Why is There No Solution for the Harmonic Oscillator with \(k = k_m\)?

In summary, the given equation with homogeneous boundary conditions has a solution in the form of \(y(x) = \frac{2}{\ell}\sum_{n = 1}^{\infty} \frac{\sin(k_nx)}{k^2 - k_n^2}\), where \(\phi(x)\) is also represented in a similar form. If \(k = k_m\), then there is no solution unless \(\phi(x)\) is orthogonal to the function \(u_m(x) = \sqrt{\frac{2}{\ell}}\sum_{n = 1}^{\infty}\frac{\sin(k_nx)}{k^2 - k_n^2}\), where
  • #1
Dustinsfl
2,281
5
Given \((\mathcal{L} + k^2)y = \phi(x)\) with homogeneous boundary conditions \(y(0) = y(\ell) = 0\) where
\begin{align}
y(x) & = \frac{2}{\ell}\sum_{n = 1}^{\infty} \frac{\sin(k_nx)}{k^2 - k_n^2},\\
\phi(x) & = \frac{2}{\ell}\sum_{n = 1}^{\infty}\sin(k_nx),\\
u_n(x) &= \sqrt{\frac{2}{\ell}}\sum_{n = 1}^{\infty}\frac{\sin(k_nx)}{k^2 - k_n^2},
\end{align}
\(\mathcal{L} = \frac{d^2}{dx^2}\), and \(k_n = \frac{n\pi}{\ell}\).
If \(k = k_m\), there is no solution unless \(\phi(x)\) is orthogonal to \(u_m(x)\).

Why is this?
 
Last edited:
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  • #2
I end up getting
\[
\sum_n\frac{k_m^2 - k_n^2}{k_m^2 - k_n^2}\sin(k_nx)=\sum_n\sin(k_nx)
\]
If \(k_m = k_n\), I have an indeterminant form. If \(k_m\neq k_n\), equality holds. How does orthgonality play a role? For \(k_m\) different from \(k_n\), it seems to not matter unless I am missing someting.
 

FAQ: Why is There No Solution for the Harmonic Oscillator with \(k = k_m\)?

1. What is a harmonic oscillator?

A harmonic oscillator is a type of system in which the motion or behavior of a particle or object can be described by a simple harmonic motion equation, such as an oscillating spring or a pendulum. It is characterized by a restoring force that is directly proportional to the displacement from its equilibrium position.

2. What does it mean when a harmonic oscillator has no solution?

A harmonic oscillator has no solution when there is no value of time or displacement that satisfies the equation of motion. This can occur when the system is subject to external forces that cause the restoring force to exceed the maximum displacement possible, or when there is a lack of initial conditions given to solve the equation.

3. What factors can affect the solution of a harmonic oscillator?

The solution of a harmonic oscillator can be affected by various factors such as the amplitude of oscillation, the frequency of oscillation, the mass of the object, the strength of the restoring force, and the presence of any external forces or damping effects.

4. How is a harmonic oscillator typically represented mathematically?

A harmonic oscillator is typically represented by a second-order differential equation, such as the simple harmonic motion equation, x'' + ω2x = 0, where x is the displacement from equilibrium and ω is the angular frequency. This equation can be solved using various mathematical methods.

5. What are some real-life examples of harmonic oscillators?

Some examples of harmonic oscillators in real life include a swinging pendulum, a mass attached to a spring and bouncing up and down, the vibrations of a guitar string, and the motion of a simple pendulum. These systems exhibit simple harmonic motion and can be described using the equations of a harmonic oscillator.

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