940000: APPLIED MATHEMATICS
2006 –2007 Semester I
N. Iwamoto ©2006
HOMEWORK NO. 6
Thursday, 20 July – 12 AM, Thursday, 27 July 2006
Submit to the box in front of the Business Office ( 学務係 ) .
1.
[Electrostatic potential of a parallel pipe: 2D Laplace equation/2D Cartesian coordinates]
Consider a pipe with a cross section of a rectangle with sides a and b (0 < x < a, 0 < y < b). The z-axis is taken to be parallel to the pipe. Suppose that there are no electric charges. Since the system is uniform in the z direction, all the quantities are z-independent.
(a) Determine the electrostatic potential Ψ by solving the two-dimensional Laplace equation,
[(∂ 2 /∂x 2 ) + (∂ 2 /∂y 2 )] Ψ(x, y) = 0, under the following boundary conditions
(1) Ψ = 0 at x = 0 and x = a.
(2) Ψ = V at y = 0 for 0 < x <a, where V is a constant.
(3)
Ψ → 0 for large y.
(b) Show that many terms are necessary to express Ψ(x, y) when y << a/π, while only one term is sufficient when y >> a/π.
(c) What is the (approximate) value of the potential Ψ(x, y) at point (x, y) = (1 cm, 20 cm), when a =
10 cm and V = 100 volt?
Hint: Separate variables. Ψ(x, y) is expressed in the form of a series.
2.
[A particle inside a disk: 2D Helmholtz equation/2 D polar coordinates]
Consider a particle of mass m confined in a two-dimensional disk of radius a.
The Schrödinger equation
[-(ħ 2 /2m) Δ - E] ψ = 0 may be transformed into the Helmholtz equation
[Δ + k 2 ] ψ = 0, where k 2 = 2mE/ħ 2 . Use the two-dimensional polar coordinates (ρ, φ). By separating variables, look for a solution of the form
ψ(ρ, φ) = P(ρ) Φ(φ).
(a) Show that P(ρ) satisfies Bessel’s equation
(b) Solve for Φ(φ) under the condition that Φ(φ) is single-valued with respect to φ, i.e., Φ(φ + 2π) =
Φ(φ).
3.
[A particle in a cylinder: 3D Helmholtz equation]
Consider a particle of mass m confined in a right cylinder of radius a and length b.
The Schrödinger equation
1

[-(ħ 2 /2m) Δ - E] ψ = 0 may be transformed into the Helmholtz equation
[Δ + k 2 ] ψ = 0, where k 2 = 2mE/ħ 2 .
(a) Find the wavefunctions and energy eigenvalues under the boundary conditions that the wavefunction vanish on the surface of the cylinder, i.e.,
(1) ψ(a, φ, z) = 0
(2) ψ(ρ, φ, 0) = 0
(3)
ψ(ρ, φ, b) = 0,
Here, the cylindrical coordinates (ρ, φ, z) are used. Denote the k-th zero of the Bessel function of order n to be j n,k
.
(b) What is the lowest energy eigenvalue when a = b = 10 Å. The mass of the electron is m = 9.11x10
-28 g or mc 2 = 511 keV and the Planck constant is ħ = 1.054x10
-27 erg s.
4.
[A particle inside a sphere: 3D Helmholtz equation]
Consider a particle of mass m confined into a sphere of radius a. The Schrödinger equation
[-(ħ 2 /2m) Δ - E] ψ = 0 may be transformed into the Helmholtz equation
[Δ + k 2 ] ψ = 0, where k 2 = 2mE/ħ 2 . Use the three-dimensional polar coordinates (r, θ, φ).
(a) Find first three lowest energy levels of this particle. Use the boundary condition that the wavefunction vanish on the surface of the sphere: ψ(a, θ, φ) = 0.
(b) Find the corresponding wave functions.
(c) Sketch the nodes of the wavefunctions.
(d) What are the values (in eV) of these energy levels for an electron inside a sphere of radius
10 Ǻ? The mass of the electron is m e
= 9.11x10
-28 g or m e c 2 = 511 keV with c the speed of light and the Planck constant is ħ = 1.054x10
-27 erg s.
5.
[A circular membrane/drum head: 2D Helmholtz equation]
Consider a circular membrane or a drum head of radius R. A small-amplitude oscillation of the membrane is described by the two-dimensional wave equation,
[Δ – (1/c 2 ) ∂ 2 /∂t 2 ] u = 0, where u (x, y) is the displacement and c is the propagation speed of the oscillation. Taking the displacement of the form u(x, y, t) = u(x, y) exp (-iωt), and putting this into the above equation, one obtains
[Δ + k 2 ] u(x, y) = 0, where k = ω/c is the wavenumber.
2

(a) Find the first four low-frequency eigenmodes of the membrane. Find the wavenumber, frequency, displacement for each mode. Use the two-dimensional polar coordinates (r, θ). The outer boundary of the membrane is assumed to be fixed, i.e., the boundary condition is u(a, θ) = 0.
(b) Sketch the node of the membrane for each mode.
Hint. Some of the zeros of the Bessel functions are
J
0
(x) = 0: x = 2.40, 5.52, 8.65, …
J
1
(x) = 0: x = 3.83, 7.02, 10.17, …
J
2
(x) = 0: x = 5.14, 8.42, 11.62, …
3