Mathematical Methods for Physics and Engineering - Matematica.NET

MULTIPLE INTEGRALS6.6 The function(Ψ(r) =A 2 − Zr )e −Zr/2aagives the form of the quantum-mechanical wavefunction representing the electronin a hydrogen-like atom of atomic number Z, when the electron is in its firstallowed spherically symmetric excited state. Here r is the usual spherical polarcoordinate, but, because of the spherical symmetry, the coordinates θ and φ donot appear explicitly in Ψ. Determine the value that A (assumed real) must haveif the wavefunction is to be correctly normalised, i.e. if the volume integral of|Ψ| 2 over all space is to be equal to unity.6.7 In quantum mechanics the electron in a hydrogen atom in some particular stateis described by a wavefunction Ψ, which is such that |Ψ| 2 dV is the probability offinding the electron in the infinitesimal volume dV . In spherical polar coordinatesΨ=Ψ(r, θ, φ) anddV = r 2 sin θdrdθdφ. Two such states are described by( ) 1/2 ( ) 3/2 1 1Ψ 1 =2e −r/a 0,4π a 0( ) 1/2 ( ) 3/2 31Ψ 2 = − sin θe iφ re −r/2a 0√ .8π2a 0 a 0 3(a) Show that each Ψ i is normalised, i.e. the integral over all space ∫ |Ψ| 2 dV isequal to unity – physically, this means that the electron must be somewhere.(b) The (so-called) dipole matrix element between the states 1 and 2 is given bythe integral∫p x = Ψ ∗ 1qr sin θ cos φ Ψ 2 dV ,where q is the charge on the electron. Prove that p x has the value −2 7 qa 0 /3 5 .6.8 A planar figure is formed from uniform wire and consists of two equal semicirculararcs, each with its own closing diameter, joined so as to form a letter ‘B’. Thefigure is freely suspended from its top left-hand corner. Show that the straightedge of the figure makes an angle θ with the vertical given by tan θ =(2+π) −1 .6.9 A certain torus has a circular vertical cross-section of radius a centredonahorizontal circle of radius c (> a).(a) Find the volume V and surface area A of the torus, and show that they canbe written asV = π24 (r2 o − ri 2 )(r o − r i ), A = π 2 (ro 2 − ri 2 ),where r o and r i are, respectively, the outer and inner radii of the torus.(b) Show that a vertical circular cylinder of radius c, coaxial with the torus,divides A in the ratioπc +2a : πc − 2a.6.10 A thin uniform circular disc has mass M and radius a.(a) Prove that its moment of inertia about an axis perpendicular to its planeand passing through its centre is 1 2 Ma2 .(b) Prove that the moment of inertia of the same disc about a diameter is 1 4 Ma2 .208