power P in terms of q, m, r, and Q. (b) By what factor is P changed if we double q (leaving m, r, and Q un- changed)? (c) What if we double Q (with m, r, and q unchanged)? 11.12 atom was that they failed to predict the correct fre- quency for the radiation emitted. According to classi- cal electromagnetic theory, the frequency of emitted radiation should equal the frequency of the orbiting electron. (a) Calculate the orbital frequencies, forb (1) and forb (2), of a classical electron in the n= n3 2 Bohr orbits of a hydrogen atom. (b) Now find the frequency f,(2→1) of the actual photon emitted in the 2 1 transition. Show that f,(2 1) is not equal to either forb 2) or forb 1) (or their ayerage or 11.18 (a) The first excited st 2.11 eV above the ground diation can cause transit els? What sort of radiatic (b) Answer the same que in hydrogen, which are 4.5 x 10 eV apart. (TH ting discussed in Sectic questions for the lowest which are 0.48 MeV apa One of the difficulties with classical models of the 11.19 The atoms of a certain gy levels: E = 0, E2 Es = 12.4, all measure frared light with wavele 3200 nm shines through cause? If the gas was se ground state, would yo %3D 1 and portions. The transition zone between the near and far fields necessarily con- at lial field. When portions of the field lines are offset from near langed position moves outward portiransverse component, as shown in Fig. 11.1(b) and (c). While the ra- dial component of the electric field falls like 1/r it can be shown that the transverse component falls like 1/r. Consequently, at large distances it is the transverse component that dominates and carries radiated energy away from (b) the charge. The total power P radiated by any single charge q (moving nonrelativis- tically) can be shown to be ms ns 2kq a P = (c) (11.1) 3c FIGURE 11.1 ol- (a) Electric field lines from a static where a is the charge's acceleration. This formula accurately describes the charge are radial. (b) When the ply so, power radiated by any macroscopic system of moving charges. For example, in charge is given an abrupt kick to the right, changes in its electric field propagate outward at speed c distant portions of the field still point outward from the original TV or radio broadcasting, electric charges are made to oscillate inside the rods of an antenna, and the resulting radiated power is given by (11.1). (See Prob- n- lem 11.2.) Notice that the power (11.1) depends on the acceleration a. Thus a charge moving at constant velocity does not radiate. We should also mention that with an assembly of many accelerating charges, the fields produced by the uferent charges can sometimes interfere destructively, with no net radiated power. For example, consider a uniform ring of charge rotating at a constant (open circle) position. (c) The transverse disturbance linking near and far fields continues to move radially outward as the charge ats #. a- coasts forward. by er. steady current loop and does not radiate any power.