# Double- well potential and Mexican potential

Is double well potential related to Maxican hat potential? I have found on Quantum Field Theory in a Nutshell by A. Zee

He wrote the double well potential as : $V (φ) = (λ/4)(φ^ 2 − v^2)^2$. Can anyone give me an argument for writing the equation like that.

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You get the Mexican hat by spinning the double well potential about its axis of symmetry. You need a two component scalar field or equivalently a complex field. The potential you wrote is the only symmetric quartic potential with more than one miminum. –  Michael Brown Apr 19 '13 at 8:15

Cross section of the Mexican hat potential could give one-dimensional double-quantum well potential profile.

In A.Zee (ed. 2) I found the Mexican hat potential at the page 224, Eq. 3: $$V(\vec{\varphi})=-\frac{1}{2} \mu^2 \vec{\varphi}^2+\frac{\lambda}{4}(\vec{\varphi}^2)^2$$

In that formula $\vec{\varphi}$ is a positional vector in N-dimensional space. It is clear that this formula is just a sum of two polynomial functions with different orientation of branches relative each other. In 1D case it gives two quantum wells separated by a parabolic barrier.

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Can you some mathematics on it please? –  Unlimited Dreamer Apr 20 '13 at 14:44
OK, but I did not find the formula given by you in the post. –  freude Apr 20 '13 at 17:00