在計算量子系統穩定態之能階和波函數時，常使用不同的近似法來求得近似解，像是變分法(variational method)、微擾理論(perturbation theory)，都是常見的獲得近似解方式。以變分法計算近似解時，常引入一連串彼此獨立的變分參數(variational parameter)來增加近似解之準確性，然而隨著變分參數增加，計算量會變得十分龐大。為了使計算效率提升，本篇論文設計特定形式之試探函數(trial function)，此試探函數只有一個變分參數，大幅降低計算之複雜度，藉此提高計算近似解之效率。另外，本篇論文引入新的物理量(ΔE)^2，此物理量可以用來表示近似解之準確度，藉由調整變分參數來最小化(ΔE)^2，除了可以得到基態之近似解，亦可以得到激發態之近似能量和波函數。因此本篇論文將以「最小化(ΔE)^2」為基礎原則，發展「多項式法」(polynomial method)和「基底函數法」(basis set method)兩種方式來獲得量子系統穩定態之近似能量和波函數。接著以不同位能系統作為計算實例，從簡單的一維多項式位能系統，逐漸拓展至二維之複雜物理系統，計算系統之近似能量和波函數，並和精確解進行比較，以驗證新方法之可行性。最後驗證此方法可用於各種複雜的系統，且都有不錯之準確度，是能夠獲得精確近似解之優良計算方式。

There are many approximation methods for solutions to the time-independent Schrödinger equation, such as the variational method and perturbation theory. For the variational calculations, we usually introduce a set of independent variational parameters to find the optimal approximation. As the number of variational parameters increases, the variational calculations inevitably entail higher-order multidimensional searches for the optimal values of variational parameters. This can be notoriously difficult numerically. Thus, to increase the efficiency of calculation, we introduce a special form of the trial function. The trial function contains only one variational parameter, which can lower the complexation of calculation. Furthermore, we introduce a new physical quantity,〖(ΔE)〗^2, which can present the accuracy of the approximation. By adjusting variational parameters and minimizing (ΔE)^2, we can not only find the approximate solution to the ground state wave function but also find the approximate solution to the excited state wave functions and energies. Therefore, by following the principle of “minimizing (ΔE)^2”, we develop the polynomial method and the basis set method to find the approximate energies and wave functions of quantum systems. We choose several quantum systems as calculation examples, from a simple one-dimensional polynomial potential system to the complicated vibrational states of methyl iodide, and calculate the approximate energies and wave functions. By comparing the computational results with the exact solutions, we find our results are in excellent agreement with the exact solutions with great accuracy. Therefore, our new calculation method is suited for many complex systems and is an excellent way to find the approximation solutions to the bound states of quantum systems.

(1) Stubbins, C., Das, D. Phys. Rev. A 1993, 47, 4506.
(2) Fessatidis, V., Mancini, J. D., Prie, J. D., Zhou, Y., Majewski, A. Phys. Rev. A
1999, 60, 1713.
(3) Kato, T., J. Phy. Soc. Japan 1949, 4, 334.
(4) Simon, B. Bull. Math. Sci. 2018, 8, 121.
(5) Light, J. C., Hamilton, I. P., Lill, J. V. J. Chem. Phys. 1985, 82, 1400.
(6) Biswas, S. N., Datta, K., Saxena, R. P., Srivastava, P. K., Varma, V. S. Phys. Rev. D 1971, 12, 3617.
(7) Killingbeck, J. J. Phys. A 1986, 19, 2903.
(8) Chou, C. C., Biamonte, M. T., Bodmann, B.G., Kouri, D. J. J. Phys. A 2012, 45,
505302.
(9) Trahan, C. J., Wyatt, R. E., Chem. Phys. Lett. 2004, 385, 280.
(10) Lee, T. Y., Chou, C. C. J. Phys. Chem. A 2018, 122, 1451.
(11) Chou, C.C. Ann. Phys. 2015, 362, 57.