根据文件内容，文中提到几个关键图形（如图1、图2、图3和图4），这些图像是基于特定物理模型和数学表达式的数值模拟结果。由于实际文件中并没有提供生成这些图像的具体Python代码，我们将基于文中描述的理论和公式，使用Python和一些常用的科学计算库（例如NumPy和Matplotlib）来近似复现这些图像。 ### 图像复原代码 #### 图1：线性和非线性情况下的绝热能级 ```python import numpy as np import matplotlib.pyplot as plt def calculate_energies(g, v=0.2, c=0): """计算能量级别""" # 转换参数 g_normalized = g / v c_normalized = c / v # 计算四次方程的根 def quartic_eq(x): return x**4 - 2*g_normalized*x**3 + (g_normalized**2 - c_normalized**2 - 1)*x**2 - 2*g_normalized*x + g_normalized**2 # 使用numpy.roots求解多项式方程 x = np.roots([1, -2*g_normalized, g_normalized**2 - c_normalized**2 - 1, -2*g_normalized, g_normalized**2]) # 只保留实数解 real_solutions = [sol.real for sol in x if np.isreal(sol)] return [sol * v for sol in real_solutions] # 设置g范围 g_values = np.linspace(-2, 2, 400) v = 0.2 # 线性情况的能量级别 linear_levels = [calculate_energies(g, v=v, c=0) for g in g_values] nonlinear_levels_c_less_v = [calculate_energies(g, v=v, c=0.1*v) for g in g_values] nonlinear_levels_c_equal_v = [calculate_energies(g, v=v, c=v) for g in g_values] nonlinear_levels_c_greater_v = [calculate_energies(g, v=v, c=1.5*v) for g in g_values] plt.figure(figsize=(10, 6)) for levels in linear_levels: plt.plot(g_values, levels, 'k--', label='Linear' if levels == linear_levels[0] else "") for levels in nonlinear_levels_c_less_v: plt.plot(g_values, levels, 'b-', label=r'$c < v$' if levels == nonlinear_levels_c_less_v[0] else "") for levels in nonlinear_levels_c_equal_v: plt.plot(g_values, levels, 'r-', label=r'$c = v$' if levels == nonlinear_levels_c_equal_v[0] else "") for levels in nonlinear_levels_c_greater_v: plt.plot(g_values, levels, 'g-', label=r'$c > v$' if levels == nonlinear_levels_c_greater_v[0] else "") plt.xlabel(r'$g$') plt.ylabel('Energy') plt.title('Adiabatic Energy Levels for Linear and Nonlinear Cases') plt.legend() plt.grid(True) plt.show() ``` #### 图2：不同交互强度下的隧穿概率 ```python def tunneling_probability(a, v=0.2, c=0): """计算隧穿概率""" if c <= v: return np.exp(-np.pi * v**2 / (2 * a)) else: # 这里我们简化处理，实际上需要更复杂的数值模拟 return 1 - np.exp(-np.pi * v**2 / (2 * a)) a_values = np.logspace(-5, -1, 500) # 不同c值下的隧穿概率 probabilities_c_0_1v = [tunneling_probability(a, v=v, c=0.1*v) for a in a_values] probabilities_c_0_4v = [tunneling_probability(a, v=v, c=0.4*v) for a in a_values] plt.figure(figsize=(10, 6)) plt.loglog(a_values, probabilities_c_0_1v, 'b-', label=r'$c = 0.1v$') plt.loglog(a_values, probabilities_c_0_4v, 'r-', label=r'$c = 0.4v$') # 线性情况下的对比 linear_probabilities = [np.exp(-np.pi * v**2 / (2 * a)) for a in a_values] plt.loglog(a_values, linear_probabilities, 'k--', label='Linear') plt.xlabel(r'$a$') plt.ylabel('Tunneling Probability') plt.title('Numerical Results for Tunneling Probability') plt.legend() plt.grid(True) plt.show() ``` #### 图3：亚临界交互值下非绝热跃迁概率的指数依赖性 ```python # 仅展示c小于v的情况，使用图2中定义的函数 a_values_subcritical = np.logspace(-5, -1, 500) probabilities_subcritical = [tunneling_probability(a, v=v, c=0.1*v) for a in a_values_subcritical] plt.figure(figsize=(10, 6)) plt.loglog(a_values_subcritical, probabilities_subcritical, 'b-', label=r'Subcritical $c < v$') plt.xlabel(r'$a$') plt.ylabel('Nonadiabatic Transition Probability') plt.title('Exponential Dependence of Nonadiabatic Transition Probability on Speed of Level Crossing') plt.legend() plt.grid(True) plt.show() ``` #### 图4：动态能级与绝热能级比较 ```python def dynamic_energy_levels(g, a=0.001, v=0.2, c=0.1*v): """计算动态能级""" # 这里的实现较为简化，实际应通过解薛定谔方程获得 t_values = np.linspace(-10, 10, 400) g_t = a * t_values energies = [calculate_energies(gt, v=v, c=c) for gt in g_t] return t_values, energies # 绘制不同c值下的图像 fig, axes = plt.subplots(1, 3, figsize=(15, 5)) for idx, c_value in enumerate([0.1*v, v, 1.5*v]): t_vals, dyn_energies = dynamic_energy_levels(g_values, a=0.001, v=v, c=c_value) for energies in dyn_energies: axes[idx].plot(t_vals, energies, 'ks', markersize=2, label='Dynamic Levels') # 添加对应的绝热能级作为参考 for g in g_values: adiabatic_levels = calculate_energies(g, v=v, c=c_value) axes[idx].plot([min(t_vals), max(t_vals)], [adiabatic_levels]*2, 'r-', label='Adiabatic Levels') axes[idx].set_title(f'Dynamic vs Adiabatic Levels ($c = {c_value/v:.2f}v$)') axes[idx].set_xlabel('Time') axes[idx].set_ylabel('Energy') axes[idx].legend() axes[idx].grid(True) plt.tight_layout() plt.show() ``` ### 知识点 1. **非线性Schroedinger方程**：描述玻色-爱因斯坦凝聚体在光晶格中的行为，理解其如何转化为简化模型。 2. **四次方程求解**：使用numpy.roots求解四次方程，得到系统的能量本征值。 3. **数值模拟方法**：通过调整参数并计算相应的物理量来进行数值模拟，生成理论预测的图像。