Exact solutions of a quartic potential

摘要

We find that the analytical solutions to quantum system with a quartic potential V(x) = ax(2) + bx(4) (arbitrary a and b > 0 are real numbers) are given by the triconfluent Heun functions H-T(alpha, beta, gamma; z). The properties of the wave functions, which are strongly relevant for the potential parameters a and b, are illustrated. It is shown that the wave functions are shrunk to the origin for a given b when the potential parameter a increases, while the wave peak of wave functions is concaved to the origin when the negative potential parameter vertical bar a vertical bar increases or parameter b decreases for a given negative potential parameter a. The minimum value of the double well case (a < 0) is given by V-min = -a(2)/(4b) at x = +/-root vertical bar a vertical bar/2b.