Spreadsheets and computer-algebra systems can easily be used to solve equations of the form f(x) = 0. For example, suppose we want to solve ex = 2 - x2. A computer-made graph shows that the function ex - 2 + x2 equals zero at only two points, one positive and one negative. In Excel, enter an initial guess of 0 for x in cell A1 and = exp(A1)-2+A1^2 in cell A3. Then click the Data tab and in the Analysis group, choose Solver. In the Solver Parameters box, enter A3 next to Set Objective. After To click on Value of and enter 0. Enter A1 after By Changing Cells; then click Options and change Precision to 1E-14. Click on OK and then click on Solve. Excel gives us the result 0.537274449173857, with 2 × 10-15 in cell A3. To find the negative root, start with -1 in cell A1 and use the Solver.
(a) Use a spreadsheet or Mathcad to solve Prob. 2.24 for the bound-state energies of a particle in a well.
(b) For the double-well potential of Prob. 4.37, application of the procedures used in Section 2.4 shows that the allowed bound-state energy levels satisfy

where V0r ‰¡ V0/ (h2/ml2), Er ‰¡ E / (h2/ml2), and p = - 1 for the even wave functions and p = 1 for the odd wave functions. The hyperbolic tangent function, defined by tanh z ‰¡ (ez - e-z)/(ez + e-z), can be produced by typing TANH in Excel. Use (4.90) and a spreadsheet or Mathcad to find the lowest four double-well energies for each of the V0r values of Prob. 4.37.

Transcribed Image Text:

(Vor E)/E12 tan[ (E,/8)2](tV E,)/8]2 )P (4.90)