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I've come across a problem while studying the WKB method. I want to calculate the eigenvalues of a symmetric quartic double well potential. It could be any potential. I chose it to be $$V(x) = x^4 - 4x^2 +4$$ The hamiltonian with $\hbar$ = $m$ = $1$ gives $$H = -\frac{1}{2}\frac{d^2}{dx^2} + V(x)$$ and I plan to find the eigenvalues of the bound states given by the potential $V(x)$ represented bellow

$\hskip1.7in$ enter image description here

with returning points $x_2 > x_1$ and roots $x=\pm \sqrt{2}$

The quatization* problem for the double potential well with respect to even (odd) solutions is, for $x>0$ $$\theta \simeq (n + \frac{1}{2}) \pi \mp \frac{1}{2} e^{- \phi} \tag{1}$$

with $$\theta = \int_{x_1}^{x_2} p(x') dx'$$ $$\phi = \int_{0}^{x_1} |p(x')| dx'$$ $$p(x) = \sqrt{2m(E_n - V(x))} = \sqrt{2m(E_n - (x^4 - 4x^2 + 4))}$$

( *Introduction to Quantum Mechanics by David J. Griffiths, problem $8.15$ )

My problem lies exactly in solving eq. ($1$), since it involves integrals of the square-root of a quartic function $$\int_a^b \sqrt{2m(E_n - (x^4 - 4x^2 + 4))} dx$$

I used Mathematica but it couldn't compute a solution.

Is there any approximation or trick I could use to solve it analytically? If not, any software that could do the computation?

PS: After solving numerically the $Schr\ddot{o}dinger$ equation for the ground state eigenvalue I obtained $E_0 \simeq 1.8$ with $\hbar$ = $m$ = $1$ as stated above. With the WKB method I'm hoping to obtain a similar result.

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    $\begingroup$ Why can’t you use numerical integration? $\endgroup$
    – G. Smith
    Commented Apr 11, 2020 at 3:34
  • $\begingroup$ Because E_n is an unknown. The main goal of eq. (1) is to get an expressions for E_n. At least for the ground eigenvalue E_0, I would consider a battle won. $\endgroup$
    – EinRock
    Commented Apr 11, 2020 at 4:21
  • $\begingroup$ You integrate for various $E_n$ until you find one that works. There are systematic techniques for narrowing in on the right choice, similar to when you try to find the root of a polynomial numerically, $\endgroup$
    – G. Smith
    Commented Apr 11, 2020 at 4:28
  • $\begingroup$ I get $E_0=1.74646$. $\endgroup$
    – G. Smith
    Commented Apr 11, 2020 at 4:44
  • $\begingroup$ Look it up in Gradshtein&Ryzhik, though it is quite possible that such integrals are not solvable. $\endgroup$
    – Roger V.
    Commented Apr 11, 2020 at 5:43

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You can do this in Mathematica using NIntegrate to do numerical integration and either NSolve or FindRoot to find the energy that satisfies equation (1).

In this way I found the $n=0$ energies $E_0^\text{even}\approx 1.74646$ and $E_0^\text{odd}\approx 2.07823$ with a few lines of code. For the former, $\theta\approx 1.43953$ and $\phi\approx 1.33739$; for the latter, $\theta\approx 1.73284$ and $\phi\approx 1.12677$. Since this is only an approximation, it seemed pointless to go beyond standard precision.

The next energies with $n=1$ appear to be "above the hump", where I don't think your equations apply, because they give nonsense.

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  • $\begingroup$ Integrals of square roots of quartics are Elliptic functions. There is probably an analytic solution to your integral if you look up a book about such functions. $\endgroup$
    – mike stone
    Commented Apr 11, 2020 at 12:45
  • $\begingroup$ I'm having trouble using NSolve / FindRoot. Could you tell what am I doing wrong? The Code: f[Ei_?NumericQ] := NIntegrate[ Sqrt[2 (Ei - (x^4 - 4 x^2 + 4))], {x, Sqrt[2 - Sqrt[Ei]], Sqrt[2 + Sqrt[Ei]]}]; g[Ei_?NumericQ] := NIntegrate[ Sqrt[2 ((x^4 - 4 x^2 + 4) - Ei)], {x, 0., Sqrt[2 - Sqrt[Ei]]}]; FindRoot[f[x] - Pi/2 + Exp[-g[x]]/2 == 0, {x, 1.5}]; $\endgroup$
    – EinRock
    Commented Apr 11, 2020 at 18:19
  • $\begingroup$ I cut and pasted your code into Mathematica, except for the final semicolon. The output for me is $\{x\to 1.74646\}$. If you really have a semicolon at the end you're not going to see this. $\endgroup$
    – G. Smith
    Commented Apr 11, 2020 at 21:33
  • $\begingroup$ I just did it myself again, in a new notebook, and yes, worked out perfectly. Must have been some previous functions or variables that needed to be Clear[]. A rookie mistake as I am. Thanks G. Smith $\endgroup$
    – EinRock
    Commented Apr 11, 2020 at 22:50

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