3. A quantum-mechanical particle on the line with an infinite potential out- side the interval (0, 1) ("particle in a box") is given by Schrödinger's equation u, = iuxx on (0, 1) with Dirichlet conditions at the ends. Separate the variables and use (8) to find its representation as a series.
3. A quantum-mechanical particle on the line with an infinite potential out- side the interval (0, 1) ("particle in a box") is given by Schrödinger's equation u, = iuxx on (0, 1) with Dirichlet conditions at the ends. Separate the variables and use (8) to find its representation as a series.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
[Second Order Equations] How do you solve question 3?
Transcribed Image Text:2
λn = (17²)², X₂(x);
Xn(x) = sin
nπX
1
(n = 1, 2, 3, ...)
(8)
Transcribed Image Text:1. (a) Use the Fourier expansion to explain why the note produced by a
violin string rises sharply by one octave when the string is clamped
exactly at its midpoint.
Explain why the note rises when the string is tightened.
(b)
2. Consider a metal rod (0 < x < 1), insulated along its sides but not at its
ends, which is initially at temperature = 1. Suddenly both ends are plunged
into a bath of temperature = 0. Write the differential equation, boundary
conditions, and initial condition. Write the formula for the temperature
u(x, t) at later times. In this problem, assume the infinite series expansion
1
4
5.
6.
П
π.χ 1 3πx 1
sin + = sin + sin
1 3 1 5
5πx
1
3. A quantum-mechanical particle on the line with an infinite potential out-
side the interval (0, 1) (“particle in a box") is given by Schrödinger's
equation u, = iuxx on (0, 1) with Dirichlet conditions at the ends. Separate
the variables and use (8) to find its representation as a series.
4. Consider waves in a resistant medium that satisfy the problem
U₁₁ = c²uxx-ru₁ for 0 < x < 1
u= 0 at both ends
u(x, 0) = (x) u₁(x, 0) = (x),
where r is a constant, 0 < r < 2лc/l. Write down the series expansion
of the solution.
Do the same for 2лc/l <r < 4лc/l.
Separate the variables for the equation tu₁ = Uxx +2u with the boundary
conditions u(0, t) = u(π, t) = 0. Show that there are an infinite number
of solutions that satisfy the initial condition u(x, 0) = 0. So uniqueness
is false for this equation!
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