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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.

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter6: Applications Of The Derivative
Section6.CR: Chapter 6 Review
Problem 26CR
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[Second Order Equations] How do you solve question 3?

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!
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!
2 λn = (17²)², X₂(x); Xn(x) = sin nπX 1 (n = 1, 2, 3, ...) (8)
Transcribed Image Text:2 λn = (17²)², X₂(x); Xn(x) = sin nπX 1 (n = 1, 2, 3, ...) (8)
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