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Generalized Green function of harmonic oscillator

  • Thread starter Dustinsfl
  • Start date
  • #1
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Homework Statement


The generalized Green function is
$$
G_g(x, x') = \sum_{n\neq m}\frac{u_n(x)u_n(x')}{k_m^2 - k_n^2}.
$$
Show [itex]G_g[/itex] satisfies the equation
$$
(\mathcal{L} + k_m^2)G_g(x, x') = \delta(x - x') - u_m(x)u_m(x')
$$
where [itex]\delta(x - x') = \frac{2}{\ell}\sum_{n = 1}^{\infty}\sin(k_nx)\sin(k_nx')[/itex]
and the condition that
$$
\int_0^{\ell}u_m(x)G_g(x, x')dx = 0.
$$


Homework Equations





The Attempt at a Solution


From a previous problem, I found
$$
u_n(x) = \sqrt{\frac{2}{\ell}}\sin(k_nx).
$$
I then end up with
\begin{gather}
(\mathcal{L} + k_m^2)G_g(x, x') = \frac{2}{\ell}\sum_{n = 1}^{\infty}\sin(k_nx)\sin(k_nx') - u_m(x)u_m(x') \\
\sum_{n\neq m}^{\infty}\sin(k_nx)\sin(k_nx') = \left(\sum_{n = 1}^{\infty}\sin(k_nx)\sin(k_nx')\right) - \sin(k_mx)\sin(k_mx')
\end{gather}
What is going wrong?

Homework Statement





Homework Equations





The Attempt at a Solution


Homework Statement





Homework Equations





The Attempt at a Solution

 

Answers and Replies

  • #2
vela
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Homework Statement



The generalized Green function is
$$G_g(x, x') = \sum_{n\neq m}\frac{u_n(x)u_n(x')}{k_m^2 - k_n^2}.$$ Show [itex]G_g[/itex] satisfies the equation
$$(\mathcal{L} + k_m^2)G_g(x, x') = \delta(x - x') - u_m(x)u_m(x')$$ where [itex]\delta(x - x') = \frac{2}{\ell}\sum_{n = 1}^{\infty}\sin(k_nx)\sin(k_nx')[/itex] and the condition that
$$\int_0^{\ell}u_m(x)G_g(x, x')dx = 0.$$


Homework Equations





The Attempt at a Solution



From a previous problem, I found
$$u_n(x) = \sqrt{\frac{2}{\ell}}\sin(k_nx).$$ I then end up with
\begin{gather}
(\mathcal{L} + k_m^2)G_g(x, x') = \frac{2}{\ell}\sum_{n = 1}^{\infty}\sin(k_nx)\sin(k_nx') - u_m(x)u_m(x') \\
\sum_{n\neq m}^{\infty}\sin(k_nx)\sin(k_nx') = \left(\sum_{n = 1}^{\infty}\sin(k_nx)\sin(k_nx')\right) - \sin(k_mx)\sin(k_mx')
\end{gather}
What is going wrong?
Why do you think anything is going wrong? Please explain what you did. I can't tell if you just plugged in the expression for ##\delta(x-x')## to get the first line or if you evaluated the lefthand side and got the righthand side or did something else.
 
  • #3
vanhees71
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Gold Member
2019 Award
14,803
6,298
The Fourier method to solve differential equations is always to make an ansatz of the unknown function in terms of a Fourier series. In your case
[tex]G(x,x')=\sum_{n=1}^{\infty} A_n(x') u_n(x).[/tex]
Then you apply the operator, [itex]\mathcal{L}+k_m^2[/itex] to this ansatz, expand the inhomogeneity of the equation also in terms of a Fourier series and then compare the coefficients on both sides, which lets you solve for [itex]A_n(x')[/itex].

Note that in this case, where [itex]\lambda=-k_m^2[/itex] is an eigenvalue of [itex]\mathcal{L}[/itex], the inhomoeneity must be perpendicular to the eigenfunction [itex]u_m[/itex] for consistency. That's why you have to subtract the part parallel to this eigenmode. Note also that any solution of the inhomogeneous equation is only determined up to a function proportional to this eigenmode!
 

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