Mathematical Methods for Physics and Engineering : A Comprehensive Guide

MULTIPLE INTEGRALS

6.6 The function

Ψ(r)=A

(

2 −

Zr

a

)

e−Zr/^2 a

gives the form of the quantum-mechanical wavefunction representing the electron
in a hydrogen-like atom of atomic numberZ, when the electron is in its first
allowed spherically symmetric excited state. Hereris the usual spherical polar
coordinate, but, because of the spherical symmetry, the coordinatesθandφdo
not appear explicitly in Ψ. Determine the value thatA(assumed real) must have
if 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 state
is described by a wavefunction Ψ, which is such that|Ψ|^2 dVis the probability of
finding the electron in the infinitesimal volumedV. In spherical polar coordinates
Ψ=Ψ(r, θ, φ)anddV=r^2 sinθdrdθdφ. Two such states are described by

Ψ 1 =

(

1

4 π

) 1 / 2 (

1

a 0

) 3 / 2

2 e−r/a^0 ,

Ψ 2 =−

(

3

8 π

) 1 / 2

sinθeiφ

(

1

2 a 0

) 3 / 2

re−r/^2 a^0

a 0

√

3

.

(a) Show that each Ψiis normalised, i.e. the integral over all space

∫

|Ψ|^2 dVis

equal 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 by

the integral

px=

∫

Ψ∗ 1 qrsinθcosφΨ 2 dV ,

whereqis the charge on the electron. Prove thatpxhas the value− 27 qa 0 / 35.

6.8 A planar figure is formed from uniform wire and consists of two equal semicircu-
lar arcs, each with its own closing diameter, joined so as to form a letter ‘B’. The
figure is freely suspended from its top left-hand corner. Show that the straight
edge of the figure makes an angleθwith the vertical given by tanθ=(2+π)−^1.
6.9 A certain torus has a circular vertical cross-section of radiusacentredona
horizontal circle of radiusc(>a).

(a) Find the volumeVand surface areaAof the torus, and show that they can

be written as

V=

π^2

4

(r^2 o−r^2 i)(ro−ri),A=π^2 (r^2 o−r^2 i),

whereroandriare, respectively, the outer and inner radii of the torus.

(b) Show that a vertical circular cylinder of radiusc, coaxial with the torus,

dividesAin the ratio

πc+2a:πc− 2 a.

6.10 A thin uniform circular disc has massMand radiusa.

(a) Prove that its moment of inertia about an axis perpendicular to its plane

and passing through its centre is^12 Ma^2.

(b) Prove that the moment of inertia of the same disc about a diameter is^14 Ma^2.