The Chemistry Maths Book, Second Edition

390 Chapter 13Second-order differential equations. Some special functions 17.Find the Legendre polynomialP 6 (x)(i)by means of t ...

14 Partial differential equations 14.1 Concepts An equation that contains partial derivatives is a partial differential equation ...

392 Chapter 14Partial differential equations 14.2 General solutions We have seen that the general solution of an ordinary differ ...

14.3 Separation of variables 393 The partial derivatives are Therefore, as required. The function can be written as f(x, 1 t) 1 ...

394 Chapter 14Partial differential equations (14.4) We show that a solution of this equation can be written as f(x, 1 y) 1 = 1 X ...

14.4 The particle in a rectangular box 395 and the partial differential equation (14.4) in two variables has been reduced to two ...

396 Chapter 14Partial differential equations As for the one-dimensional case, the constant value ofVinside the box ensures that ...

14.4 The particle in a rectangular box 397 andC x 1 = 1 −p 2 π 2 2 a 2 . Similarly, (14.21b) describes the motion of the particl ...

398 Chapter 14Partial differential equations that is, the degenerate eigenfunctions are interchanged (whenp 1 ≠ 1 q).*The symmet ...

14.5 The particle in a circular box 399 or, multiplying throughout by− 2 mr 2 2A 2 , setting (14.33) and rearranging, (14.34) Th ...

400 Chapter 14Partial differential equations The radial equation WithC 1 = 1 n 2 , the radial equation (14.37) is (14.41) The eq ...

14.6 The hydrogen atom 401 (the radial function depends only on the modulus of n). The symmetries of these wave functions, using ...

402 Chapter 14Partial differential equations Separation of variables The first step in the solution of the partial differential ...

14.6 The hydrogen atom 403 To separate the angular variables we now write Y(θ, φ) 1 = 1 Θ(θ) 1 × 1 Φ(φ) (14.57) Substitution of ...

404 Chapter 14Partial differential equations The Θequation (14.60) The equation is transformed into the associated Legendre equa ...

14.6 The hydrogen atom 405 are called spherical harmonics. They occur whenever a physical problem in three dimensions is formula ...

406 Chapter 14Partial differential equations or L 2 Y l,m 1 = 1 l(l 1 + 1 1)A 2 Y l,m (14.72) in whichL 2 is the quantum-mechani ...

14.6 The hydrogen atom 407 Then and the radial equation becomes (14.77) This is identical to equation (13.45) for the associated ...

408 Chapter 14Partial differential equations The graphs of these are shown in Figure 14.4. The number of radial nodes (spherical ...

14.6 The hydrogen atom 409 ψ n,l,m (r,θ,φ) 1 = 1 R n,l (r)Θ l,m (θ)Φ m (φ) (14.83) and the numbers n,land mare called the princi ...