【深耕 Python】Quantum Computing 量子计算机（6）计算＜m|V|n＞数值积分

写在前面

往期量子计算机博客：

【深耕 Python】Quantum Computing 量子计算机（1）图像绘制基础

【深耕 Python】Quantum Computing 量子计算机（2）绘制电子运动平面波

【深耕 Python】Quantum Computing 量子计算机（3）重要数学公式一览

【深耕 Python】Quantum Computing 量子计算机（4）量子物理概念（一）

【深耕 Python】Quantum Computing 量子计算机（5）量子物理概念（二）

一、积分定义式

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二、势能公式

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三、相关常数设置

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四、计算数值积分的Python代码

import math
from scipy import integrateh = 6.6260896 * 10 ** -34
hbar = h / (2.0 * math.pi)me = 9.10938215 * 10 ** -31eV = 1.60217733 * 10 ** -19L = 1.0 * 10 ** -9x_min = -L / 2.0
x_max = L / 2.0n_max = 10Ex = 1.0 * 10 ** 10def verphi(n, x):kn = math.pi * (n + 1) / Lreturn math.sqrt(2.0 / L) * math.sin(kn * (x + L / 2.0))def V(x, Ex):return math.e * Ex * xdef integral_matrixElement(x, n1, n2, Ex):return verphi(n1, x) * V(x, Ex) * verphi(n2, x) / eVfor n1 in range(n_max + 1):for n2 in range(n_max + 1):result = integrate.quad(integral_matrixElement, x_min, x_max, args=(n1, n2, Ex))real = result[0]imag = 0print("(" + str(n1) + ", " + str(n2) + ") " + str(real))

积分结果：

(0, 0) 6341.738760657484
(0, 1) -3.056058246357965e+19
(0, 2) 2609.307906875312
(0, 3) -2.4448465970863724e+18
(0, 4) 6870.623927366621
(0, 5) -6.735801849115569e+17
(0, 6) 4827.5805244781905
(0, 7) -2.771934917331456e+17
(0, 8) -14813.886749646688
(0, 9) -1.4031488734426288e+17
(0, 10) -91.85343359174219
(1, 0) -3.056058246357965e+19
(1, 1) 8677.375330811079
(1, 2) -3.3005429060666024e+19
(1, 3) 7559.214463884988
(1, 4) -3.1184267819979377e+18
(1, 5) 8397.13098087583
(1, 6) -9.507736766447043e+17
(1, 7) -11203.391019872324
(1, 8) -4.175083790774072e+17
(1, 9) 2629.7575721982435
(1, 10) -2.2101670420734822e+17
(2, 0) 3305.077150828044
(2, 1) -3.3005429060666024e+19
(2, 2) 4131.0895468288745
(2, 3) -3.3679009245577576e+19
(2, 4) 18393.399430849167
(2, 5) -3.395620273731079e+18
(2, 6) -9305.413780188943
(2, 7) -1.0910885639889608e+18
(2, 8) -425.52945379666164
(2, 9) -4.982101959405064e+17
(2, 10) -2797.4992591219802
(3, 0) -2.444846597086372e+18
(3, 1) 7540.93172754931
(3, 2) -3.3679009245577572e+19
(3, 3) 10871.226664979697
(3, 4) -3.395620273731073e+19
(3, 5) 4636.680523494656
(3, 6) -3.535935161075353e+18
(3, 7) 5349.394079704496
(3, 8) -1.1717903808520607e+18
(3, 9) 18557.06635991661
(3, 10) -5.48843113631636e+17
(4, 0) 6465.045377234563
(4, 1) -3.1184267819979377e+18
(4, 2) 18407.930328892166
(4, 3) -3.395620273731072e+19
(4, 4) 5278.177294867377
(4, 5) -3.4096517624654983e+19
(4, 6) -10654.575490810419
(4, 7) -3.6166369779384207e+18
(4, 8) -3997.905026982768
(4, 9) -1.2224232985431992e+18
(4, 10) -2426.58163543784
(5, 0) -6.735801849115574e+17
(5, 1) 6890.926988234758
(5, 2) -3.3956202737310833e+18
(5, 3) 4534.457182746286
(5, 4) -3.4096517624654983e+19
(5, 5) 3642.5609606579337
(5, 6) -3.4177219441518076e+19
(5, 7) 10615.200195340132
(5, 8) -3.66726989562957e+18
(5, 9) 25011.51925503606
(5, 10) -1.2562620057692713e+18
(6, 0) 6183.626968714323
(6, 1) -9.507736766447043e+17
(6, 2) -9718.696863996012
(6, 3) -3.5359351610753536e+18
(6, 4) -10105.641181736304
(6, 5) -3.4177219441518076e+19
(6, 6) 3897.1842703463662
(6, 7) -3.4227852359209206e+19
(6, 8) -8550.64335308177
(6, 9) -3.7011086028556754e+18
(6, 10) -9318.227951041345
C:\Users\lycbu\Desktop\Python 量子计算机\Day 7\quantumWell_StarkEffect_step1.py:36: IntegrationWarning: The occurrence of roundoff error is detected, which prevents the requested tolerance from being achieved.  The error may be underestimated.result = integrate.quad(integral_matrixElement, x_min, x_max, args=(n1, n2, Ex))
(7, 0) -2.7719349173314662e+17
(7, 1) -11746.815843769886
(7, 2) -1.0910885639889618e+18
(7, 3) 4746.182689685192
(7, 4) -3.6166369779384197e+18
(7, 5) 10939.680098040822
(7, 6) -3.42278523592092e+19
(7, 7) 30312.39957168161
(7, 8) -3.426169106643533e+19
(7, 9) 59630.42734047828
(7, 10) -3.7248355357271296e+18
(8, 0) -14763.958321932972
(8, 1) -4.175083790774057e+17
(8, 2) 376.26024772313724
(8, 3) -1.171790380852062e+18
(8, 4) -2891.430230791178
(8, 5) -3.667269895629572e+18
(8, 6) -9035.816467070696
(8, 7) -3.4261691066435334e+19
(8, 8) 19862.976591607174
(8, 9) -3.4285417999306818e+19
(8, 10) -54641.972971282514
(9, 0) -1.4031488734426315e+17
(9, 1) 1654.2331507959864
(9, 2) -4.982101959405055e+17
(9, 3) 17653.776341955723
(9, 4) -1.222423298543198e+18
(9, 5) 27534.81059036405
(9, 6) -3.7011086028556774e+18
(9, 7) 58372.822664076666
(9, 8) -3.4285417999306818e+19
(9, 9) 5452.609965541867
(9, 10) -3.430269460197712e+19
(10, 0) -269.7846480739375
(10, 1) -2.2101670420734925e+17
(10, 2) -4534.741401922178
(10, 3) -5.488431136316375e+17
(10, 4) -2159.696443591254
(10, 5) -1.2562620057692713e+18
(10, 6) -8252.286148308438
(10, 7) -3.724835535727129e+18
(10, 8) -55910.908016360045
(10, 9) -3.430269460197712e+19
(10, 10) -275.15859882778125Process finished with exit code 0