What Parameter Should Be Used in Variational Approximation for This Hamiltonian?

[LCSphysicist]

Homework Statement

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Relevant Equations

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Hello. I should find the energy aproximatelly using the variational approximation for this physical hamiltonian: $bx^4 + p²/2m$ Imediatally, i thought that the better trial wave function would be the one correspondent to the ground state of the harmonic quantum oscilator. THe problem is, in fact, that i don't know what parameter to use in order to vary it! I mean, $$\psi = (mw/\pi \hbar)^{1/4}e^{-mwx^2/2 \hbar}$$. So, what parameter should i use? w=w(\alpha)? or m = m(\alpha)? How do i know by the beginning what parameter should i use to vary it?

 

[anuttarasammyak]

The semiclassical treatment to find approximate energy eigenvalue as described in https://www.ks.uiuc.edu/Services/Class/PHYS480/LectNotes/well/p480_n.pdf
is familiar way to me. I am not sure taking a harmonic wave function do work well. Anyway why don't you take product $m\omega/\hbar$ as a parameter for variation?