The Chemistry Maths Book, Second Edition

350 Chapter 12Second-order differential equations. Constant coefficients or, settingk 2 m 1 = 1 ω 2 , in the form of equation (1 ...

12.5 The harmonic oscillator 351 ............................................................................................... ...

352 Chapter 12Second-order differential equations. Constant coefficients equilibrium position, potential energy is converted int ...

12.6 The particle in a one-dimensional box 353 we have (12.47) with general solution, in trigonometric form, ψ(x) 1 = 1 d 1 1 co ...

354 Chapter 12Second-order differential equations. Constant coefficients The total probability that the particle be in the box i ...

12.6 The particle in a one-dimensional box 355 where the differential operator (12.55) is called the Hamiltonian operator, or si ...

356 Chapter 12Second-order differential equations. Constant coefficients It follows therefore that (12.58) The functions are sai ...

12.7 The particle in a ring 357 Equations (12.62) and (12.64) are the periodic boundary value problem discussed in Section 12.4, ...

358 Chapter 12Second-order differential equations. Constant coefficients EXAMPLE 12.11Show that the functions ψ 1 and ψ 2 satisf ...

12.8 Inhomogeneous linear equations 359 The general case If the end-points of the linear box discussed in Section 12.6 are joine ...

360 Chapter 12Second-order differential equations. Constant coefficients This is equal to 2x 2 if (2a 2 1 + 13 a 1 1 + 12 a 0 ) ...

12.8 Inhomogeneous linear equations 361 The reduced (homogeneous) equation is y′′ 1 + 13 y′ 1 + 12 y 1 = 10 and has characterist ...

362 Chapter 12Second-order differential equations. Constant coefficients By Table 12.1, case 1 , the choice of particular integr ...

12.9 Forced oscillations 363 12.9 Forced oscillations An important equation in the theory of forced oscillations in mechanical a ...

364 Chapter 12Second-order differential equations. Constant coefficients Then and This is equal toA 1 cos 1 ωtwhend 1 = 10 andc ...

12.10 Exercises 365 x p (t) t Figure 12.9 0 Exercise 39 12.10 Exercises Section 12.2 1.Show thate − 2 x ande 2 x 23 are particul ...

366 Chapter 12Second-order differential equations. Constant coefficients 21.Solve subject to the conditionθ(t 1 + 12 πτ) 1 = ...

12.10 Exercises 367 (i)Use Kirchhoff ’s voltage law (Section 11.7) to show that the currentI(t) in the circuit is given by the i ...

13 Second-order differential equations. Some special functions 13.1 Concepts We saw in Sections 12.5 to 12.7 how three very diff ...

13.2 The power-series method 369 13.2 The power-series method Many important second-order linear differential equations y′′ 1 + ...