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Given A particle in a potential well: $V=-V_0 \exp(-x^2 / L^2)$

Goal Use the variational method to approximate the ground state energy

My proposal The well (For $L=1$ and $V_0=10$) has the following shape: enter image description here

My Idea is to approximate this with (some?) eigenfunctions from the infinite potential well (particle in a pipe) with width W (to be chosen) because it is also approximately constrained at boundaries.

(Another idea might be to try and derive the functions for a triangular well? and use those as an approximation.)

So I would choose

  • the N first eigenfunctions of the chosen class of eigenfunctions $\phi_n$
  • a width W = L

Such that the approximating wave function is:

$$\hat{\psi} = \sum_{n=1}^{N} c_n \phi_n$$ with

I now have to calculate the average energy

$$\langle E \rangle = \langle \psi|\hat{H}| \psi \rangle$$

and minimize this energy in the coefficients $c_n$

Update Actually, the harmonic oscillator wave function might be a better approximation.

I would use

$$\hat{\psi} = C \exp(-a x^2)$$ as a trial function.

then minimize $\langle E \rangle$ in C and a

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  • $\begingroup$ The Gaussian wavefunction in the update has only one parameter, once you choose $a$, $C$ is fixed by normalisation. $\endgroup$ – fqq Jan 19 '16 at 11:47
  • $\begingroup$ You are absolutely right $\endgroup$ – tgoossens Jan 19 '16 at 11:54
  • $\begingroup$ In the case where you considered using the infinite square well wave functions, you can improve the estimation by minimizing the energy with respect to both the coefficients and the width of the well. $\endgroup$ – Spencer Jan 19 '16 at 12:53
  • $\begingroup$ Please note that Physics.StackExchange is not a homework help site. Please read this Meta post on asking homework-like questions and this Meta post for "check my work" problems. $\endgroup$ – Kyle Kanos Jan 20 '16 at 3:05
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Yes, that is one form of the variational method. You can also take a single wavefunction with parameters (for example a Gaussian with variable width) and minimize its energy with respect to the parameters.

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