Consider a one-particle system with V = 1/4 b2/c - bx2 + cx4, where b and c are positive constants. If we use h, m, and b to find A and B in Er = E/A and xr = x/B, we will get the same results as for the harmonic oscillator, except that k is replaced by b. Thus, Eq. (4.74) gives B = m-1/4b-1/4h1/2. The equation for V in this problem shows that [bx2] = [cx4], so [c] = [b] /L2 and we write c = ab/B2 = ab/m-1/2b-1/2U, where a is a dimensionless constant.
(a) Verify that Vr = 1/(4a) - x2r + ax4r .
(b) Use a spreadsheet or graphing calculator to plot Vr versus xr for a = 0.05. (The form of Vr roughly resembles the potential energy for the inversion of the NH3 molecule.)
(c) For a = 0.05, use the Numerov method to find all eigenvalues with Er 6 10.
Use either a program similar to that in Table 4.1, a spreadsheet, or a computer-algebra system such as Mathcad. If negative eigenvalues are being sought using Excel 2010, you must uncheck the Make Unconstrained Variables Non-Negative box in the Solver Parameters box.