![](\"https:\/\/gnnclub-1311496010.cos.ap-beijing.myqcloud.com\/wp-content\/uploads\/2024\/07\/20240729200204171.png\")
Deep Learning Math<\/h1>\n\n
Deep Learning Math<\/h1>\n\u91cf\u5b50\u529b\u5b66\u662f\u7814\u7a76\u5fae\u89c2\u7269\u7406\u4e16\u754c\u4e2d\u7c92\u5b50\u884c\u4e3a\u7684\u79d1\u5b66\uff0c\u5c24\u5176\u662f\u5728\u539f\u5b50\u548c\u4e9a\u539f\u5b50\u5c3a\u5ea6\u4e0a\u3002\u5b66\u4e60\u51e0\u4f55\u6df1\u5ea6\u5b66\u4e60\u4e4b\u524d\uff0c\u4e86\u89e3\u91cf\u5b50\u529b\u5b66\u4e2d\u7403\u8c10\u51fd\u6570\u548c\u5f84\u5411\u6a21\u578b\u7684\u6982\u5ff5\u662f\u6709\u5176\u91cd\u8981\u6027\u7684\uff0c\u7279\u522b\u662f\u5f53\u7814\u7a76\u6216\u5e94\u7528\u9700\u8981\u5904\u7406\u4e09\u7ef4\u6570\u636e\u6216\u8fdb\u884c\u590d\u6742\u7684\u7a7a\u95f4\u5206\u6790\u65f6\u3002<\/p>\n
\u7403\u8c10\u51fd\u6570\u63d0\u4f9b\u4e86\u4e00\u79cd\u6709\u6548\u7684\u65b9\u6cd5\u6765\u8868\u793a\u7403\u9762\u4e0a\u7684\u51fd\u6570\uff0c\u8fd9\u5728\u5904\u7406\u4e09\u7ef4\u51e0\u4f55\u7ed3\u6784\uff0c\u5982\u5efa\u6a21\u5929\u4f53\uff0c\u539f\u5b50\uff0c\u5206\u5b50\u7b49\u7403\u5f62\u7ed3\u6784\u65f6\u975e\u5e38\u6709\u7528\u3002\u901a\u8fc7\u7403\u8c10\u51fd\u6570\uff0c\u53ef\u4ee5\u5728\u7403\u9762\u4e0a\u5c55\u5f00\u590d\u6742\u7684\u5f62\u72b6\u548c\u6a21\u5f0f\uff0c\u4e3a\u6df1\u5ea6\u5b66\u4e60\u6a21\u578b\u63d0\u4f9b\u4e00\u79cd\u5f3a\u5927\u7684\u65b9\u5f0f\u6765\u6355\u6349\u548c\u5b66\u4e60\u8fd9\u4e9b\u5f62\u72b6\u7684\u7279\u5f81\u3002\u53e6\u4e00\u65b9\u9762\uff0c\u5728\u5904\u7406\u4e09\u7ef4\u7a7a\u95f4\u6570\u636e\u65f6\uff0c\u65cb\u8f6c\u4e0d\u53d8\u6027\u662f\u4e00\u4e2a\u5173\u952e\u95ee\u9898\u3002\u7403\u8c10\u51fd\u6570\u5177\u6709\u5929\u7136\u7684\u65cb\u8f6c\u4e0d\u53d8\u6027\u7279\u6027\uff0c\u8fd9\u610f\u5473\u7740\u5b83\u4eec\u53ef\u4ee5\u5e2e\u52a9\u6784\u5efa\u5bf9\u65cb\u8f6c\u4e0d\u654f\u611f\u7684\u6a21\u578b\uff0c\u63d0\u9ad8\u6a21\u578b\u6cdb\u5316\u80fd\u529b\u3002<\/p>\n
\u5f84\u5411\u6a21\u578b\u5bf9\u4e8e\u63cf\u8ff0\u7a7a\u95f4\u4e2d\u70b9\u4e0e\u70b9\u4e4b\u95f4\u7684\u5173\u7cfb\u975e\u5e38\u6709\u6548\u3002\u5728\u51e0\u4f55\u6df1\u5ea6\u5b66\u4e60\u4e2d\uff0c\u5efa\u6a21\u7269\u4f53\u5185\u90e8\u6216\u4e0d\u540c\u7269\u4f53\u4e4b\u95f4\u7684\u7a7a\u95f4\u5173\u7cfb\u662f\u5f88\u91cd\u8981\u7684\u3002\u5f84\u5411\u57fa\u51fd\u6570\u80fd\u591f\u6355\u6349\u8fd9\u4e9b\u590d\u6742\u7684\u7a7a\u95f4\u6a21\u5f0f\uff0c\u5e76\u5728\u8bf8\u5982\u5206\u5b50\u52a8\u529b\u5b66\u6a21\u62df\u3001\u86cb\u767d\u8d28\u7ed3\u6784\u9884\u6d4b\u7b49\u9886\u57df\u53d1\u6325\u4f5c\u7528\u3002\u66f4\u91cd\u8981\u7684\u662f\uff0c\u51e0\u4f55\u6df1\u5ea6\u5b66\u4e60\u7ecf\u5e38\u9700\u8981\u5728\u4e0d\u540c\u7684\u5c3a\u5ea6\u4e0a\u5206\u6790\u6570\u636e\u3002\u5f84\u5411\u6a21\u578b\u53ef\u4ee5\u63d0\u4f9b\u4e00\u4e2a\u81ea\u7136\u7684\u6846\u67b6\u6765\u5b9e\u73b0\u8fd9\u4e00\u70b9\uff0c\u56e0\u4e3a\u5b83\u4eec\u53ef\u4ee5\u5728\u4e0d\u540c\u7684\u5c3a\u5ea6\u4e0a\u8c03\u6574\uff0c\u4ece\u800c\u5141\u8bb8\u6a21\u578b\u540c\u65f6\u5b66\u4e60\u5c40\u90e8\u548c\u5168\u5c40\u7279\u5f81\u3002\u8fd9\u5bf9\u4e8e\u7406\u89e3\u548c\u5904\u7406\u590d\u6742\u7684\u4e09\u7ef4\u7ed3\u6784\u7279\u522b\u6709\u4ef7\u503c\u3002<\/p>\n
![](\"https:\/\/img.icons8.com\/bubbles\/50\/000000\/checklist.png\")
Agenda<\/h3>\n\u91cf\u5b50\u529b\u5b66\u7b80\u4ecb<\/p>\n<\/li>\n
\u5bf9\u79f0\u6027(Symmetry)<\/p>\n<\/li>\n
\u4e0d\u53d8\u6027(Invariance)<\/p>\n<\/li>\n
\u7b49\u53d8\u6027(Equivariance)<\/p>\n<\/li>\n
S0(3) Group<\/p>\n<\/li>\n
\u7fa4\u7684\u8868\u793a(Representation of Groups)<\/p>\n<\/li>\n<\/ul>\n
\u91cf\u5b50\u529b\u5b66\u662f\u73b0\u4ee3\u7269\u7406\u5b66\u7684\u4e00\u5927\u57fa\u7840\u7406\u8bba\uff0c\u63cf\u8ff0\u4e86\u5fae\u89c2\u4e16\u754c\u4e2d\u7c92\u5b50\u7684\u884c\u4e3a\u3002\u5b83\u7684\u53d1\u5c55\u6539\u53d8\u4e86\u6211\u4eec\u5bf9\u81ea\u7136\u754c\u57fa\u672c\u89c4\u5f8b\u7684\u7406\u89e3\u3002\u91cf\u5b50\u529b\u5b66\u7684\u5f62\u6210\u662f20\u4e16\u7eaa\u521d\u4e00\u7cfb\u5217\u9769\u547d\u6027\u53d1\u73b0\u7684\u7ed3\u679c\u3002\u4ee5\u4e0b\u662f\u4e00\u4e9b\u5173\u952e\u4eba\u7269\u548c\u4e8b\u4ef6\uff1a<\/p>\n
\u9a6c\u514b\u65af\u00b7\u666e\u6717\u514b\uff08Max Planck\uff09\uff1a1900\u5e74\uff0c\u666e\u6717\u514b\u63d0\u51fa\u4e86\u91cf\u5b50\u5047\u8bf4\u4e0e\u666e\u6717\u514b\u5e38\u6570\uff0c\u4ee5\u89e3\u91ca\u9ed1\u4f53\u8f90\u5c04\u95ee\u9898\u3002\u4ed6\u63d0\u51fa\u80fd\u91cf\u662f\u4ee5\u79bb\u6563\u7684\u91cf\u5b50\u5f62\u5f0f\u4f20\u9012\u7684\uff0c\u8fd9\u4e00\u7406\u8bba\u6807\u5fd7\u7740\u91cf\u5b50\u529b\u5b66\u7684\u8bde\u751f\u3002\u666e\u6717\u514b\u7684\u7814\u7a76\u63ed\u793a\u4e86\u80fd\u91cf\u91cf\u5b50\u5316\u7684\u6982\u5ff5\uff0c\u7a81\u7834\u4e86\u7ecf\u5178\u7269\u7406\u5b66\u7684\u5c40\u9650\u3002<\/p>\n<\/li>\n
\u963f\u5c14\u4f2f\u7279\u00b7\u7231\u56e0\u65af\u5766\uff08Albert Einstein\uff09\uff1a1905\u5e74\uff0c\u7231\u56e0\u65af\u5766\u63d0\u51fa\u4e86\u5149\u91cf\u5b50\u7406\u8bba\uff0c\u89e3\u91ca\u4e86\u5149\u7535\u6548\u5e94\u3002\u4ed6\u63d0\u51fa\u5149\u53ef\u4ee5\u88ab\u770b\u4f5c\u662f\u7531\u5149\u5b50\u7ec4\u6210\u7684\uff0c\u5149\u5b50\u7684\u80fd\u91cf\u4e0e\u5176\u9891\u7387\u6210\u6b63\u6bd4\u3002\u7231\u56e0\u65af\u5766\u7684\u5149\u91cf\u5b50\u5047\u8bbe\u4e0d\u4ec5\u9a8c\u8bc1\u4e86\u666e\u6717\u514b\u7684\u91cf\u5b50\u5047\u8bf4\uff0c\u8fd8\u89e3\u91ca\u4e86\u4e3a\u4ec0\u4e48\u67d0\u4e9b\u9891\u7387\u7684\u5149\u80fd\u591f\u91ca\u653e\u7535\u5b50\uff0c\u800c\u5176\u4ed6\u9891\u7387\u7684\u5149\u5219\u4e0d\u80fd\u3002<\/p>\n<\/li>\n
\u5c3c\u5c14\u65af\u00b7\u73bb\u5c14\uff08Niels Bohr\uff09\uff1a\u73bb\u5c14\u57281913\u5e74\u63d0\u51fa\u4e86\u73bb\u5c14\u6a21\u578b\uff0c\u7528\u4ee5\u89e3\u91ca\u6c22\u539f\u5b50\u7684\u5149\u8c31\u3002\u4ed6\u7684\u6a21\u578b\u7ed3\u5408\u4e86\u91cf\u5b50\u7406\u8bba\u548c\u7ecf\u5178\u7269\u7406\u5b66\uff0c\u5047\u8bbe\u7535\u5b50\u5728\u539f\u5b50\u6838\u5468\u56f4\u7684\u8f68\u9053\u4e0a\u8fd0\u52a8\u65f6\u5177\u6709\u79bb\u6563\u7684\u80fd\u7ea7\u3002\u73bb\u5c14\u6a21\u578b\u6210\u529f\u89e3\u91ca\u4e86\u6c22\u539f\u5b50\u53d1\u5c04\u548c\u5438\u6536\u5149\u7684\u7279\u5b9a\u9891\u7387\u3002<\/p>\n<\/li>\n
\u7ef4\u5c14\u7eb3\u00b7\u6d77\u68ee\u5821\uff08Werner Heisenberg\uff09\uff1a1925\u5e74\uff0c\u6d77\u68ee\u5821\u63d0\u51fa\u4e86\u77e9\u9635\u529b\u5b66\uff0c\u8fd9\u662f\u4e00\u79cd\u63cf\u8ff0\u91cf\u5b50\u7cfb\u7edf\u7684\u6570\u5b66\u65b9\u6cd5\u3002\u4ed6\u7684\u77e9\u9635\u529b\u5b66\u6452\u5f03\u4e86\u7ecf\u5178\u8f68\u9053\u7684\u6982\u5ff5\uff0c\u8f6c\u800c\u4f7f\u7528\u7b97\u7b26\u548c\u77e9\u9635\u6765\u63cf\u8ff0\u91cf\u5b50\u6001\u7684\u53d8\u5316\u3002\u6d77\u68ee\u5821\u7684\u5de5\u4f5c\u5960\u5b9a\u4e86\u91cf\u5b50\u529b\u5b66\u7684\u6570\u5b66\u57fa\u7840\uff0c\u5e76\u5f15\u5165\u4e86\u4e0d\u786e\u5b9a\u6027\u539f\u7406\u3002<\/p>\n<\/li>\n
\u57c3\u5c14\u6e29\u00b7\u859b\u5b9a\u8c14\uff08Erwin Schr\u00f6dinger\uff09\uff1a1926\u5e74\uff0c\u859b\u5b9a\u8c14\u63d0\u51fa\u4e86\u859b\u5b9a\u8c14\u65b9\u7a0b\uff0c\u63cf\u8ff0\u4e86\u91cf\u5b50\u6001\u968f\u65f6\u95f4\u7684\u6f14\u5316\u3002\u4ed6\u7684\u6ce2\u52a8\u65b9\u7a0b\u662f\u91cf\u5b50\u529b\u5b66\u7684\u6838\u5fc3\uff0c\u63d0\u4f9b\u4e86\u4e00\u79cd\u63cf\u8ff0\u7c92\u5b50\u6ce2\u51fd\u6570\u7684\u65b9\u6cd5\u3002\u859b\u5b9a\u8c14\u65b9\u7a0b\u4e0d\u4ec5\u80fd\u591f\u89e3\u91ca\u7535\u5b50\u5728\u539f\u5b50\u4e2d\u7684\u884c\u4e3a\uff0c\u8fd8\u53ef\u4ee5\u5e94\u7528\u4e8e\u66f4\u590d\u6742\u7684\u91cf\u5b50\u7cfb\u7edf\u3002\u8fd9\u4e00\u65b9\u7a0b\u662f\u91cf\u5b50\u529b\u5b66\u7684\u57fa\u7840\uff0c\u7c7b\u4f3c\u4e8e\u7ecf\u5178\u529b\u5b66\u4e2d\u7684\u725b\u987f\u7b2c\u4e8c\u5b9a\u5f8b\u3002<\/p>\n<\/li>\n<\/ul>\n
\u91cf\u5b50\u529b\u5b66\u7684\u6838\u5fc3\u95ee\u9898\u4e4b\u4e00\u5c31\u662f\u6c42\u89e3\u859b\u5b9a\u8c14\u65b9\u7a0b<\/strong>\u3002\u6c42\u89e3\u859b\u5b9a\u8c14\u65b9\u7a0b\u53ef\u4ee5\u5f97\u5230\u7cfb\u7edf\u7684\u6ce2\u51fd\u6570\uff0c\u4ece\u800c\u63a8\u5bfc\u51fa\u7cfb\u7edf\u7684\u80fd\u91cf\u672c\u5f81\u503c\u548c\u672c\u5f81\u6001\u3001\u6982\u7387\u5bc6\u5ea6\u5206\u5e03\u3001\u5404\u79cd\u7269\u7406\u91cf\u7684\u671f\u671b\u503c\u3001\u7cfb\u7edf\u7684\u52a8\u6001\u6f14\u5316\u884c\u4e3a\u4ee5\u53ca\u91cf\u5b50\u6001\u7684\u53e0\u52a0\u548c\u7ea0\u7f20\u3002\u8fd9\u4e9b\u7ed3\u679c\u4e0d\u4ec5\u5e2e\u52a9\u6211\u4eec\u7406\u89e3\u5fae\u89c2\u7c92\u5b50\u7684\u884c\u4e3a\uff0c\u8fd8\u5e7f\u6cdb\u5e94\u7528\u4e8e\u7269\u7406\u3001\u5316\u5b66\u548c\u6750\u6599\u79d1\u5b66\u7b49\u9886\u57df\uff0c\u4e3a\u73b0\u4ee3\u79d1\u5b66\u6280\u672f\u7684\u53d1\u5c55\u63d0\u4f9b\u4e86\u91cd\u8981\u7684\u7406\u8bba\u57fa\u7840\u3002<\/p>\n \u4f20\u7edf\u8ba1\u7b97\u65b9\u6cd5\u5305\u62ec\u5bc6\u5ea6\u6cdb\u51fd\u7406\u8bba\uff08DFT\uff09\u3001\u54c8\u7279\u91cc-\u798f\u514b\u65b9\u6cd5\uff08Hartree-Fock\uff09\u4ee5\u53ca\u540e\u54c8\u7279\u91cc-\u798f\u514b\u65b9\u6cd5\uff08\u5982\u914d\u7f6e\u76f8\u4e92\u4f5c\u7528CI\u548c\u8026\u5408\u7c07CC\uff09\u3002\u8fd9\u4e9b\u65b9\u6cd5\u5177\u6709\u9ad8\u5ea6\u7684\u51c6\u786e\u6027\uff0c\u4f46\u8ba1\u7b97\u590d\u6742\u5ea6\u6781\u9ad8<\/strong>\u3002\u901a\u5e38\u968f\u7cfb\u7edf\u89c4\u6a21\u7684\u6307\u6570\u589e\u957f\u3002\u4f8b\u5982\uff0cCI\u548cCC\u65b9\u6cd5\u7684\u8ba1\u7b97\u65f6\u95f4\u548c\u5185\u5b58\u9700\u6c42\u968f\u7535\u5b50\u6570\u76ee\u589e\u52a0\u800c\u5448\u6307\u6570\u7ea7\u589e\u957f\uff0c\u4f7f\u5f97\u5927\u89c4\u6a21\u7cfb\u7edf\u7684\u6a21\u62df\u51e0\u4e4e\u4e0d\u53ef\u80fd\u5b9e\u73b0\u3002<\/p>\n \u56e0\u6b64\uff0c\u4f20\u7edf\u65b9\u6cd5\u901a\u5e38\u9002\u7528\u4e8e\u5c0f\u5206\u5b50\u6216\u4e2d\u7b49\u89c4\u6a21\u7cfb\u7edf\uff0c\u5bf9\u4e8e\u5927\u5206\u5b50\u3001\u6750\u6599\u548c\u751f\u7269\u5927\u5206\u5b50\uff0c\u8ba1\u7b97\u53d8\u5f97\u6781\u4e3a\u56f0\u96be\u3002\u5373\u4f7f\u662f\u4e2d\u7b49\u89c4\u6a21\u7cfb\u7edf\uff0c\u8ba1\u7b97\u4e5f\u53ef\u80fd\u8017\u8d39\u6570\u5c0f\u65f6\u3001\u6570\u5929\u751a\u81f3\u6570\u5468\u7684\u65f6\u95f4\uff0c\u4e0d\u9002\u5408\u5feb\u901f\u8fed\u4ee3\u548c\u5927\u89c4\u6a21\u7b5b\u9009<\/strong>\u3002<\/p>\n \u7531\u4e8e\u4f20\u7edf\u91cf\u5b50\u529b\u5b66\u548c\u91cf\u5b50\u5316\u5b66\u65b9\u6cd5\u7684\u8ba1\u7b97\u9650\u5236\uff0c\u79d1\u5b66\u5bb6\u4eec\u5f00\u59cb\u5bfb\u6c42\u65b0\u7684\u8ba1\u7b97\u65b9\u6cd5\u3002\u6df1\u5ea6\u5b66\u4e60\u6a21\u578b\uff0c\u5982SchNet\u7b49\uff0c\u901a\u8fc7\u5b66\u4e60\u5927\u91cf\u91cf\u5b50\u529b\u5b66\u8ba1\u7b97\u6570\u636e\uff0c\u80fd\u591f\u663e\u8457\u63d0\u9ad8\u8ba1\u7b97\u6548\u7387\u548c\u65f6\u95f4<\/strong>\u3002<\/p>\n \u8fd9\u4e9b\u6df1\u5ea6\u5b66\u4e60\u6a21\u578b\u5e76\u975e\u51ed\u7a7a\u6784\u5efa\uff0c\u800c\u662f\u5efa\u7acb\u5728\u91cf\u5b50\u529b\u5b66\u7684\u575a\u5b9e\u57fa\u7840\u4e0a\uff0c\u7279\u522b\u662f\u51e0\u4f55\u7279\u6027\u548c\u5bf9\u79f0\u6027<\/strong>\u3002\u8fd9\u4e9b\u57fa\u7840\u5305\u62ec\u7403\u8c10\u51fd\u6570\u3001Wigner-D\u77e9\u9635\u3001CG\u7cfb\u6570\u548c\u5f84\u5411\u6a21\u578b<\/strong>\u7b49\u3002<\/p>\n \u7403\u8c10\u51fd\u6570\uff08Spherical Harmonics\uff09<\/p>\n Wigner-D\u77e9\u9635<\/p>\n CG\u7cfb\u6570\uff08Clebsch-Gordan Coefficients\uff09<\/p>\n \u5f84\u5411\u6a21\u578b\uff08Radial Model\uff09<\/p>\n \u7403\u8c10\u51fd\u6570\u5728\u51e0\u4f55\u6df1\u5ea6\u5b66\u4e60\u4e2d\u626e\u6f14\u7740\u6838\u5fc3\u89d2\u8272\uff0c\u5b83\u4eec\u901a\u8fc7\u63d0\u4f9b\u4e00\u79cd\u5f3a\u5927\u7684\u6570\u5b66\u5de5\u5177\u96c6\uff0c\u5e2e\u52a9\u7814\u7a76\u8005\u548c\u5f00\u53d1\u8005\u4ee5\u66f4\u6709\u6548\u3001\u66f4\u76f4\u89c2\u7684\u65b9\u5f0f\u5904\u7406\u548c\u5206\u6790\u4e09\u7ef4\u51e0\u4f55\u6570\u636e\u3002\u7403\u8c10\u51fd\u6570\u662f\u5b9a\u4e49\u5728\u7403\u9762\u4e0a\u7684\u51fd\u6570\uff0c\u5728\u51e0\u4f55\u6df1\u5ea6\u5b66\u4e60\u4e2d\u7684\u91cd\u8981\u6027\u4e3b\u8981\u4f53\u73b0\u5728\u5b83\u4eec\u5bf9\u4e8e\u4e09\u7ef4\u51e0\u4f55\u6570\u636e\u7684\u5206\u6790\u548c\u5904\u7406\u80fd\u529b\uff0c\u5c24\u5176\u662f\u5728\u5904\u7406\u7403\u9762\u6216\u7403\u5f62\u51e0\u4f55\u7ed3\u6784\u65f6\u7ecf\u5e38\u9700\u8981\u6267\u884c\u5404\u79cd\u590d\u6742\u7684\u4e09\u7ef4\u51e0\u4f55\u64cd\u4f5c\uff0c\u5982\u65cb\u8f6c\u3001\u7f29\u653e\u548c\u53d8\u5f62\u3002\u7403\u8c10\u51fd\u6570\u7684\u6570\u5b66\u7279\u6027\u4f7f\u5f97\u8fd9\u4e9b\u64cd\u4f5c\u53ef\u4ee5\u4ee5\u975e\u5e38\u81ea\u7136\u548c\u6570\u5b66\u4e0a\u4f18\u96c5\u7684\u65b9\u5f0f\u5b9e\u73b0\u3002<\/p>\n \u7403\u5750\u6807\u7cfb$(r, \\theta, \\phi)$ \u7528\u4e8e\u5728\u7a7a\u95f4\u4e2d\u63cf\u8ff0\u70b9\u7684\u4f4d\u7f6e\u3002\u5176\u4e2d, $r$ \u8868\u793a\u5f84\u5411\u8ddd\u79bb, $\\theta$ (\u5929\u9876\u89d2)\u662f\u4ece\u6b63 $z$ \u8f74\u5411\u4e0b\u7684\u89d2\u5ea6, \u800c $\\phi$ (\u65b9\u4f4d\u89d2) \u662f\u5728 $\\mathrm{x}-\\mathrm{y}$ \u5e73\u9762\u4e0a\u4ece\u6b63 $x$ \u8f74\u6d4b\u91cf\u7684\u89d2\u5ea6\u3002<\/p>\n<\/li>\n \u7403\u8c10\u51fd\u6570\u5728\u8fd9\u79cd\u5750\u6807\u7cfb\u4e0b\u5b9a\u4e49, \u80fd\u591f\u6709\u6548\u63cf\u8ff0\u7403\u9762\u4e0a\u7684\u590d\u6742\u51e0\u4f55\u548c\u7269\u7406\u8fc7\u7a0b\u3002\u5177\u4f53\u800c\u8a00, \u7403\u5750\u6807\u7cfb\u4e0e\u7b1b\u5361\u5c14\u5750\u6807\u7cfb $(x, y, z)$ \u4e4b\u95f4\u5b58\u5728\u8f6c\u6362\u5173\u7cfb\u3002\u7ed9\u5b9a\u7403\u5750\u6807\u4e2d\u7684\u70b9 $(r, \\theta, \\phi)$, \u53ef\u4ee5\u901a\u8fc7\u4e0b\u5217\u8f6c\u6362\u65b9\u7a0b\u8f6c\u6362\u4e3a\u7b1b\u5361\u5c14\u5750\u6807: \u8fd9\u79cd\u8f6c\u6362\u5173\u7cfb\u8868\u660e\uff0c\u867d\u7136\u7403\u8c10\u51fd\u6570\u672c\u8eab\u5728\u7403\u5750\u6807\u7cfb\u4e0b\u5b9a\u4e49\uff0c\u63cf\u8ff0\u7403\u9762\u4e0a\u7684\u70b9\u6216\u8005\u51fd\u6570\uff0c\u5b83\u4eec\u4e5f\u53ef\u4ee5\u901a\u8fc7\u8fd9\u79cd\u5173\u7cfb\u4e0e\u7b1b\u5361\u5c14\u5750\u6807\u7cfb\u4e2d\u7684\u70b9\u8054\u7cfb\u8d77\u6765\uff0c\u4ece\u800c\u5728\u5b9e\u9645\u5e94\u7528\u4e2d\u63cf\u8ff03D\u7a7a\u95f4\u4e2d\u7684\u7269\u7406\u73b0\u8c61\u6216\u6570\u5b66\u51fd\u6570\u3002<\/p>\n \n \u4e3a\u4f8b\u901a\u4fd7\u7684\u89e3\u91ca\u7403\u8c10\u51fd\u6570\u7684\u6982\u5ff5\u548c\u5e94\u7528\uff0c\u53ef\u4ee5\u901a\u8fc7\u7c7b\u6bd4\u6cf0\u52d2\u516c\u5f0f\u6765\u5e2e\u52a9\u7406\u89e3\u3002\u6cf0\u52d2\u7ea7\u6570\u662f\u901a\u8fc7\u591a\u9879\u5f0f\u7684\u5f62\u5f0f\u5728\u67d0\u4e00\u70b9\u9644\u8fd1\u5c55\u5f00\u4e00\u4e2a\u51fd\u6570\u3002\u8fd9\u5728\u4e00\u7ef4\u6216\u4e8c\u7ef4\u7684\u5e73\u9762\u4e0a\u975e\u5e38\u6709\u6548\u3002\u5177\u4f53\u6765\u8bf4\uff0c\u6cf0\u52d2\u7ea7\u6570\u662f\u57fa\u4e8e\u51fd\u6570\u5728\u67d0\u4e00\u70b9\u5904\u7684\u5bfc\u6570\u4fe1\u606f\u6765\u8fdb\u884c\u5c55\u5f00\u7684\uff0c\u53ef\u4ee5\u8fd1\u4f3c\u4efb\u4f55\u5728\u8be5\u70b9\u5904\u5177\u6709\u65e0\u9650\u9636\u5bfc\u6570\u7684\u51fd\u6570\u3002<\/p>\n \u6cf0\u52d2\u7ea7\u6570\u7684\u4e00\u822c\u5f62\u5f0f\u4e3a: \n \u7403\u8c10\u51fd\u6570\u53ef\u4ee5\u88ab\u770b\u4f5c\u662f\u6cf0\u52d2\u7ea7\u6570\u5728\u7403\u9762\u4e0a\u7684\u4e00\u79cd\u63a8\u5e7f\u3002\u6cf0\u52d2\u7ea7\u6570\u901a\u8fc7\u591a\u9879\u5f0f\u7684\u5f62\u5f0f\u8fd1\u4f3c\u4e00\u4e2a\u51fd\u6570\uff0c\u5728\u4e00\u7ef4\u6216\u4e8c\u7ef4\u5e73\u9762\u4e0a\u975e\u5e38\u6709\u6548\u3002\u800c\u7403\u8c10\u51fd\u6570\u5219\u662f\u5728\u4e09\u7ef4\u7403\u9762\u4e0a\u5bf9\u51fd\u6570\u8fdb\u884c\u5c55\u5f00\uff0c\u4ece\u800c\u5b9e\u73b0\u7c7b\u4f3c\u7684\u6548\u679c\u3002\u4e0d\u540c\u7684\u662f\uff0c\u6cf0\u52d2\u7ea7\u6570\u4f7f\u7528\u591a\u9879\u5f0f\u4f5c\u4e3a\u57fa\u51fd\u6570\uff0c\u800c\u7403\u8c10\u51fd\u6570\u4f7f\u7528\u7684\u662f\u5b9a\u4e49\u5728\u7403\u9762\u4e0a\u7684\u4e00\u7ec4\u6b63\u4ea4\u57fa\u51fd\u6570\u3002\u53e6\u4e00\u65b9\u9762\uff0c\u6cf0\u52d2\u7ea7\u6570\u4e3b\u8981\u7528\u4e8e\u4e00\u7ef4\u6216\u4e8c\u7ef4\u5e73\u9762\u4e0a\u51fd\u6570\u7684\u5c40\u90e8\u8fd1\u4f3c\uff1b\u7403\u8c10\u51fd\u6570\u7528\u4e8e\u4e09\u7ef4\u7403\u9762\u4e0a\u7684\u51fd\u6570\u5c55\u5f00\uff0c\u7c7b\u4f3c\u5bf9\u7403\u9762\u4e0a\u51fd\u6570\u7684\u5206\u89e3\u3002<\/p>\n \u5085\u91cc\u53f6\u5206\u89e3\u662f\u5c06\u4e00\u4e2a\u5468\u671f\u51fd\u6570\u8868\u793a\u4e3a\u4e00\u7cfb\u5217\u6b63\u5f26\u548c\u4f59\u5f26\u51fd\u6570\uff08\u6216\u6307\u6570\u51fd\u6570\uff09\u7684\u548c\u3002\u5085\u91cc\u53f6\u5206\u89e3\u7279\u522b\u9002\u7528\u4e8e\u5904\u7406\u5468\u671f\u6027\u4fe1\u53f7\u548c\u51fd\u6570\u3002\u5176\u5f62\u5f0f\u4e3a: \u5085\u91cc\u53f6\u5206\u89e3\u7684\u5173\u952e\u601d\u60f3\u662f\u5229\u7528\u4e00\u7ec4\u6b63\u4ea4\u57fa\u51fd\u6570\uff08\u6b63\u5f26\u548c\u4f59\u5f26\u51fd\u6570\uff09\u6765\u8868\u793a\u4e00\u4e2a\u51fd\u6570\u3002<\/p>\n \u5bf9\u6bd4\u4e4b\u524d\uff0c\u4e24\u8005\u90fd\u6d89\u53ca\u4f7f\u7528\u4e00\u7ec4\u6b63\u4ea4\u57fa\u51fd\u6570\u6765\u8868\u793a\u4e00\u4e2a\u51fd\u6570\u7684\u5c55\u5f00\u3002\u5085\u91cc\u53f6\u5206\u89e3\u5904\u7406\u7684\u662f\u4e00\u7ef4\u6216\u4e8c\u7ef4\u7684\u5468\u671f\u51fd\u6570\uff0c\u800c\u7403\u8c10\u51fd\u6570\u5904\u7406\u7684\u662f\u4e09\u7ef4\u7403\u9762\u4e0a\u7684\u51fd\u6570\u3002\u4e24\u8005\u90fd\u57fa\u4e8e\u6b63\u4ea4\u57fa\u51fd\u6570\uff0c\u901a\u8fc7\u5c55\u5f00\u6765\u8fd1\u4f3c\u6216\u8868\u793a\u590d\u6742\u7684\u51fd\u6570\u3002<\/p>\n \n \u6700\u540e\u5f15\u51fa\u7403\u8c10\u51fd\u6570\u7684\u5b9a\u4e49, \u7403\u8c10\u51fd\u6570\u662f\u7403\u5750\u6807\u7cfb\u4e2d\u7684\u89e3\u6790\u51fd\u6570, \u901a\u5e38\u8868\u793a\u4e3a $Y_l^m(\\theta, \\phi)$,\u5176\u4e2d $l$ \u662f\u6b63\u6574\u6570, $m$ \u662f\u6574\u6570\u4e14 $-l \\leq m \\leq l$ \u3002\u8fd9\u91cc, $\\theta$ \u662f\u4ece\u6b63 $z$ \u8f74\u5411\u4e0b\u6d4b\u91cf\u7684\u5929\u9876\u89d2, $\\phi$ \u662f\u5728 $\\mathrm{xy}$ \u5e73\u9762\u4e0a, \u4ece\u6b63 $x$ \u8f74\u6d4b\u91cf\u7684\u65b9\u4f4d\u89d2\u3002<\/p>\n \u5177\u4f53\u7684, \u7403\u8c10\u51fd\u6570\u7684\u4e00\u822c\u5f62\u5f0f\u662f: \u5176\u4e2d, $P_l^m(\\cos \\theta)$ \u662f\u5173\u8054\u52d2\u8ba9\u5fb7\u591a\u9879\u5f0f, $e^{i m \\phi}$ \u662f\u590d\u6307\u6570\u51fd\u6570, \u8868\u793a\u65cb\u8f6c\u3002 $l$ \u79f0\u4e3a\u7403\u8c10\u51fd\u6570\u7684\u9636, $m$ \u79f0\u4e3a\u6b21\u6570\u3002\u5173\u8054\u52d2\u8ba9\u5fb7\u591a\u9879\u5f0f $P_l^m(x)$ \u662f\u52d2\u8ba9\u5fb7\u591a\u9879\u5f0f $P_l(x)$ \u7684\u5bfc\u6570, \u5e76\u4e58\u4ee5\u4e00\u4e2a\u5e38\u6570\u56e0\u5b50\u6765\u6ee1\u8db3\u6b63\u4ea4\u6027\u6761\u4ef6\u3002\u5b83\u4eec\u53ef\u4ee5\u901a\u8fc7\u7f57\u5fb7\u91cc\u683c\u65af\u516c\u5f0f\u7ed9\u51fa: \u5176\u4e2d, $P_l(x)$ \u662f\u52d2\u8ba9\u5fb7\u591a\u9879\u5f0f, \u53ef\u4ee5\u901a\u8fc7\u7f57\u5fb7\u91cc\u683c\u65af\u516c\u5f0f\u5f97\u5230: \u81f3\u4e8e\u7403\u8c10\u51fd\u6570\u7684\u4e00\u822c\u5f62\u5f0f\u6765\u6e90\u4e8e\u52d2\u8ba9\u5fb7\u591a\u9879\u5f0f\u7684\u63a8\u5bfc\uff0c\u8fd9\u91cc\u4e0d\u505a\u8fdb\u4e00\u6b65\u7684\u63a8\u5bfc\u89e3\u91ca, \u4e0b\u9762\u4e3b\u8981\u8bb2\u89e3\u5176\u4f5c\u7528\u4ee5\u53ca\u7403\u8c10\u51fd\u6570\u516c\u5f0f\u4e2d\u9636\u6570 $l$ \u548c\u6b21\u6570 $m$ \u7684\u4ee3\u8868\u7684\u610f\u4e49\u3002\u5982\u56fe\u6240\u793a, \u662f\u5bf9\u7403\u8c10\u51fd\u6570\u7684\u53c2\u6570 $l$ \u548c $m$ \u8fdb\u884c\u7684\u53ef\u89c6\u5316\u3002<\/p>\n \n $$ \n \u4e3a\u4e86\u65b9\u4fbf\u7406\u89e3l\u548cm\uff0c\u53ef\u4ee5\u8003\u8651\u4e00\u4e2a\u4f7f\u7528\u7403\u8c10\u51fd\u6570\u62df\u5408\u5929\u4f53\u7684\u573a\u666f<\/p>\n \n $l$ (\u9636\u6570) \u51b3\u5b9a\u4e86\u5927\u6d32\u548c\u5927\u6d0b\u7684\u201c\u5e73\u9762\u5f62\u72b6\u201d\u590d\u6742\u7a0b\u5ea6\u3002<\/p>\n $m$\u6b21\u6570\u51b3\u5b9a\u5927\u6d32\u548c\u5927\u6d0b\u7ed5\u7403\u9762\u8f74\u7ebf\uff08\u4f8b\u5982\uff0c\u5317\u6781\u5230\u5357\u6781\uff09\u7684\u65cb\u8f6c\u5bf9\u79f0\u6027\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c$m$\u51b3\u5b9a\u4e86\u5728\u7ed5\u5929\u4f53\u4e00\u5468\u65f6\uff0c\u5927\u6d32\u548c\u5927\u6d0b\u91cd\u590d\u51fa\u73b0\u7684\u6b21\u6570\u3002<\/p>\n \u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u4e00\u4e2a\u7403\u8c10\u51fd\u6570\u7684\u63cf\u8ff0\u90fd\u662f\u5728\u7403\u9762\u4e0a\u7684\uff0c\u201c\u7403\u9762\u201d\u610f\u5473\u7740\u6ca1\u6709\u6d77\u62d4\u4fe1\u606f\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u5982\u679c\u8fd9\u4e2a\u5929\u4f53\u7684\u6240\u6709\u8868\u9762\u6d77\u62d4\u90fd\u4e3a0\uff0c\u90a3\u4e48\u4e00\u4e2a\u7403\u8c10\u51fd\u6570\u53ef\u4ee5\u63cf\u8ff0\u8be5\u5929\u4f53\u8868\u9762\u5927\u6d32\u5927\u6d0b\u7684\u5f62\u72b6\u3002\u5982\u679c\u60f3\u7528\u7403\u8c10\u51fd\u6570\u6765\u6a21\u62df\u4e00\u4e2a\u51f9\u51f8\u4e0d\u5e73\u7684\u7acb\u4f53\u5929\u4f53\uff0c\u53ef\u4ee5\u5c06\u7403\u9762\u4e0a\u6bcf\u4e00\u70b9\u7684\u9ad8\u5ea6 $H(\\theta, \\phi)$ \u8868\u793a\u4e3a\u4e00\u7cfb\u5217\u7403\u8c10\u51fd\u6570\u7684\u52a0\u6743\u548c: \u5176\u4e2d, $a_{l m}$ \u662f\u7cfb\u6570, \u8868\u793a\u6bcf\u4e2a\u7403\u8c10\u51fd\u6570\u5728\u603b\u9ad8\u5ea6\u5206\u5e03\u4e2d\u7684\u6743\u91cd\u3002\u8fd9\u5b9e\u9645\u4e0a\u662f\u901a\u8fc7\u4e0d\u540c\u9636\u548c\u6b21\u7684\u7403\u8c10\u51fd\u6570\u7ebf\u6027\u7ec4\u5408\u6765\u6a21\u62df\u9ad8\u5ea6\u53d8\u5316\u3002<\/p>\n \u5728\u7fa4\u8bba\u7684\u6559\u5b66\u4e2d\uff0c\u6211\u4eec\u8ba8\u8bba\u4e86\u7fa4\u8bba\u4e2d\u7684\u5bf9\u79f0\u6027\u5bf9\u51e0\u4f55\u6570\u636e\u7684\u91cd\u8981\u6027\uff0c\u5b83\u5f80\u5f80\u4f5c\u4e3a\u51e0\u4f55\u6df1\u5ea6\u5b66\u4e60\u6a21\u578b\u7684\u4e00\u79cd\u5f52\u7eb3\u504f\u7f6e\u3002\u800c\u7403\u8c10\u51fd\u6570\u5728\u63cf\u8ff0\u51e0\u4f55\u6570\u636e\u65f6\u5c55\u73b0\u51fa\u4e86\u4e00\u79cd\u7279\u6b8a\u7684\u6027\u8d28\uff1a\u5b83\u4eec\u5bf9\u4e09\u7ef4\u7a7a\u95f4\u4e2d\u7684\u65cb\u8f6c\u5177\u6709\u65cb\u8f6c\u4e0d\u53d8\u6027\u3002\u8fd9\u662f\u56e0\u4e3a\u7403\u8c10\u51fd\u6570\u81ea\u8eab\u6784\u6210\u4e86\u7403\u9762\u4e0a\u7684\u4e00\u4e2a\u6b63\u4ea4\u57fa\uff0c\u4e3a\u5728\u65cb\u8f6c\u4e0b\u4ecd\u80fd\u4fdd\u6301\u4e00\u81f4\u6027\u7684\u7403\u9762\u6a21\u5f0f\u63d0\u4f9b\u4e86\u4e00\u79cd\u81ea\u7136\u7684\u63cf\u8ff0\u65b9\u5f0f\u3002<\/p>\n<\/li>\n \u5177\u4f53\u800c\u8a00\uff0c\u5728\u7403\u9762\u4e0a\u5e94\u7528SO(3)\u7fa4\u4e2d\u7684\u4efb\u4f55\u65cb\u8f6c\u65f6\uff0c\u7403\u8c10\u51fd\u6570\u7684\u53d8\u6362\u662f\u53ef\u9884\u6d4b\u7684\u3002\u4e00\u4e2a\u7403\u8c10\u51fd\u6570\u65cb\u8f6c\u540e\uff0c\u53ef\u4ee5\u88ab\u8868\u793a\u4e3a\u539f\u59cb\u7403\u8c10\u51fd\u6570\u96c6\u5408\u7684\u4e00\u4e2a\u7ebf\u6027\u7ec4\u5408\u3002\u8fd9\u8868\u660e\u7403\u8c10\u51fd\u6570\u80fd\u591f\u4ee5\u4e00\u79cd\u7cfb\u7edf\u7684\u65b9\u5f0f\u6355\u83b7\u65cb\u8f6c\u64cd\u4f5c\u7684\u5f71\u54cd\uff0c\u800c\u4e14\u8fd9\u79cd\u5f71\u54cd\u53ef\u4ee5\u901a\u8fc7\u5206\u6790\u539f\u59cb\u51fd\u6570\u7684\u65b0\u7ec4\u5408\u6765\u7406\u89e3\u3002\u8fd9\u4e2a\u7279\u6027\u4f7f\u5f97\u7403\u8c10\u51fd\u6570\u6210\u4e3a\u65cb\u8f6c\u7fa4SO(3)\u7684\u4e00\u4e2a\u91cd\u8981\u8868\u793a\u3002<\/p>\n<\/li>\n \u5728\u6570\u5b66\u4e0a\uff0c\u8868\u793a\u662f\u4e00\u79cd\u901a\u8fc7\u7fa4\u7684\u5143\u7d20\uff08\u672c\u4f8b\u4e2d\u662f\u65cb\u8f6c\uff09\u6765\u63cf\u8ff0\u51fd\u6570\u53d8\u6362\u884c\u4e3a\u7684\u65b9\u6cd5\u3002\u7531\u4e8e\u7403\u8c10\u51fd\u6570\u5728\u65cb\u8f6c\u4e0b\u7684\u8fd9\u79cd\u53ef\u9884\u6d4b\u53d8\u6362\u65b9\u5f0f\uff0c\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u5b83\u4eec\u4f5c\u4e3a\u4e00\u7ec4\u57fa\u7840\u5de5\u5177\uff0c\u6765\u63cf\u8ff0\u548c\u5206\u6790\u5b9a\u4e49\u5728\u7403\u9762\u4e0a\u7684\u4efb\u4f55\u51fd\u6570\u3002\u53ea\u8981\u8fd9\u4e2a\u51fd\u6570\u6ee1\u8db3\u65cb\u8f6c\u5bf9\u79f0\u6027\uff0c\u5373\u5b83\u7684\u5f62\u6001\u5728\u65cb\u8f6c\u4e0b\u4fdd\u6301\u4e0d\u53d8\u6216\u4ee5\u53ef\u9884\u6d4b\u7684\u65b9\u5f0f\u53d8\u5316\uff0c\u5b83\u5c31\u53ef\u4ee5\u901a\u8fc7\u7403\u8c10\u51fd\u6570\u7684\u7ebf\u6027\u7ec4\u5408\u6765\u8868\u793a\u3002\u6b63\u5982\u5728\u4e00\u7ef4\u7a7a\u95f4\u4e2d\u4efb\u4f55\u51fd\u6570\u90fd\u53ef\u4ee5\u901a\u8fc7\u5085\u7acb\u53f6\u7ea7\u6570\u5c55\u5f00\u4e3a\u6b63\u5f26\u548c\u4f59\u5f26\u51fd\u6570\u7684\u7ebf\u6027\u7ec4\u5408\u4e00\u6837\uff0c\u5728\u7403\u9762\u4e0a\uff0c\u51fd\u6570\u4e5f\u53ef\u4ee5\u5c55\u5f00\u4e3a\u7403\u8c10\u51fd\u6570\u7684\u7ebf\u6027\u7ec4\u5408\u3002\u8fd9\u4e2a\u5c55\u5f00\u4e0d\u4ec5\u5305\u542b\u4e86\u51fd\u6570\u7684\u7a7a\u95f4\u5206\u5e03\uff0c\u8fd8\u9690\u542b\u5730\u7f16\u7801\u4e86\u51fd\u6570\u5728\u65cb\u8f6c\u4e0b\u7684\u884c\u4e3a\u3002 \u5982\u679c\u901a\u8fc7\u8bad\u7ec3\u51e0\u4f55\u6df1\u5ea6\u5b66\u4e60\u6a21\u578b\u540e\uff0c\u5f97\u5230\u4e86\u7403\u9762\u51fd\u6570\u5bf9\u5e94\u7684\u7cfb\u6570\uff0c\u5c31\u53ef\u4ee5\u91cd\u5efa\u8be5\u51fd\u6570\u5728\u7403\u9762\u4e0a\u7684\u5b8c\u6574\u4fe1\u606f\uff0c\u5305\u62ec\u5b83\u5982\u4f55\u54cd\u5e94\u65cb\u8f6c\u53d8\u6362\u3002<\/p>\n<\/li>\n \u7efc\u4e0a\u6240\u8ff0\uff0c\u7403\u8c10\u51fd\u6570\u4e2d\u7684l\u548cm\u53c2\u6570\u5171\u540c\u6784\u6210\u4e86\u4e00\u79cd\u7cfb\u7edf\u5316\u7684\u65b9\u6cd5\uff0cl\u63d0\u4f9b\u4e86\u4ece\u7c97\u7cd9\u5230\u7cbe\u7ec6\u7684\u591a\u5c3a\u5ea6\u89c6\u89d2\uff0c\u800cm\u5219\u63d0\u4f9b\u4e86\u5728\u6bcf\u4e2a\u5c3a\u5ea6\u4e0a\u6cbf\u65b9\u4f4d\u89d2\u65b9\u5411\u7684\u7ec6\u8282\u548c\u5bf9\u79f0\u6027\u3002\u8fd9\u79cd\u7ec4\u5408\u4e3a\u51e0\u4f55\u5f62\u72b6\u548c\u6a21\u5f0f\u7684\u8bc6\u522b\u3001\u5206\u7c7b\u548c\u5206\u6790\u63d0\u4f9b\u4e86\u5f3a\u5927\u7684\u5de5\u5177\uff0c\u4f7f\u5f97\u7403\u8c10\u51fd\u6570\u5728\u5904\u7406\u590d\u6742\u7403\u9762\u6570\u636e\uff0c\u5c24\u5176\u662f\u5728\u51e0\u4f55\u6df1\u5ea6\u5b66\u4e60\u9886\u57df\u4e2d\uff0c\u6210\u4e3a\u4e86\u4e00\u4e2a\u975e\u5e38\u6709\u4ef7\u503c\u7684\u5de5\u5177\u3002<\/p>\n<\/li>\n<\/ul>\n \u5f53\u6211\u4eec\u5bf9\u5206\u5b50\u7684\u7a7a\u95f4\u4f4d\u7f6e\u8fdb\u884c\u5efa\u6a21\uff0c\u5c24\u5176\u662f\u4f7f\u7528\u7403\u8c10\u51fd\u6570\u65f6\uff0c\u6211\u4eec\u9700\u8981\u8003\u8651\u5206\u5b50\u5728\u4e0d\u540c\u65b9\u5411\u4e0a\u7684\u884c\u4e3a\u3002\u8fd9\u5305\u62ec\u5206\u5b50\u7684\u65cb\u8f6c\u4ee5\u53ca\u5982\u4f55\u63cf\u8ff0\u65cb\u8f6c\u540e\u7684\u72b6\u6001\u3002Wigner D \u77e9\u9635\u5728\u8fd9\u4e00\u8fc7\u7a0b\u4e2d\u8d77\u5230\u4e86\u81f3\u5173\u91cd\u8981\u7684\u4f5c\u7528\u3002<\/p>\n \u9636\u6570$l$\u662f\u7403\u8c10\u51fd\u6570 $Y_{l m}(\\theta, \\phi)$ \u4e2d\u7684\u91cd\u8981\u53c2\u6570\uff0c\u5bf9\u4e8e\u4e00\u4e2a\u56fa\u5b9a\u7684$l$ \uff0c\u6211\u4eec\u6709\u4e00\u7ec4 $2 l+1$ \u4e2a\u7403\u8c10\u51fd\u6570\uff0c\u8fd9\u4e9b\u51fd\u6570\u53ef\u4ee5\u89c6\u4e3a\u63cf\u8ff0\u4e86\u5206\u5b50\u7684\u7a7a\u95f4\u5206\u5e03\u3002<\/p>\n \u5f53\u5206\u5b50\u65cb\u8f6c\u65f6\uff0c\u5176\u7535\u5b50\u5206\u5e03\u6216\u5176\u4ed6\u7a7a\u95f4\u5c5e\u6027\u4f1a\u968f\u4e4b\u53d8\u5316\u3002\u6211\u4eec\u53ef\u4ee5\u7528\u65cb\u8f6c\u7b97\u7b26 $R(\\alpha, \\beta, \\gamma)$ \u6765\u63cf\u8ff0\u8fd9\u79cd\u65cb\u8f6c\uff0c\u5176\u4e2d $\\alpha, \\beta, \\gamma$ \u662f\u6b27\u62c9\u89d2\uff0c\u8868\u793a\u7ed5\u4e0d\u540c\u8f74\u7684\u65cb\u8f6c\u3002<\/p>\n Wigner $\\mathrm{D}$ \u77e9\u9635 $D_{m^{\\prime} m}^j(\\alpha, \\beta, \\gamma)$ \u662f $S O(3)$ \u7fa4\u7684\u4e00\u4e2a\u4e0d\u53ef\u7ea6\u8868\u793a\u3002\u5bf9\u4e8e\u9636\u6570$l$ \uff0c\u53ef\u4ee5\u4f7f\u7528 Wigner D \u77e9\u9635\u63cf\u8ff0\u7403\u8c10\u51fd\u6570\u5728\u65cb\u8f6c\u4e0b\u7684\u53d8\u5316\u3002\u8bbe $|l, m\\rangle$ \u8868\u793a\u65cb\u8f6c\u884c\u4e3a\uff0c\u4e0e\u4e4b\u5bf9\u5e94\u7684\u7403\u8c10\u51fd\u6570\u4e3a $Y_{l m}$ \u3002 \u5176\u4e2d $\\theta^{\\prime}$ \u548c $\\phi^{\\prime}$ \u662f\u65cb\u8f6c\u540e\u7684\u89d2\u5750\u6807\u3002<\/p>\n Wigner $\\mathrm{D}$ \u77e9\u9635 $D_{m^{\\prime} m}^j(\\alpha, \\beta, \\gamma)$ \u63cf\u8ff0\u4e86\u91cf\u5b50\u6001\u5728\u65cb\u8f6c\u4e0b\u7684\u53d8\u5316\u3002\u5728\u795e\u7ecf\u7f51\u7edc\u4e2d\uff0cWigner D \u77e9\u9635\u53ef\u4ee5\u7528\u4e8e\u786e\u4fdd\u7279\u5f81\u5728\u65cb\u8f6c\u64cd\u4f5c\u4e0b\u7684\u8f6c\u6362\u3002\u8fd9\u610f\u5473\u7740\uff0c\u5982\u679c\u8f93\u5165\u6570\u636e\u65cb\u8f6c\u4e86\uff0c\u76f8\u5e94\u7684\u7279\u5f81\u4e5f\u4f1a\u6309\u7167 Wigner D \u77e9\u9635\u8fdb\u884c\u65cb\u8f6c\u53d8\u6362\uff0c\u4ece\u800c\u4fdd\u6301\u7b49\u53d8\u6027\u3002<\/p>\n \u5177\u4f53\u6765\u8bf4\uff1a<\/p>\n \u4e3a\u4ec0\u4e48 Wigner-D \u77e9\u9635\u4fdd\u8bc1\u7b49\u53d8\u6027\uff1f<\/strong><\/p>\n \u5728\u91cf\u5b50\u529b\u5b66\u4e2d\uff0c\u6211\u4eec\u7528\u6570\u5b66\u5de5\u5177\u6765\u63cf\u8ff0\u65cb\u8f6c\u3002\u4e00\u4e2a\u91cd\u8981\u7684\u5de5\u5177\u5c31\u662f Wigner-D \u77e9\u9635\u3002\u5b83\u544a\u8bc9\u6211\u4eec\u5728\u91cf\u5b50\u4e16\u754c\u91cc\uff0c\u5f53\u6211\u4eec\u65cb\u8f6c\u4e00\u4e2a\u7269\u4f53\u65f6\uff0c\u91cf\u5b50\u6001\uff08\u53ef\u4ee5\u7406\u89e3\u4e3a\u63cf\u8ff0\u7269\u4f53\u72b6\u6001\u7684\u6570\u5b66\u5bf9\u8c61\uff09\u662f\u5982\u4f55\u53d8\u5316\u7684\u3002<\/p>\n Wigner-D \u77e9\u9635\u662f\u4e13\u95e8\u7528\u6765\u63cf\u8ff0\u65cb\u8f6c\u7684\u3002\u5c31\u50cf\u4f60\u7528\u4e00\u628a\u5c3a\u5b50\u53ef\u4ee5\u7cbe\u786e\u6d4b\u91cf\u957f\u5ea6\uff0cWigner-D \u77e9\u9635\u80fd\u7cbe\u786e\u63cf\u8ff0\u65cb\u8f6c\u540e\u7684\u91cf\u5b50\u6001\u3002\u6240\u6709\u7684\u65cb\u8f6c\u90fd\u53ef\u4ee5\u7528\u4e00\u5957\u56fa\u5b9a\u7684\u6570\u5b66\u89c4\u5219\u63cf\u8ff0\uff0cWigner-D \u77e9\u9635\u9075\u5faa\u8fd9\u4e9b\u89c4\u5219\uff08SO3\u7fa4\u4e0d\u53ef\u7ea6\u8868\u793a\uff0c\u6b27\u62c9\u89d2\u7b49\uff09\u3002\u8fd9\u4e9b\u89c4\u5219\u786e\u4fdd\u4e86\u4efb\u4f55\u91cf\u5b50\u6001\u5728\u65cb\u8f6c\u540e\u90fd\u80fd\u7528\u540c\u6837\u7684\u65b9\u5f0f\u63cf\u8ff0\u3002\u5f53\u4f60\u7528 Wigner-D \u77e9\u9635\u6765\u65cb\u8f6c\u4e00\u4e2a\u91cf\u5b50\u6001\u65f6\uff0c\u5b83\u867d\u7136\u4f1a\u6539\u53d8\u6001\u7684\u5177\u4f53\u8868\u8fbe\uff0c\u4f46\u4e0d\u4f1a\u6539\u53d8\u7cfb\u7edf\u7684\u7269\u7406\u6027\u8d28\u3002\u5c31\u50cf\u4f60\u628a\u4e00\u4e2a\u7403\u65cb\u8f6c\u540e\uff0c\u7403\u4e0a\u7684\u70b9\u6362\u4e86\u4f4d\u7f6e\uff0c\u4f46\u7403\u7684\u5f62\u72b6\u6ca1\u53d8\u3002<\/p>\n \u5177\u4f53\u6765\u8bf4\uff0c\u5047\u8bbe\u4f60\u6709\u4e00\u4e2a\u91cf\u5b50\u6001 $|\\psi\\rangle$ \uff0c\u8868\u793a\u4e3a\u4e00\u4e2a\u5411\u91cf\u3002\u5f53\u4f60\u65cb\u8f6c\u8fd9\u4e2a\u6001\u65f6\uff0c\u6bd4\u5982\u7ed5\u67d0\u4e2a\u8f74\u65cb\u8f6c\uff0c\u65cb\u8f6c\u540e\u7684\u6001\u4f1a\u53d8\u6210\u4e00\u4e2a\u65b0\u6001 $\\left|\\psi^{\\prime}\\right\\rangle$ \u3002\u8fd9\u4e2a\u65b0\u6001\u53ef\u4ee5\u7528Wigner-D\u77e9\u9635 $D$ \u548c\u539f\u6765\u7684\u6001 $|\\psi\\rangle$ \u8868\u793a: \u8fd9\u5c31\u50cf\u662f\u4f60\u7528\u4e00\u4e2a\u51fd\u6570 $f(x)$ \u6765\u63cf\u8ff0\u4e00\u4e2a\u7403\u4e0a\u7684\u70b9\uff0c\u65cb\u8f6c\u540e\u8fd9\u4e9b\u70b9\u53d8\u6210 $f^{\\prime}(x)$ \uff0c\u4f46\u51fd\u6570\u5f62\u5f0f\u4fdd\u6301\u4e00\u6837\uff0c\u53ea\u662f\u8f93\u5165 (\u70b9\u7684\u4f4d\u7f6e) \u53d8\u4e86\u3002\u53cd\u4e4b\uff0c\u5982\u679c\u65cb\u8f6c\u8fd9\u4e2a\u91cf\u5b50\u6001\u800c\u4e0d\u4f7f\u7528 Wigner-D \u77e9\u9635\uff0c\u6211\u4eec\u4e0d\u80fd\u4fdd\u8bc1\u53d8\u6362\u540e\u7684\u91cf\u5b50\u6001 $\\left|\\psi^{\\prime}\\right\\rangle$ \u4e0e\u539f\u6001 $|\\psi\\rangle$ \u4e4b\u95f4\u7684\u5173\u7cfb\u6ee1\u8db3\u65cb\u8f6c\u5bf9\u79f0\u6027\u7684\u8981\u6c42\uff0c\u6bd4\u5982\u7528\u4e00\u4e2a\u968f\u610f\u7684\u77e9\u9635 $M$ \u6765\u53d8\u6362 \u8fd9\u79cd\u53d8\u6362 $M$ \u4e0d\u4e00\u5b9a\u6ee1\u8db3\u65cb\u8f6c\u5bf9\u79f0\u6027\u7684\u8981\u6c42\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u53ef\u80fd\u5bfc\u81f4\u5bf9\u8c61\u53d1\u751f\u5f62\u53d8\u3002\u4ece\u8fd9\u4e2a\u89d2\u5ea6\u89e3\u91ca\uff0cWigner-D \u77e9\u9635\u63d0\u4f9b\u4e86\u4e00\u79cd\u4e25\u683c\u7684\u3001\u6807\u51c6\u5316\u7684\u7b49\u53d8\u6027\uff0c\u5373\u8f93\u5165\u53d8\u4e86\u591a\u5c11\uff0c\u8f93\u51fa\u5c31\u53d8\u4e86\u591a\u5c11\u3002\u800c\u4e0d\u7528 Wigner-D \u77e9\u9635\u53ef\u80fd\u53ea\u80fd\u5b9e\u73b0\u4e00\u79cd\u5e7f\u4e49\u7684\u3001\u4e0d\u4e25\u683c\u7684\u7b49\u53d8\u6027\uff0c\u5373\u8f93\u51fa\u4e5f\u4f1a\u968f\u7740\u8f93\u5165\u7684\u53d8\u6362\u800c\u53d8\u6362\uff0c\u4f46\u8fd9\u4e2a\u53d8\u6362\u5c3a\u5ea6\u4e0d\u4e00\u81f4\uff0c\u53ef\u80fd\u5bfc\u81f4\u539f\u7269\u4f53\u7684\u5f62\u53d8\u3002\u56e0\u6b64\uff0cWigner-D \u77e9\u9635\u5728\u91cf\u5b50\u529b\u5b66\u4e2d\u7684\u91cd\u8981\u6027\u5728\u4e8e\u5b83\u80fd\u591f\u786e\u4fdd\u7cfb\u7edf\u5728\u65cb\u8f6c\u53d8\u6362\u4e2d\u7684\u4e25\u683c\u5bf9\u79f0\u6027\u548c\u7269\u7406\u4e00\u81f4\u6027\u3002<\/p>\n \u5728\u4e0a\u8ff0\u5185\u5bb9\u4e2d\uff0c\u6211\u4eec\u77e5\u9053\uff0c\u7403\u8c10\u51fd\u6570\u53ef\u4ee5\u8868\u5f81\u51e0\u4f55\u7a7a\u95f4\u4e2d\u7684\u5bf9\u8c61\u5f62\u72b6\u548c\u72b6\u6001\u3002\u57fa\u4e8e\u7403\u8c10\u51fd\u6570\u8868\u793a\u7684\u4e0d\u540c\u5bf9\u8c61\u4e4b\u95f4\u7684\u4ea4\u4e92\u9700\u8981\u9075\u5faa\u7403\u8c10\u51fd\u6570\u7684\u4e00\u4e9b\u6027\u8d28\uff0c\u4f8b\u5982\u5bf9\u79f0\u6027\u548c\u6b63\u4ea4\u6027\u3002\u56e0\u6b64\uff0c\u8fd9\u4e9b\u5bf9\u8c61\u4e4b\u95f4\u7684\u4fe1\u606f\u4ea4\u4e92\u4e0e\u878d\u5408\u4e0d\u80fd\u76f4\u63a5\u4f7f\u7528\u5e38\u89c1\u7684\u52a0\u6cd5\u3001\u4e58\u6cd5\u6216\u6c42\u5e73\u5747\u7b49\u5e38\u89c4\u64cd\u4f5c\uff0c\u800c\u662f\u9700\u8981\u501f\u52a9\u514b\u83b1\u5e03\u4ec0-\u6208\u767b\uff08CG\uff09\u7cfb\u6570\u3002CG\u7cfb\u6570\u7684\u4f7f\u7528\u786e\u4fdd\u7279\u5f81\u8026\u5408\u8fc7\u7a0b\u9075\u5faa\u7403\u8c10\u51fd\u6570\u7684\u5bf9\u79f0\u6027\u548c\u6b63\u4ea4\u6027\uff0c\u4ece\u800c\u6700\u5927\u9650\u5ea6\u5730\u4fdd\u7559\u4e86\u539f\u59cb\u4fe1\u606f\u7684\u5b8c\u6574\u6027\u548c\u51c6\u786e\u6027\u3002<\/p>\n \u5177\u4f53\u6765\u8bf4\uff0cCG\u7cfb\u6570\u662f\u5c06\u4e24\u4e2a\u89d2\u52a8\u91cf\uff08\u6216\u66f4\u4e00\u822c\u5730\uff0c\u4e24\u4e2a\u7403\u8c10\u51fd\u6570\uff09\u91cf\u5b50\u6001\u7684\u76f4\u63a5\u79ef (tensor product)\uff0c \u8868\u793a\u4e3a\u4e0d\u540c\u603b\u89d2\u52a8\u91cf\u91cf\u5b50\u6001\u7684\u7ebf\u6027\u7ec4\u5408\u7684\u7cfb\u6570\u3002\u5047\u8bbe\u6211\u4eec\u6709\u4e24\u4e2a\u89d2\u52a8\u91cf $l_1$ \u548c $l_2$ \uff0c\u4ee5\u53ca\u5bf9\u5e94\u7684\u78c1\u91cf\u5b50\u6570 $m_1$ \u548c $m_2$ \uff0c\u5219\u5b83\u4eec\u7684\u76f4\u63a5\u79ef\u53ef\u4ee5\u8868\u793a\u4e3a: \u603b\u89d2\u52a8\u91cf $L$ \u7684\u6001\u53ef\u4ee5\u8868\u793a\u4e3a: CG\u7cfb\u6570 $C_{l_1, m_1 ; l_2, m_2}^{L, M}$ \u662f\u6ee1\u8db3\u4e0b\u5217\u5173\u7cfb\u7684\u7cfb\u6570: \u5bf9\u79f0\u60271\uff1a\u4ea4\u6362 $l_1$ \u548c $l_2$ \u4ee5\u53ca $m_1$ \u548c $m_2$ \u7684\u4f4d\u7f6e: \u5bf9\u79f0\u60272: \u6539\u53d8 $m_1$ \u548c $m_2$ \u7684\u6b63\u8d1f\u7b26\u53f7\uff0c\u540c\u65f6\u603b\u89d2\u52a8\u91cf\u78c1\u91cf\u5b50\u6570 $M$ \u4e5f\u6539\u53d8\u7b26\u53f7: \u6b63\u4ea4\u60271 \u5bf9\u4e8e\u56fa\u5b9a\u7684 $l_1, l_2, l$ \u548c\u4e0d\u540c\u7684\u78c1\u91cf\u5b50\u6570\u7ec4\u5408 $m_1, m_2, M$ : \u6b63\u4ea4\u60272\uff1a \u5bf9\u4e8e\u56fa\u5b9a\u7684 $l_1, l_2, m_1, m_2$ \u548c\u4e0d\u540c\u7684\u603b\u89d2\u52a8\u91cf\u7ec4\u5408 $l, M$ : \u6b63\u4ea4\u6027\u516c\u5f0f\u8868\u660e\uff0c\u5bf9\u4e8e\u56fa\u5b9a\u7684\u89d2\u52a8\u91cf $l_1, l_2$ \u548c\u603b\u89d2\u52a8\u91cf $l$ \uff0c\u4e0d\u540c\u7684\u78c1\u91cf\u5b50\u6570\u7ec4\u5408 $m_1, m_2$ \u7684CG\u7cfb\u6570\u5728\u5185\u79ef\u65f6\u4ea7\u751f\u7684\u7ed3\u679c\u662f\u4e00\u4e2a\u514b\u7f57\u5185\u514b $\\delta$ \u51fd\u6570\u3002\u8fd9\u610f\u5473\u7740\uff0c\u5f53 $l$ \u548c $M$ \u4e0d\u540c\u65f6\uff0c\u5185\u79ef\u4e3a\u96f6 (\u5b83\u4eec\u662f\u6b63\u4ea4\u7684\uff09\uff1b\u5f53 $l=l^{\\prime}$ \u4e14 $M=M^{\\prime}$ \u65f6\uff0c\u5185\u79ef\u4e3a 1 (\u5b83\u4eec\u662f\u6807\u51c6\u5316\u7684)\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u8fd9\u4e2a\u516c\u5f0f\u786e\u4fdd\u4e86\u4e0d\u540c\u603b\u89d2\u52a8\u91cf\u6001 $|l, M\\rangle$ \u5728\u78c1\u91cf\u5b50\u6570\u7ec4\u5408 $\\left|l_1, m_1\\right\\rangle \\otimes\\left|l_2, m_2\\right\\rangle$ \u57fa\u5e95\u4e0a\u7684\u6295\u5f71\u662f\u6b63\u4ea4\u4e14\u6807\u51c6\u5316\u7684\u3002M\u540c\u7406\u3002<\/p>\n \u6b64\u5916\uff0cCG\u7cfb\u6570\u8fd8\u9700\u8981\u6ee1\u8db3\u4e09\u89d2\u4e0d\u7b49\u5f0f\u8fd9\u6765\u6e90\u4e8e\u89d2\u52a8\u91cf\u52a0\u6cd5\u89c4\u5219\u4e2d\u7684\u9009\u62e9\u5b9a\u5219\u3002\u5177\u4f53\u6765\u8bf4\uff0c$\\mathrm{CG}$ \u7cfb\u6570\u53ea\u5728 $\\left|j_1-j_2\\right| \\leq j \\leq j_1+j_2$ \u65f6\u4e0d\u4e3a\u96f6\uff0c\u5e76\u4e14:$m=m_1+m_2$<\/p>\n CG\u7cfb\u6570\u7684\u8ba1\u7b97\u53ef\u4ee5\u901a\u8fc7\u89e3\u6790\u8868\u8fbe\u5f0f\u6216\u67e5\u8868\u83b7\u5f97\u3002\u4e00\u4e2a\u5178\u578b\u7684\u8ba1\u7b97\u516c\u5f0f\u662f:<\/p>\n $$ \u8fd9\u91cc\u7684 $k$ \u662f\u4e00\u4e2a\u6c42\u548c\u53d8\u91cf\uff0c\u5176\u8303\u56f4\u7531\u6bcf\u4e2a\u9879\u7684\u9636\u4e58\u90e8\u5206\u7684\u975e\u8d1f\u6027\u51b3\u5b9a\u3002<\/p>\n \u5728\u6df1\u5ea6\u5b66\u4e60\u4e2d\uff0c\u6211\u4eec\u53ef\u4ee5\u7c7b\u6bd4CG\u7cfb\u6570\u4e3a\u7279\u5f81\u878d\u5408\u8fc7\u7a0b\u4e2d\u7684\u4e00\u79cd\u6743\u91cd\u7cfb\u6570\u3002\u5177\u4f53\u6765\u8bf4\uff0c\u5728\u6df1\u5ea6\u5b66\u4e60\u4e2d\uff0c\u7279\u5f81\u878d\u5408\u6d89\u53ca\u5c06\u6765\u81ea\u4e0d\u540c\u5c42\u6216\u4e0d\u540c\u6e90\u7684\u7279\u5f81\u8fdb\u884c\u7ec4\u5408\uff0c\u4ee5\u5f62\u6210\u66f4\u5177\u8868\u8fbe\u529b\u7684\u7279\u5f81\u8868\u793a\u3002\u7c7b\u4f3c\u4e8eCG\u7cfb\u6570\u5728\u89d2\u52a8\u91cf\u8026\u5408\u4e2d\u7684\u4f5c\u7528\uff0c\u8fd9\u4e9b\u878d\u5408\u8fc7\u7a0b\u9700\u8981\u9075\u5faa\u4e00\u5b9a\u7684\u5bf9\u79f0\u6027\u548c\u6b63\u4ea4\u6027\u539f\u5219\uff0c\u4ee5\u786e\u4fdd\u4fe1\u606f\u878d\u5408\u7684\u6709\u6548\u6027\u3002<\/p>\n \u5bf9\u79f0\u6027\uff1a\u5728\u7279\u5f81\u878d\u5408\u8fc7\u7a0b\u4e2d\uff0c\u5bf9\u79f0\u6027\u53ef\u4ee5\u7c7b\u6bd4\u4e3a\u7279\u5f81\u7ec4\u5408\u7684\u987a\u5e8f\u4e0d\u5f71\u54cd\u6700\u7ec8\u7ed3\u679c\u3002\u4f8b\u5982\uff0c\u7ed3\u5408\u56fe\u50cf\u7279\u5f81\u548c\u6587\u672c\u7279\u5f81\u7684\u591a\u6a21\u6001\u6a21\u578b\u4e2d\uff0c\u5148\u5904\u7406\u56fe\u50cf\u7279\u5f81\u518d\u5904\u7406\u6587\u672c\u7279\u5f81\uff0c\u6216\u5148\u5904\u7406\u6587\u672c\u7279\u5f81\u518d\u5904\u7406\u56fe\u50cf\u7279\u5f81\uff0c\u90fd\u5e94\u5f97\u5230\u76f8\u540c\u7684\u878d\u5408\u7ed3\u679c\u3002<\/p>\n \u6b63\u4ea4\u6027\uff1a\u7279\u5f81\u878d\u5408\u8fc7\u7a0b\u4e2d\u7684\u6b63\u4ea4\u6027\u786e\u4fdd\u4e0d\u540c\u7279\u5f81\u5728\u878d\u5408\u65f6\u4e0d\u4e22\u5931\u5404\u81ea\u7684\u72ec\u7acb\u6027\uff0c\u907f\u514d\u5197\u4f59\u4fe1\u606f\u7684\u51fa\u73b0\u3002\u6b63\u5982CG\u7cfb\u6570\u786e\u4fdd\u4e0d\u540c\u89d2\u52a8\u91cf\u6001\u4e4b\u95f4\u7684\u6b63\u4ea4\u6027\uff0c\u6df1\u5ea6\u5b66\u4e60\u4e2d\u7684\u7279\u5f81\u878d\u5408\u5e94\u5c3d\u91cf\u4fdd\u6301\u8f93\u5165\u7279\u5f81\u7684\u72ec\u7acb\u6027\u548c\u4e92\u8865\u6027\u3002<\/p>\n \u5bf9\u4e8e\u4e0a\u8ff0\u4ee3\u7801\uff0c\u7b80\u5355\u89e3\u91ca\u4e00\u4e0bcg_coeff \u00d7 feature1 + (1 - cg_coeff) \u00d7 feature2\u7684\u4f5c\u7528\u3002<\/p>\n \u9996\u5148\uff0c\u76f4\u63a5\u5c06\u4e24\u4e2a\u7279\u5f81\u7b80\u5355\u76f8\u52a0\u53ef\u80fd\u4f1a\u5bfc\u81f4\u4ee5\u4e0b\u95ee\u9898\uff1a<\/p>\n \u56e0\u6b64CG\u7cfb\u6570\u672c\u8eab\u7684\u6b63\u4ea4\u6027\u53ef\u4ee5\u907f\u514d\u4ea4\u4e92\u65f6\u7684\u4fe1\u606f\u5197\u4f59\uff0c\u56e0\u6b64\u5728\u7279\u5f81\u878d\u5408\u4e2d\uff0c\u6211\u4eec\u501f\u7528CG\u7cfb\u6570\u7684\u6982\u5ff5\uff0c\u901a\u8fc7\u4f7f\u7528\u6743\u91cd\u7cfb\u6570\u6765\u786e\u4fdd\u4e0d\u540c\u7279\u5f81\u7684\u72ec\u7acb\u6027\uff0c\u907f\u514d\u4fe1\u606f\u5197\u4f59\u3002\u4f46\u662f\uff0c\u4ec5\u901a\u8fc7\u7ebf\u6027\u7ec4\u5408\u7684\u65b9\u5f0fcg_coeff feature1 + (1 - cg_coeff) <\/em> feature2 \u5e76\u4e0d\u80fd\u5b8c\u5168\u4fdd\u8bc1\u6b63\u4ea4\u6027\uff0c\u5b83\u53ea\u662f\u4e00\u4e2a\u501f\u9274\u6b63\u4ea4\u6027\u601d\u60f3\u7684\u5c1d\u8bd5\u3002<\/p>\n 1.$V_1$\u548c$V_2$<\/strong><\/p>\n \u7531\u4e8e $V_1$ \u4ee3\u8868\u7684\u662f\u89d2\u52a8\u91cf $l_1$ \u7684\u7279\u5f81\uff0c\u6839\u636e\u7403\u8c10\u51fd\u6570\uff0c\u5176\u7279\u5f81\u7ef4\u5ea6\u6570\u4e3am\u7684\u53d6\u503c\u8303\u56f4\uff1a $2l_1 + 1 = 2 \\times 1 + 1 = 3$\uff0c\u56e0\u6b64\u7b2c\u4e00\u7ef4\u5ea6\u4e3a3\u3002C1=4 \u8868\u793a\u901a\u9053\u6570 $C_1$ \u4e3a4\uff0c\u8fd9\u662f\u7279\u5f81\u77e9\u9635\u7684\u7b2c\u4e8c\u7ef4\u5ea6\uff0c\u8868\u793a\u67094\u4e2a\u901a\u9053\u3002<\/p>\n \u540c\u7406\uff0c\u7531\u4e8e $V2$ \u4ee3\u8868\u7684\u662f\u89d2\u52a8\u91cf $l_2$ \u7684\u7279\u5f81\uff0c\u5176\u7279\u5f81\u7ef4\u5ea6\u6570\u4e3a $2l_2 + 1 = 2 \\times 2 + 1 = 5$\uff0c\u56e0\u6b64\u7b2c\u4e00\u7ef4\u5ea6\u4e3a5\u3002C2=6 \u8868\u793a\u901a\u9053\u6570 $C_2$ \u4e3a6\u3002\u8fd9\u662f\u7279\u5f81\u77e9\u9635\u7684\u7b2c\u4e8c\u7ef4\u5ea6\uff0c\u8868\u793a\u67096\u4e2a\u901a\u9053\u3002<\/p>\n 2.\u8f93\u51fa\u7ed3\u679c ' $V$ '<\/strong><\/p>\n \u5f62\u72b6\u4e3a ' $(4,6,8)$ ' \u7684\u77e9\u9635\u3002<\/p>\n \u5728\u51fd\u6570 \u5176\u4e2d$l = (l1 + l2) = 1 + 2 = 3$ \uff0c\u56e0\u6b64\u8f93\u51fa\u7279\u5f81\u77e9\u9635\u7684\u7b2c\u4e00\u4e2a\u7ef4\u5ea6\u4e3a $2l + 1 = 2 \\times 3 + 1 = 7$\u3002 3.\u53ef\u5b66\u4e60\u53c2\u6570\u77e9\u9635 ' $w$ '<\/strong><\/p>\n $W[c_1, c_2, c]$\uff1a 4.CG\u7cfb\u6570 Q'<\/strong><\/p>\n Q\u662fCG\u7cfb\u6570\uff0c$Q \\cdot V 1 \\cdot V 2$\u786e\u4fdd\u4e86\u53ea\u6709\u5728\u7279\u5b9a\u89d2\u52a8\u91cf$l$\u8026\u5408\u89c4\u5219\u4e0b\uff08\u5bf9\u79f0\u6027\uff0c\u6b63\u4ea4\u6027\uff0c\u4e09\u89d2\u4e0d\u7b49\u5f0f\uff09\uff0c\u7279\u5f81\u624d\u80fd\u8fdb\u884c\u6709\u6548\u7684\u4ea4\u4e92\u548c\u878d\u5408\uff08\u4e0d\u7b26\u5408\u89d2\u52a8\u91cf$l$\u8026\u5408\u89c4\u5219\u65f6\uff0cQ\u4e3a0\uff09\uff0c\u4ece\u800c\u907f\u514d\u4e86\u4e0d\u76f8\u5173\u7279\u5f81\u4e4b\u95f4\u7684\u76f8\u4e92\u5e72\u6270\u3002<\/p>\n \u5728\u6781\u5750\u6807\u7684\u80cc\u666f\u4e0b, \u7ed3\u5408\u4e4b\u524d\u5173\u4e8e\u7403\u8c10\u51fd\u6570\u7684\u8bb2\u89e3, \u53ef\u4ee5\u53d1\u73b0\u5176\u53ea\u80fd\u8868\u5f81\u7403\u5750\u6807\u7cfb\u4e0b\u7684\u6781\u89d2 $\\theta$ \u548c\u65b9\u4f4d\u89d2 $\\phi$ \u4fe1\u606f\u3002\u5176\u516c\u5f0f\u8868\u793a\u4e5f\u53ef\u4ee5\u8bc1\u660e\u8fd9\u4e00\u70b9: $Y_l^m(\\theta, \\phi)$ \u3002<\/p>\n \u56e0\u6b64, \u5982\u679c\u60f3\u5bf9\u6781\u5750\u6807\u7cfb\u4e0b\u5168\u90e8\u7684\u7269\u7406\u91cf (\u8ddd\u79bb $r$, \u6781\u89d2 $\\theta$ \u548c\u65b9\u4f4d\u89d2 $\\phi$ ) \u8fdb\u884c\u5b8c\u6574\u7684\u63cf\u8ff0\uff0c\u8fd9\u91cc\u8fd8\u9700\u8981\u4e00\u4e2a\u8868\u5f81\u8ddd\u79bb\u7684\u6570\u5b66\u6a21\u578b\uff0c\u5373\u5f84\u5411\u6a21\u578b\u3002\u8fd9\u662f\u4e00\u79cd\u7528\u4e8e\u5206\u6790\u6216\u63cf\u8ff0\u7a7a\u95f4\u5bf9\u8c61\uff08\u5982\u5206\u5b50\u3001\u539f\u5b50\u3001\u7c92\u5b50\u6216\u5929\u4f53\uff09\u4e2d\u5fc3\u5411\u5916\u5ef6\u5c55\u7279\u6027\u7684\u6570\u5b66\u6a21\u578b\uff0c\u7528\u4e8e\u7406\u89e3\u548c\u9884\u6d4b\u5bf9\u8c61\u7684\u884c\u4e3a\u548c\u6027\u8d28\u3002\u5f84\u5411\u6a21\u578b\u7684\u5173\u952e\u5728\u4e8e\u5b83\u63d0\u4f9b\u4e86\u4e00\u79cd\u7b80\u5316\u4f46\u5f3a\u5927\u7684\u65b9\u5f0f\u6765\u5206\u6790\u548c\u7406\u89e3\u5177\u6709\u7403\u5f62\u5bf9\u79f0\u6027\u6216\u8fd1\u4f3c\u5bf9\u79f0\u6027\u7cfb\u7edf\u7684\u7269\u7406\u6027\u8d28\u3002\u901a\u8fc7\u5c06\u590d\u6742\u7684\u4e09\u7ef4\u95ee\u9898\u7b80\u5316\u4e3a\u4e00\u7ef4\u95ee\u9898\uff08\u7a7a\u95f4\u5206\u5e03\u7b80\u5316\u4e3a\u8ddd\u79bb\u8868\u793a\uff09\uff0c\u7814\u7a76\u4eba\u5458\u53ef\u4ee5\u66f4\u5bb9\u6613\u5730\u83b7\u5f97\u7cfb\u7edf\u6027\u8d28\u7684\u6df1\u5165\u6d1e\u89c1\uff0c\u5e76\u5f00\u53d1\u51fa\u51c6\u786e\u7684\u7406\u8bba\u6a21\u578b\u548c\u6570\u503c\u6a21\u62df\u65b9\u6cd5\u3002<\/p>\n \u5f84\u5411\u6a21\u578b\u901a\u5e38\u6d89\u53ca\u4e00\u4e2a\u6216\u591a\u4e2a\u5f84\u5411\u51fd\u6570\uff0c\u8fd9\u4e9b\u51fd\u6570\u8868\u8fbe\u4e86\u67d0\u4e2a\u6216\u67d0\u4e9b\u7269\u7406\u91cf\u5982\u4f55\u968f\u7740\u8ddd\u79bb\u4e2d\u5fc3\u70b9\u7684\u8ddd\u79bb\u800c\u53d8\u5316\u3002\u8fd9\u79cd\u63cf\u8ff0\u53ef\u4ee5\u662f\u7b80\u5355\u7684\u89e3\u6790\u51fd\u6570\uff0c\u4e5f\u53ef\u4ee5\u662f\u590d\u6742\u7684\u6570\u503c\u6a21\u578b\uff0c\u5177\u4f53\u53d6\u51b3\u4e8e\u7814\u7a76\u5bf9\u8c61\u7684\u6027\u8d28\u548c\u6240\u9700\u7684\u7cbe\u786e\u5ea6\u3002\u4e00\u4e2a\u5e38\u89c1\u7684\u5f84\u5411\u51fd\u6570\u4e3a\u9ad8\u65af\u5f84\u5411\u51fd\u6570\u3002<\/p>\n \u5148\u901a\u4fd7\u7684\u7406\u89e3\u4e00\u4e9b\u9ad8\u65af\u5f84\u5411\u51fd\u6570\u7684\u4f5c\u7528\u3002\u60f3\u8c61\u91cf\u5b50\u529b\u5b66\u7684\u573a\u666f\uff0c\u6bcf\u4e2a\u539f\u5b50\u6216\u5206\u5b50\u90fd\u88ab\u4e00\u56e2\u4e91\u96fe\u5305\u56f4\uff0c\u8fd9\u56e2\u4e91\u96fe\u4ee3\u8868\u7740\u7535\u8377\u7684\u5206\u5e03\u3002\u5728\u73b0\u5b9e\u4e16\u754c\u4e2d\uff0c\u7535\u8377\u5e76\u4e0d\u662f\u96c6\u4e2d\u5728\u4e00\u4e2a\u70b9\u4e0a\uff0c\u800c\u662f\u5206\u5e03\u5728\u4e00\u4e2a\u533a\u57df\u5185\u3002\u9ad8\u65af\u5f84\u5411\u51fd\u6570\u5c31\u662f\u7528\u6765\u63cf\u8ff0\u8fd9\u79cd\u201c\u4e91\u96fe\u201d\u6216\u7535\u8377\u5206\u5e03\u7684\u6570\u5b66\u5de5\u5177\u3002\u8fd9\u79cd\u4e91\u96fe\u7684\u5f62\u72b6\u7c7b\u4f3c\u4e8e\u9ad8\u65af\u51fd\u6570\u8fd9\u79cd\u949f\u5f62\u66f2\u7ebf\uff0c\u5b83\u5728\u539f\u5b50\u6216\u5206\u5b50\u7684\u4e2d\u5fc3\u6700\u5bc6\u96c6\uff0c\u5411\u5916\u6269\u6563\u65f6\u9010\u6e10\u53d8\u5f97\u7a00\u758f\u3002<\/p>\n \u73b0\u5728\uff0c\u5f53\u6211\u4eec\u8c08\u8bba\u4e24\u4e2a\u539f\u5b50\u6216\u5206\u5b50\u4e4b\u95f4\u7684\u76f8\u4e92\u4f5c\u7528\u65f6\uff0c\u53ef\u4ee5\u60f3\u8c61\u5b83\u4eec\u5404\u81ea\u7684\u4e91\u96fe\u5f00\u59cb\u76f8\u4e92\u91cd\u53e0\u3002\u9ad8\u65af\u955c\u50cf\u51fd\u6570\u5e2e\u52a9\u6211\u4eec\u8ba1\u7b97\u8fd9\u79cd\u91cd\u53e0\u7684\u7a0b\u5ea6\uff0c\u4ece\u800c\u4e86\u89e3\u539f\u5b50\u4e4b\u95f4\u7684\u5438\u5f15\u6216\u6392\u65a5\u529b\u5927\u5c0f\u3002\u901a\u8fc7\u5206\u6790\u8fd9\u4e9b\u4e91\u96fe\uff08\u5373\u7535\u8377\u5206\u5e03\uff09\u5982\u4f55\u968f\u7740\u8ddd\u79bb\u53d8\u5316\u800c\u6539\u53d8\uff0c\u6211\u4eec\u53ef\u4ee5\u5efa\u7acb\u4e00\u4e2a\u6a21\u578b\u6765\u63cf\u8ff0\u8fd9\u79cd\u76f8\u4e92\u4f5c\u7528\u7684\u5f3a\u5ea6\u3002 \u9ad8\u65af\u51fd\u6570\u7684\u4e00\u822c\u5f62\u5f0f\u4e3a $G(r)=A \\exp \\left(-\\alpha r^2\\right)$, \u5176\u4e2d $r$ \u662f\u4ece\u7535\u8377\u5206\u5e03\u4e2d\u5fc3\u5230\u67d0\u70b9\u7684\u8ddd\u79bb, $A$\u662f\u632f\u5e45\uff08\u6216\u9ad8\u5ea6\uff09\u7cfb\u6570, \u800c $\\alpha$ \u662f\u4e00\u4e2a\u6b63\u5e38\u6570, \u63a7\u5236\u7740\u9ad8\u65af\u51fd\u6570\u7684\u5bbd\u5ea6\u6216\u6269\u6563\u7a0b\u5ea6\u3002<\/p>\n $A$ \u5728\u5f53\u524d\u573a\u666f\u4e0b\u53ef\u4ee5\u770b\u4f5c\u7535\u8377\u5206\u5e03\u603b\u91cf\u3002<\/p>\n $\\alpha$ \u503c\u8d8a\u5927\uff1a\u5f53 $\\alpha$ \u7684\u503c\u589e\u5927\u65f6, $-\\alpha r^2$ \u7684\u503c\uff08\u5bf9\u4e8e\u4efb\u4f55\u7ed9\u5b9a\u7684 $r>0$ \uff09\u4f1a\u53d8\u5f97\u66f4\u8d1f, \u4f7f\u5f97 $\\exp \\left(-\\alpha r^2\\right)$ \u8fd9\u90e8\u5206\u5feb\u901f\u8d8b\u8fd1\u4e8e\u96f6\u3002\u8fd9\u610f\u5473\u7740\u7535\u8377\u5206\u5e03\u8fc5\u901f\u968f\u8ddd\u79bb\u51cf\u5c0f, \u5bfc\u81f4\u7535\u8377\u4e3b\u8981\u96c6\u4e2d\u5728\u4e2d\u5fc3\u9644\u8fd1, \u5f62\u6210\u4e00\u4e2a\u66f4 \u201c\u5c16\u9510\u201d \u548c\u96c6\u4e2d\u7684\u5206\u5e03\u3002<\/p>\n<\/li>\n $\\alpha$ \u503c\u8d8a\u5c0f\uff1a\u76f8\u53cd\uff0c\u5f53 $\\alpha$ \u8f83\u5c0f\u65f6, $-\\alpha r^2$ \u7684\u8d1f\u503c\u51cf\u5c0f, \u4f7f\u5f97 $\\exp \\left(-\\alpha r^2\\right)$ \u8fd9\u90e8\u5206\u4e0b\u964d\u5f97\u66f4\u6162,\u7535\u8377\u5206\u5e03\u5728\u7a7a\u95f4\u4e2d\u66f4\u52a0 \u201c\u6241\u5e73\u201d \u548c\u5206\u6563\u3002<\/p>\n<\/li>\n<\/ul>\n $r^2$ \u662f\u8ddd\u79bb\u7684\u5e73\u65b9, \u662f\u9ad8\u65af\u51fd\u6570\u4e2d\u786e\u4fdd\u7535\u8377\u5206\u5e03\u5bf9\u79f0\u6027\u7684\u5173\u952e\u90e8\u5206\u3002\u8fd9\u91cc\u7684 $r$ \u8868\u793a\u4ece\u7535\u8377\u5206\u5e03\u7684\u4e2d\u5fc3 (\u901a\u5e38\u662f\u539f\u5b50\u6838\u6216\u5206\u5b50\u7684\u4e2d\u5fc3) \u5230\u5206\u5e03\u5916\u4efb\u610f\u70b9\u7684\u76f4\u7ebf\u8ddd\u79bb\u3002\u7531\u4e8e $r^2$ \u540c\u65f6\u8003\u8651\u4e86\u6240\u6709\u65b9\u5411\u4e0a\u7684\u8ddd\u79bb\uff08\u65e0\u8bba\u662f $\\mathrm{x} \u3001 \\mathrm{y}$ \u8fd8\u662f $\\mathrm{z}$ \u65b9\u5411\uff09, \u5b83\u786e\u4fdd\u4e86\u5206\u5e03\u7684\u7403\u5f62\u5bf9\u79f0\u6027\u4e00\u4e00\u65e0\u8bba\u89c2\u5bdf\u70b9\u5728\u4e2d\u5fc3\u70b9\u7684\u54ea\u4e2a\u65b9\u5411\u4e0a, \u53ea\u8981\u8ddd\u79bb\u76f8\u7b49, \u7535\u8377\u5206\u5e03\u7684\u5bc6\u5ea6\u5c31\u76f8\u540c\u3002<\/p>\n \u7b80\u800c\u8a00\u4e4b, $r^2$ \u786e\u4fdd\u4e86\u7535\u8377\u5206\u5e03\u4e0d\u4f9d\u8d56\u4e8e\u65b9\u5411, \u53ea\u4f9d\u8d56\u4e8e\u4e0e\u4e2d\u5fc3\u7684\u5f84\u5411\u8ddd\u79bb, \u4ece\u800c\u521b\u9020\u4e86\u4e00\u4e2a\u5b8c\u7f8e\u7684\u7403\u5f62\u5bf9\u79f0\u5206\u5e03\u3002\u8fd9\u79cd\u5bf9\u79f0\u6027\u662f\u5206\u6790\u548c\u8ba1\u7b97\u7535\u8377\u5206\u5e03\u53ca\u5176\u76f8\u4e92\u4f5c\u7528\u65f6\u975e\u5e38\u91cd\u8981\u7684\u7279\u6027\uff0c\u56e0\u4e3a\u5b83\u7b80\u5316\u4e86\u6570\u5b66\u8868\u8fbe\u5f0f\uff0c\u540c\u65f6\u63d0\u4f9b\u4e86\u5bf9\u7269\u7406\u73b0\u8c61\u7684\u76f4\u89c2\u7406\u89e3\u3002<\/p>\n \u5f53\u6211\u4eec\u8003\u8651\u4e24\u4e2a\u7535\u8377\u5206\u5e03\uff08\u4f8b\u5982\uff0c\u4e24\u4e2a\u539f\u5b50\u6216\u5206\u5b50\uff09\u4e4b\u95f4\u7684\u76f8\u4e92\u4f5c\u7528\u65f6\uff0c\u6211\u4eec\u4f1a\u8ba1\u7b97\u8fd9\u4e24\u4e2a\u9ad8\u65af\u51fd\u6570\u7684\u91cd\u53e0\u90e8\u5206\u3002\u8fd9\u79cd\u91cd\u53e0\u90e8\u5206\u7684\u8ba1\u7b97\u6d89\u53ca\u5230\u5bf9\u4e24\u4e2a\u9ad8\u65af\u51fd\u6570\u7684\u4e58\u79ef\u8fdb\u884c\u79ef\u5206\uff0c\u6700\u7ec8\u7ed9\u51fa\u4e86\u4e00\u4e2a\u63cf\u8ff0\u76f8\u4e92\u4f5c\u7528\u80fd\u91cf\u968f\u8ddd\u79bb\u53d8\u5316\u7684\u51fd\u6570\u3002\u8fd9\u5c31\u662f\u6240\u8c13\u7684\u5f84\u5411\u6a21\u578b\uff0c\u5b83\u80fd\u5e2e\u52a9\u6211\u4eec\u4e86\u89e3\u5728\u4e0d\u540c\u8ddd\u79bb\u4e0b\uff0c\u8fd9\u4e9b\u539f\u5b50\u6216\u5206\u5b50\u4e4b\u95f4\u7684\u76f8\u4e92\u4f5c\u7528\u5f3a\u5ea6\u5982\u4f55\u53d8\u5316\uff0c\u5982\u4e0b\u6240\u793a\uff1a<\/p>\n \u5b9a\u4e49$G_1(r)$ \u548c $G_2(r)$ \u7684\u91cd\u53e0\u79ef\u5206\u516c\u5f0f\u5982\u4e0b: \u5176\u4e2d\uff0c\u5047\u8bbe\u4e24\u4e2a\u9ad8\u65af\u51fd\u6570\u5177\u6709\u76f8\u540c\u7684\u5f62\u5f0f\uff0c\u4f46\u4e2d\u5fc3\u4f4d\u7f6e\u4e0d\u540c: \u8fd9\u91cc\uff0c $d$ \u662f\u4e24\u4e2a\u9ad8\u65af\u5206\u5e03\u4e2d\u5fc3\u4e4b\u95f4\u7684\u8ddd\u79bb\u3002<\/p>\n \n \u9664\u4e86\u9ad8\u65af\u51fd\u6570\u5916\uff0c\u8d1d\u585e\u5c14\u51fd\u6570\u4e5f\u7ecf\u5e38\u88ab\u7528\u4e8e\u5f84\u5411\u6a21\u578b\u7684\u5efa\u6a21\u3002\u8d1d\u585e\u5c14\u65b9\u7a0b\u7684\u4e00\u822c\u5f62\u5f0f\u662f\uff1a<\/p>\n $$ \u5176\u4e2d:<\/p>\n \u8d1d\u585e\u5c14\u65b9\u7a0b\u7684\u89e3, \u5373\u8d1d\u585e\u5c14\u51fd\u6570, \u901a\u5e38\u8868\u793a\u4e3a $J_n(x)$ \u548c $Y_n(x)$, \u5206\u522b\u5bf9\u5e94\u7b2c\u4e00\u7c7b\u548c\u7b2c\u4e8c\u7c7b\u8d1d\u585e\u5c14\u51fd\u6570:<\/p>\n \u5176\u4e2d\uff0c\u7b2c\u4e00\u7c7b\u8d1d\u585e\u5c14\u51fd\u6570\uff0c\u5728\u4e09\u7ef4\u7a7a\u95f4\u4e2d\u5f62\u6210\u4e86\u968f\u7740\u8ddd\u79bb\u589e\u52a0\u800c\u51fa\u73b0\u632f\u8361\u8870\u51cf\u7684\u7279\u6027\u66f2\u9762\u3002\u8fd9\u79cd\u632f\u8361\u8870\u51cf\u7684\u884c\u4e3a\u9002\u5408\u4e8e\u63cf\u8ff0\u6ce2\u52a8\u548c\u632f\u52a8\u7b49\u7269\u7406\u73b0\u8c61\uff0c\u4ee5\u53ca\u4e00\u4e9b\u590d\u6742\u7684\u5206\u5b50\u95f4\u7684\u76f8\u4e92\u4f5c\u7528\u3002\u5728\u5904\u7406\u5177\u6709\u7279\u5b9a\u5bf9\u79f0\u6027\u7684\u7269\u7406\u95ee\u9898\u65f6\u66f4\u4e3a\u5e38\u89c1\uff0c\u5982\u5728\u5206\u5b50\u52a8\u529b\u5b66\u6a21\u62df\u548c\u7535\u78c1\u573a\u8ba1\u7b97\u4e2d\uff0c\u5176\u4e2d\u9700\u8981\u51c6\u786e\u63cf\u8ff0\u6ce2\u52a8\u548c\u632f\u52a8\u73b0\u8c61\u3002\u5982\u4e0b\u56fe\u6240\u793a\uff1a<\/p>\n \n \n \u8d1d\u585e\u5c14\u51fd\u6570\u7279\u522b\u662f\u96f6\u9636\u4fee\u6b63\u8d1d\u585e\u5c14\u51fd\u6570\uff08Modified Bessel function of the first kind\uff09\u901a\u5e38\u5728\u5f84\u5411\u5206\u5e03\u4e2d\u8868\u73b0\u4e3a\u5355\u8c03\u9012\u589e\u6216\u5355\u8c03\u9012\u51cf\uff0c\u800c\u4e0d\u662f\u9707\u8361\u6027\u8d28\u3002\u82e5\u9700\u8981\u4f53\u73b0\u9707\u8361\u6027\u8d28\uff0c\u53ef\u4ee5\u4f7f\u7528\u8d1d\u585e\u5c14\u51fd\u6570\u7684\u53e6\u4e00\u79cd\u5f62\u5f0f\u2014\u2014\u4f8b\u5982\uff0c\u7b2c\u4e00\u7c7b\u7b2c\u4e00\u79cd\u8d1d\u585e\u5c14\u51fd\u6570\uff08Bessel function of the first kind\uff09<\/p>\n \n \u5bf9\u6bd4\u4e4b\u4e0b\uff0c\u9ad8\u65afRBF\u5177\u6709\u4ee5\u4e0b\u7279\u6027\uff1a<\/p>\n \u5728\u5f84\u5411\u6a21\u578b\u4e2d\u4f7f\u7528\u7684\u7b2c\u4e00\u7c7b\u8d1d\u585e\u5c14\u51fd\u6570\uff0c\u5177\u6709\u4ee5\u4e0b\u7279\u6027\uff1a<\/p>\n \u5728\u91cf\u5b50\u529b\u5b66\u4e2d\uff0c\u539f\u5b50\u548c\u5206\u5b50\u7684\u6ce2\u51fd\u6570\u901a\u5e38\u5728\u7403\u5750\u6807\u7cfb\u4e0b\u89e3\u51b3\uff0c\u7279\u522b\u662f\u5bf9\u4e8e\u5177\u6709\u7403\u5f62\u6216\u5706\u67f1\u5f62\u5bf9\u79f0\u6027\u7684\u7cfb\u7edf\u3002\u8d1d\u585e\u5c14\u51fd\u6570\u53ef\u4ee5\u7528\u4f5c\u6ce2\u51fd\u6570\u7684\u5f84\u5411\u90e8\u5206\uff0c\u63cf\u8ff0\u7535\u5b50\u76f8\u5bf9\u4e8e\u539f\u5b50\u6838\u7684\u6982\u7387\u5206\u5e03\u3002\u8fd9\u79cd\u6982\u7387\u5206\u5e03\u7684\u201c\u632f\u8361\u201d\u5bf9\u4e8e\u7406\u89e3\u5316\u5b66\u952e\u7684\u5f62\u6210\u3001\u7535\u5b50\u7684\u6392\u5e03\u4ee5\u53ca\u539f\u5b50\u548c\u5206\u5b50\u7684\u7a33\u5b9a\u6027\u81f3\u5173\u91cd\u8981\u3002<\/p>\n \u603b\u7684\u6765\u8bf4\uff0c\u4e00\u822c\u5f62\u5f0f\u5730\u5f84\u5411\u57fa\u51fd\u6570\u53ef\u4ee5\u5b9a\u4e49\u4e3a\uff1a \u5176\u4e2d, $x$ \u4ee3\u8868\u7a7a\u95f4\u4e2d\u7684\u4efb\u610f\u70b9, $c$ \u662f\u5f84\u5411\u57fa\u51fd\u6570\u7684\u4e2d\u5fc3\u70b9, $|x-c|$ \u8868\u793a $x$ \u4e0e $c$ \u4e4b\u95f4\u7684\u8ddd\u79bb\u3002\u8fd9\u91cc\u7684\u8ddd\u79bb\u53ef\u4ee5\u662f\u6b27\u51e0\u91cc\u5f97\u8ddd\u79bb, \u4e5f\u53ef\u4ee5\u662f\u975e\u6b27\u51e0\u91cc\u5f97\u7a7a\u95f4\u4e2d\u7684\u5176\u4ed6\u5408\u9002\u7684\u8ddd\u79bb\u5ea6\u91cf\u3002 \u5176\u4e2d, $\\sigma$ \u662f\u4e00\u4e2a\u53c2\u6570, \u51b3\u5b9a\u4e86\u51fd\u6570\u7684\u5bbd\u5ea6, \u5373\u5f71\u54cd\u8303\u56f4\u7684\u5927\u5c0f; $|x-c|^2$ \u8868\u793a $x$ \u4e0e\u4e2d\u5fc3\u70b9 $c$ \u4e4b\u95f4\u8ddd\u79bb\u7684\u5e73\u65b9\u3002<\/p>\n \u5bf9\u4e8e\u975e\u6b27\u51e0\u91cc\u5f97\u7a7a\u95f4\uff0c\u8ddd\u79bb\u9700\u8981\u6839\u636e\u8be5\u7a7a\u95f4\u7684\u51e0\u4f55\u7279\u6027\u6765\u5b9a\u4e49\uff0c\u53ef\u80fd\u6d89\u53ca\u66f4\u590d\u6742\u7684\u8ddd\u79bb\u8ba1\u7b97\u65b9\u6cd5\u3002\u65e0\u8bba\u54ea\u79cd\u7a7a\u95f4\uff0c\u5f84\u5411\u57fa\u51fd\u6570\u7684\u6838\u5fc3\u601d\u60f3\u662f\u4fdd\u6301\u4e0d\u53d8\u7684\uff0c\u5373\u5176\u503c\u4ec5\u53d6\u51b3\u4e8e\u8ba1\u7b97\u70b9\u4e0e\u53c2\u7167\u70b9\uff08\u4e00\u822c\u4e3a\u4e2d\u5fc3\u7684\uff09\u7684\u76f8\u5bf9\u8ddd\u79bb\uff0c\u8fd9\u4f7f\u5f97\u5b83\u4eec\u5728\u591a\u79cd\u9886\u57df\u5185\uff0c\u5982\u63d2\u503c\u3001\u51fd\u6570\u903c\u8fd1\u3001\u673a\u5668\u5b66\u4e60\u7b49\uff0c\u90fd\u6709\u7740\u5e7f\u6cdb\u7684\u5e94\u7528\u3002<\/p>\n \u7403\u8c10\u51fd\u6570\u548c\u5f84\u5411\u6a21\u578b\u5728\u51e0\u4f55\u6df1\u5ea6\u5b66\u4e60\u4e2d\u7684\u7ed3\u5408\u4e3b\u8981\u4f53\u73b0\u5728\u4e24\u65b9\u9762\uff1a<\/p>\n \u5728\u6570\u636e\u9884\u5904\u7406\u9636\u6bb5\uff0c\u7403\u8c10\u51fd\u6570\u7528\u4e8e\u5bf9\u51e0\u4f55\u7ed3\u6784\u7684\u9891\u57df\u7279\u5f81\u8fdb\u884c\u63d0\u53d6\u3002\u5177\u4f53\u6765\u8bf4\uff0c\u9996\u5148\u5c06\u6570\u636e\u4ece\u7b1b\u5361\u5c14\u5750\u6807\u7cfb\uff08xyz\uff09\u8f6c\u6362\u5230\u7403\u5750\u6807\u7cfb\uff0c\u5f97\u5230\u6bcf\u4e2a\u70b9\u7684\u5f84\u5411\u8ddd\u79bb\u3001\u4fef\u4ef0\u89d2\u548c\u65b9\u4f4d\u89d2\u3002\u7136\u540e\uff0c\u57fa\u4e8e\u4fef\u4ef0\u89d2\u548c\u65b9\u4f4d\u89d2\uff0c\u53ef\u4ee5\u5229\u7528\u7403\u8c10\u51fd\u6570\u5bf9\u7403\u9762\u4e0a\u7684\u6570\u636e\u8fdb\u884c\u5c55\u5f00\u3002\u8fd9\u4e2a\u8fc7\u7a0b\u53ef\u4ee5\u770b\u4f5c\u662f\u5728\u7403\u9762\u4e0a\u5bf9\u51fd\u6570\u8fdb\u884c\u4e00\u7cfb\u5217\u7684\u9891\u57df\u5206\u89e3\uff0c\u7c7b\u4f3c\u4e8e\u5085\u91cc\u53f6\u53d8\u6362\u5728\u65f6\u95f4\u5e8f\u5217\u4e0a\u7684\u5e94\u7528\u3002\u6bcf\u4e2a\u7403\u8c10\u51fd\u6570\u90fd\u53ef\u4ee5\u6355\u83b7\u7279\u5b9a\u9891\u7387\u7684\u7a7a\u95f4\u53d8\u5316\uff0c\u4ece\u800c\u5bf9\u590d\u6742\u7684\u4e09\u7ef4\u5f62\u72b6\u8fdb\u884c\u7f16\u7801\u3002\u901a\u8fc7\u7403\u8c10\u5c55\u5f00\uff0c\u6211\u4eec\u53ef\u4ee5\u5f97\u5230\u4e00\u7cfb\u5217\u7531\u7403\u8c10\u57fa\u51fd\u6570\u52a0\u6743\u7ec4\u5408\u5f97\u5230\u7684\u51fd\u6570\u8868\u793a\uff0c\u51fd\u6570\u4e2d\u7684\u7cfb\u6570\u5219\u63cf\u8ff0\u4e86\u4e09\u7ef4\u5f62\u72b6\u5728\u4e0d\u540c\u7a7a\u95f4\u9891\u7387\u4e0a\u7684\u7279\u5f81\u3002\u8fd9\u4e9b\u7cfb\u6570\u6784\u6210\u4e86\u5f62\u72b6\u7684\u4e00\u4e2a\u7d27\u51d1\u800c\u5168\u9762\u7684\u7279\u5f81\u8868\u793a\uff0c\u53ef\u7528\u4e8e\u540e\u7eed\u7684\u5b66\u4e60\u4efb\u52a1\u3002<\/p>\n 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Deep Learning Math \u91cf\u5b50\u529b\u5b66\uff08Quantum mechanics\uff09 \u91cf\u5b50\u529b\u5b66\u662f\u7814\u7a76\u5fae\u89c2\u7269\u7406\u4e16 […]<\/p>\n","protected":false},"author":1,"featured_media":1633,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[14],"tags":[],"class_list":["post-1589","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-14"],"_links":{"self":[{"href":"http:\/\/www.gnn.club\/index.php?rest_route=\/wp\/v2\/posts\/1589","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/www.gnn.club\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/www.gnn.club\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/www.gnn.club\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/www.gnn.club\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1589"}],"version-history":[{"count":19,"href":"http:\/\/www.gnn.club\/index.php?rest_route=\/wp\/v2\/posts\/1589\/revisions"}],"predecessor-version":[{"id":1723,"href":"http:\/\/www.gnn.club\/index.php?rest_route=\/wp\/v2\/posts\/1589\/revisions\/1723"}],"wp:featuredmedia":[{"embeddable":true,"href":"http:\/\/www.gnn.club\/index.php?rest_route=\/wp\/v2\/media\/1633"}],"wp:attachment":[{"href":"http:\/\/www.gnn.club\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1589"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.gnn.club\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1589"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.gnn.club\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1589"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}\n
\n
\n
\n
\n
\u7403\u8c10\u51fd\u6570\uff08Spherical harmonics\uff09<\/h2>\n
\n\n
\n$$
\n\\begin{cases}
\nx=r \\sin \\theta \\cos \\phi \\\\
\ny=r \\sin \\theta \\sin \\phi \\\\
\nz=r \\cos \\theta
\n\\end{cases}
\n$$<\/p>\n<\/li>\n<\/ul>\n![](\"https:\/\/gnnclub-1311496010.cos.ap-beijing.myqcloud.com\/wp-content\/uploads\/2024\/07\/20240729200407132.png\")
\n<\/p>\n\u7403\u8c10\u51fd\u6570\u7c7b\u6bd4\u6cf0\u52d2\u516c\u5f0f<\/h3>\n
\n$$
\nf(x)=f(a)+f^{\\prime}(a)(x-a)+\\frac{f^{\\prime \\prime}(a)}{2!}(x-a)^2+\\frac{f^{\\prime \\prime \\prime}(a)}{3!}(x-a)^3+\\cdots
\n$$<\/p>\n![](\"https:\/\/gnnclub-1311496010.cos.ap-beijing.myqcloud.com\/wp-content\/uploads\/2024\/07\/20240729200439457.png\")
\n<\/p>\n\u7403\u8c10\u51fd\u6570\u7c7b\u6bd4\u5085\u91cc\u53f6\u5206\u89e3<\/h3>\n
\n$$
\nf(x)=a_0+\\sum_{n=1}^{\\infty}\\left(a_n \\cos (n x)+b_n \\sin (n x)\\right)
\n$$<\/p>\n![](\"https:\/\/gnnclub-1311496010.cos.ap-beijing.myqcloud.com\/wp-content\/uploads\/2024\/07\/20240729200507624.png\")
\n<\/p>\n\u7403\u8c10\u51fd\u6570\u7684\u5b9a\u4e49<\/h3>\n
\n$$
\nY_l^m(\\theta, \\phi)=P_l^m(\\cos \\theta) e^{i m \\phi}
\n$$<\/p>\n
\n$$
\nP_l^m(x)=(-1)^m\\left(1-x^2\\right)^{m \/ 2} \\frac{d^m}{d x^m} P_l(x)
\n$$<\/p>\n
\n$$
\nP_l(x)=\\frac{1}{2^l l!} \\frac{d^l}{d x^l}\\left(x^2-1\\right)^l
\n$$<\/p>\n![](\"https:\/\/gnnclub-1311496010.cos.ap-beijing.myqcloud.com\/wp-content\/uploads\/2024\/07\/20240729200542539.png\")
\n<\/p>\n
\n\u6cf0\u52d2\u516c\u5f0f\uff1a
\n\\begin{aligned}
\n& f(x)=f(a)+f^{\\prime}(a)(x-a)+\\frac{f^{\\prime \\prime}(a)}{2!}(x-a)^2+\\frac{f^{\\prime \\prime \\prime}(a)}{3!}(x-a)^3+\\cdots+\\frac{f^{(n)}(a)}{n!}(x-a)^n+ & \\cdots
\n\\end{aligned}
\n$$<\/p>\n![](\"https:\/\/gnnclub-1311496010.cos.ap-beijing.myqcloud.com\/wp-content\/uploads\/2024\/07\/20240729200615365.png\")
\n<\/p>\n![](\"https:\/\/gnnclub-1311496010.cos.ap-beijing.myqcloud.com\/wp-content\/uploads\/2024\/07\/20240729200653600.png\")
\n<\/p>\n
\n$$
\nH(\\theta, \\phi)=\\sum_{l=0}^{\\infty} \\sum_{m=-l}^l a_{l m} Y_l^m(\\theta, \\phi)
\n$$<\/p>\n\u7403\u8c10\u51fd\u6570\u5bf9\u79f0\u6027\u8ba8\u8bba<\/h3>\n
\n
Winger D \u77e9\u9635\uff08Winger D Matrix\uff09<\/h2>\n
\n
\n\u5728\u65cb\u8f6c $R(\\alpha, \\beta, \\gamma)$ \u540e\uff0c\u65b0\u7684\u7403\u8c10\u51fd\u6570\u53ef\u4ee5\u8868\u793a\u4e3a:
\n$$
\nR(\\alpha, \\beta, \\gamma) Y_{l m}(\\theta, \\phi)=\\sum_{m^{\\prime}} D_{m^{\\prime} m}^l(\\alpha, \\beta, \\gamma) Y_{l m^{\\prime}}\\left(\\theta^{\\prime}, \\phi^{\\prime}\\right)
\n$$<\/p>\n\n
import numpy as np\nfrom scipy.special import sph_harm\n\n# \u5b9a\u4e49\u7403\u8c10\u51fd\u6570\ndef compute_spherical_harmonics(l, m, theta, phi):\n return sph_harm(m, l, phi, theta)\n\nfrom scipy.special import factorial\n\n# \u8ba1\u7b97Wigner D\u77e9\u9635\ndef wigner_d_matrix(j, alpha, beta, gamma):\n d_matrix = np.zeros((2 * j + 1, 2 * j + 1), dtype=complex)\n for mp in range(-j, j + 1):\n for m in range(-j, j + 1):\n d_matrix[mp + j, m + j] = np.exp(-1j * mp * alpha) * small_d(j, mp, m, beta) * np.exp(-1j * m * gamma)\n return d_matrix\n\ndef small_d(j, mp, m, beta):\n d = 0\n for k in range(int(max(0, m - mp)), int(min(j - mp, j + m)) + 1):\n d += ((-1) ** (m - mp + k) * np.sqrt(factorial(j + mp) * factorial(j - mp) * factorial(j + m) * factorial(j - m)) \/\n (factorial(j - mp - k) * factorial(j + m - k) * factorial(k) * factorial(mp - m + k)) *\n np.cos(beta \/ 2) ** (2 * k + mp - m) * np.sin(beta \/ 2) ** (2 * j - 2 * k - mp + m))\n return d\n\n# \u793a\u4f8b: \u8ba1\u7b97 j = 1 \u7684 Wigner D \u77e9\u9635\nj = 1\nalpha, beta, gamma = np.pi \/ 3, np.pi \/ 4, np.pi \/ 6\nD_matrix = wigner_d_matrix(j, alpha, beta, gamma)\nprint("Wigner D \u77e9\u9635 (j = 1):\\n", D_matrix)\n<\/code><\/pre>\nWigner D \u77e9\u9635 (j = 1):\n [[ 2.77555756e-17+0.14644661j 2.50000000e-01+0.4330127j\n 7.39198920e-01+0.4267767j ]\n [-4.33012702e-01-0.25j -7.07106781e-01+0.j\n 4.33012702e-01-0.25j ]\n [ 7.39198920e-01-0.4267767j -2.50000000e-01+0.4330127j\n 2.77555756e-17-0.14644661j]]<\/code><\/pre>\nimport torch\nimport torch.nn as nn\n\nclass EquivariantLayer(nn.Module):\n def __init__(self, j, alpha, beta, gamma):\n super(EquivariantLayer, self).__init__()\n self.j = j\n self.D_matrix = torch.tensor(wigner_d_matrix(j, alpha, beta, gamma), dtype=torch.cfloat)\n\n def forward(self, x):\n # \u8f93\u5165 x \u7684\u5f62\u72b6\u4e3a (batch_size, 2j + 1)\n return torch.matmul(x, self.D_matrix)\n\n# \u793a\u4f8b: \u521b\u5efa\u7b49\u53d8\u5c42\u5e76\u8fdb\u884c\u524d\u5411\u4f20\u64ad\nj = 1\nalpha, beta, gamma = np.pi \/ 3, np.pi \/ 4, np.pi \/ 6\nequivariant_layer = EquivariantLayer(j, alpha, beta, gamma)\n\n# \u521b\u5efa\u4e00\u4e2a\u793a\u4f8b\u8f93\u5165\nx = torch.rand((1, 2 * j + 1), dtype=torch.cfloat)\nprint("\u8f93\u5165:\\n", x)\n\n# \u524d\u5411\u4f20\u64ad\noutput = equivariant_layer(x)\nprint("\u8f93\u51fa:\\n", output)\n<\/code><\/pre>\n\u8f93\u5165:\n tensor([[0.4397+0.0432j, 0.4085+0.8599j, 0.6284+0.4830j]])\n\u8f93\u51fa:\n tensor([[ 0.7023-0.3213j, -0.5639-0.2555j, 0.7692+0.3978j]])<\/code><\/pre>\n
\n$$
\n\\left|\\psi^{\\prime}\\right\\rangle=D|\\psi\\rangle
\n$$<\/p>\n
\n$$
\n\\left|\\psi^{\\prime}\\right\\rangle=M|\\psi\\rangle
\n$$<\/p>\n CG\u7cfb\u6570\uff08Clebsch-Gordan coefficients\uff09<\/h2>\n
\n
\n$$
\n\\left|l_1, m_1\\right\\rangle \\otimes\\left|l_2, m_2\\right\\rangle
\n$$<\/p>\n
\n$$
\n|L, M\\rangle
\n$$<\/p>\n
\n$$
\n\\left|l_1, m_1\\right\\rangle \\otimes\\left|l_2, m_2\\right\\rangle=\\sum_{L, M} C_{l_1, m_1 ; l_2, m_2}^{L, M}|L, M\\rangle
\n$$
\n\u8fdb\u4e00\u6b65\u7684\uff0cCG\u7cfb\u6570\u6ee1\u8db3\u6b63\u4ea4\u6027\u4e0e\u5bf9\u79f0\u6027\uff0c\u5373\uff1a<\/p>\n\n
\n$$
\nC_{l_1, m_1 ; l_2, m_2}^{L, M}=C_{l_2, m_2 ; l_1, m_1}^{L, M}
\n$$<\/p>\n<\/li>\n
\n$$
\nC_{l_1, m_1 ; l_2, m_2}^{L, M}=(-1)^{l_1+l_2-L} C_{l_1,-m_1: l_2,-m_2}^{L,-M_2}
\n$$<\/p>\n<\/li>\n
\n$$
\n\\sum_{m_1, m_2} C_{l_1, m_1 ; l_2, m_2}^{l, M} C_{l_1, m_1 ; l_2, m_2}^{l^{\\prime}, M^{\\prime}}=\\delta_{l, l^{\\prime}} \\delta_{M, M^{\\prime}}
\n$$<\/p>\n<\/li>\n
\n$$
\n\\sum_{l, M} C_{l_1, m_1 ; l_2, m_2}^{l, M} C_{l_1, m_1^{\\prime} ; l_2, m_2^{\\prime}}^{l, M}=\\delta_{m_1, m_1^{\\prime}} \\delta_{m_2, m_2^{\\prime}}
\n$$<\/p>\n<\/li>\n<\/ul>\n
\nCG\u7cfb\u6570=\\sqrt{2 s+1} \\sum_k \\frac{(-1)^k \\sqrt{\\left(s_1+s_2-s\\right)!\\left(s_1-s_2+s\\right)!\\left(s_2-s_1+s\\right)!}}{\\left(s_1+n_1-k\\right)!\\left(s_2+n_2-k\\right)!\\left(s-s_1-s_2+k\\right)!k!}
\n$$<\/p>\nimport sympy\nfrom sympy.physics.quantum.cg import CG\nfrom sympy import S\n\ndef clebsch_gordan(j1, m1, j2, m2, j, m):\n """\n \u8ba1\u7b97\u514b\u83b1\u5e03\u4ec0-\u6208\u767b\u7cfb\u6570 C_{j1, m1 ; j2, m2}^{j, m}\n\n \u53c2\u6570:\n j1, m1: \u7b2c\u4e00\u4e2a\u89d2\u52a8\u91cf\u7684\u91cf\u5b50\u6570\u548c\u78c1\u91cf\u5b50\u6570\n j2, m2: \u7b2c\u4e8c\u4e2a\u89d2\u52a8\u91cf\u7684\u91cf\u5b50\u6570\u548c\u78c1\u91cf\u5b50\u6570\n j, m: \u8026\u5408\u540e\u7684\u603b\u89d2\u52a8\u91cf\u7684\u91cf\u5b50\u6570\u548c\u78c1\u91cf\u5b50\u6570\n """\n cg = CG(S(j1), S(m1), S(j2), S(m2), S(j), S(m)).doit()\n return float(cg)\n\n# \u793a\u4f8b\u8ba1\u7b97\nj1, m1 = 1, 1\nj2, m2 = 1, -1\nj, m = 1, 0\ncg_coeff = clebsch_gordan(j1, m1, j2, m2, j, m) \n\ncg_coeff<\/code><\/pre>\n0.7071067811865476<\/code><\/pre>\nimport numpy as np\n\ndef feature_fusion(feature1, feature2, cg_coeff):\n """\n \u7279\u5f81\u878d\u5408\u51fd\u6570\uff0c\u4f7f\u7528CG\u7cfb\u6570\u4f5c\u4e3a\u6743\u91cd\n\n \u53c2\u6570:\n feature1: \u7b2c\u4e00\u4e2a\u7279\u5f81\u5411\u91cf\uff08\u4f8b\u5982\u56fe\u50cf\u7279\u5f81\uff09\n feature2: \u7b2c\u4e8c\u4e2a\u7279\u5f81\u5411\u91cf\uff08\u4f8b\u5982\u6587\u672c\u7279\u5f81\uff09\n cg_coeff: \u8ba1\u7b97\u5f97\u5230\u7684\u514b\u83b1\u5e03\u4ec0-\u6208\u767b\u7cfb\u6570\n """\n return cg_coeff * feature1 + (1 - cg_coeff) * feature2\n\n# \u793a\u4f8b\u7279\u5f81\nimage_feature = np.array([1.0, 0.5, 0.2])\ntext_feature = np.array([0.3, 0.7, 0.9])\n\n# \u8ba1\u7b97CG\u7cfb\u6570\ncg_coeff = clebsch_gordan(1, 1, 1, -1, 1, 0)\n\n# \u7279\u5f81\u878d\u5408\nfused_feature = feature_fusion(image_feature, text_feature, cg_coeff)\nprint(f"\u878d\u5408\u7279\u5f81: {fused_feature}")\n<\/code><\/pre>\n\u878d\u5408\u7279\u5f81: [0.79497475 0.55857864 0.40502525]<\/code><\/pre>\n\n
import numpy as np\nfrom sympy.physics.quantum.cg import CG\nfrom sympy import S\n\ndef clebsch_gordan_coeff(l1, m1, l2, m2, l, m):\n """\n \u8ba1\u7b97\u514b\u83b1\u5e03\u4ec0-\u6208\u767b\u7cfb\u6570 C_{l1, m1; l2, m2}^{l, m}\n """\n return float(CG(S(l1), S(m1), S(l2), S(m2), S(l), S(m)).doit())\n\ndef tensor_product_cg(V1, V2, W):\n """\n \u8ba1\u7b97 CG \u5f20\u91cf\u79ef\n \u53c2\u6570:\n V1, V2: \u8f93\u5165\u7279\u5f81\u77e9\u9635\uff0c\u5206\u522b\u5f62\u5982 (2*l1+1, C1) \u548c (2*l2+1, C2)\n W: \u53ef\u5b66\u4e60\u53c2\u6570\u77e9\u9635\uff0c\u5f62\u5982 (C1, C2, C)\n\n \u8fd4\u56de:\n V: \u8f93\u51fa\u7279\u5f81\u77e9\u9635\uff0c\u5f62\u5982 (2*l+1, C)\n """\n l1, C1 = V1.shape\n l2, C2 = V2.shape\n l = (l1 - 1) \/\/ 2 + (l2 - 1) \/\/ 2\n V = np.zeros((2 * l + 1, W.shape[2]))\n # \u901a\u8fc7\u591a\u91cd\u5faa\u73af\u904d\u5386\u6240\u6709\u53ef\u80fd\u7684\u7279\u5f81\u7ef4\u5ea6\u548c\u901a\u9053\u7ec4\u5408\u3002\n for m in range(-l, l + 1):\n for c in range(W.shape[2]):\n for c1 in range(C1):\n for c2 in range(C2):\n for m1 in range(-l1\/\/2, l1\/\/2 + 1):\n for m2 in range(-l2\/\/2, l2\/\/2 + 1):\n if abs(m1 + m2) <= l:\n Q = clebsch_gordan_coeff(l1\/\/2, m1, l2\/\/2, m2, l, m)\n V[m + l, c] += W[c1, c2, c] * Q * V1[m1 + l1\/\/2, c1] * V2[m2 + l2\/\/2, c2]\n return V\n\n# \u793a\u4f8b\u7279\u5f81\u77e9\u9635\nV1 = np.random.rand(3, 4) # l1=1, C1=4\nV2 = np.random.rand(5, 6) # l2=2, C2=6\nW = np.ones((4, 6, 8)) # \u53ef\u5b66\u4e60\u53c2\u6570\u77e9\u9635\n\n# \u8ba1\u7b97 CG \u5f20\u91cf\u79ef\nV = tensor_product_cg(V1, V2, W)\nprint("CG \u5f20\u91cf\u79ef\u7ed3\u679c: \\n", V)\n<\/code><\/pre>\nCG \u5f20\u91cf\u79ef\u7ed3\u679c: \n [[ 3.72821336 3.72821336 3.72821336 3.72821336 3.72821336 3.72821336\n 3.72821336 3.72821336]\n [10.59200757 10.59200757 10.59200757 10.59200757 10.59200757 10.59200757\n 10.59200757 10.59200757]\n [13.432774 13.432774 13.432774 13.432774 13.432774 13.432774\n 13.432774 13.432774 ]\n [14.80095141 14.80095141 14.80095141 14.80095141 14.80095141 14.80095141\n 14.80095141 14.80095141]\n [12.93742844 12.93742844 12.93742844 12.93742844 12.93742844 12.93742844\n 12.93742844 12.93742844]\n [12.21554941 12.21554941 12.21554941 12.21554941 12.21554941 12.21554941\n 12.21554941 12.21554941]\n [ 6.37701218 6.37701218 6.37701218 6.37701218 6.37701218 6.37701218\n 6.37701218 6.37701218]]<\/code><\/pre>\n\n
tensor_product_cg<\/code>\u4e2d\uff0c\u8ba1\u7b97\u5f97\u5230\u7684\u8f93\u51fa\u7279\u5f81\u77e9\u9635 V \u7684\u5f62\u72b6\u4e3a $(2l + 1, C)$\u3002<\/p>\n
\n\u8f93\u51fa\u7279\u5f81\u77e9\u9635 V \u7684\u5f62\u72b6\u4e3a (7, 8)\uff0c\u8868\u793a\u67097\u4e2a\u7279\u5f81\u7ef4\u5ea6\u548c8\u4e2a\u901a\u9053\u3002<\/p>\n
\n$W$ \u662f\u53ef\u5b66\u4e60\u7684\u53c2\u6570\u77e9\u9635\uff0c\u5176\u7ef4\u5ea6\u4e3a $(C1, C2, C)$\u3002
\n$c_1$ \u548c $c_2$ \u5206\u522b\u8868\u793a $V_1$ \u548c $V_2$ \u7684\u901a\u9053\u7d22\u5f15\uff0c$c$ \u8868\u793a\u8f93\u51fa\u901a\u9053\u7d22\u5f15\u3002
\n$W[c1, c2, c]$ \u8868\u793a\u4ece $c_1$ \u548c $c_2$ \u901a\u9053\u6620\u5c04\u5230 $c$ \u901a\u9053\u7684\u6743\u91cd\u53c2\u6570\u3002<\/p>\n \u5f84\u5411\u6a21\u578b\uff08Radial Model\uff09<\/h2>\n
\n
\n\u73b0\u5728\u6211\u4eec\u6765\u770b\u770b\u9ad8\u65af\u5f84\u5411\u51fd\u6570\u7684\u6570\u5b66\u5f62\u5f0f\u3002<\/p>\n\n
\n$$
\nI=\\int_{-\\infty}^{\\infty} G_1(r) G_2(r) d r
\n$$<\/p>\n
\n$$
\n\\begin{aligned}
\n& G_1(r)=A_1 \\exp \\left(-\\alpha_1 r^2\\right) \\\\
\n& G_2(r)=A_2 \\exp \\left(-\\alpha_2(r-d)^2\\right)
\n\\end{aligned}
\n$$<\/p>\nimport numpy as np\nimport matplotlib.pyplot as plt\nfrom scipy.integrate import quad\n\n# \u5b9a\u4e49\u4e24\u4e2a\u9ad8\u65af\u51fd\u6570\ndef G1(r, A1, alpha1):\n return A1 * np.exp(-alpha1 * r**2)\n\ndef G2(r, A2, alpha2, d):\n return A2 * np.exp(-alpha2 * (r - d)**2)\n\n# \u5b9a\u4e49\u91cd\u53e0\u79ef\u5206\u7684\u88ab\u79ef\u51fd\u6570\ndef integrand(r, A1, alpha1, A2, alpha2, d):\n return G1(r, A1, alpha1) * G2(r, A2, alpha2, d)\n\n# \u8bbe\u7f6e\u53c2\u6570\nA1 = 1.0\nalpha1 = 1.0\nA2 = 1.0\nalpha2 = 1.0\n\n# \u8ba1\u7b97\u4e0d\u540c\u8ddd\u79bb\u4e0b\u7684\u91cd\u53e0\u79ef\u5206\ndistances = np.linspace(0, 10, 100)\noverlaps = []\n\nfor d in distances:\n result, error = quad(integrand, -np.inf, np.inf, args=(A1, alpha1, A2, alpha2, d))\n overlaps.append(result)\n\n# \u53ef\u89c6\u5316\u7ed3\u679c\nplt.figure(figsize=(5, 3))\nplt.plot(distances, overlaps, label='Overlap Integral')\nplt.xlabel('Distance (d)')\nplt.ylabel('Overlap Integral')\nplt.title('Overlap Integral between Two Gaussian Distributions')\nplt.legend()\nplt.grid(True)\nplt.show()\n<\/code><\/pre>\n![](\"http:\/\/www.gnn.club\/wp-content\/uploads\/2024\/07\/output_27_0-2.png\")
\n<\/p>\n
\nx^2 \\frac{d^2 y}{d x^2}+x \\frac{d y}{d x}+\\left(x^2-n^2\\right) y=0
\n$$<\/p>\n\n
\n
![](\"https:\/\/gnnclub-1311496010.cos.ap-beijing.myqcloud.com\/wp-content\/uploads\/2024\/07\/20240729200736663.png\")
\n<\/p>\nimport numpy as np\nimport matplotlib.pyplot as plt\nfrom scipy.integrate import quad\nfrom scipy.special import iv # Importing the modified Bessel function of the first kind\n\n# \u5b9a\u4e49\u4e24\u4e2a\u8d1d\u585e\u5c14\u51fd\u6570\u5f84\u5411\u5206\u5e03\ndef Bessel1(r, A1, alpha1):\n return A1 * iv(0, alpha1 * r)\n\ndef Bessel2(r, A2, alpha2, d):\n return A2 * iv(0, alpha2 * np.abs(r - d))\n\n# \u5b9a\u4e49\u91cd\u53e0\u79ef\u5206\u7684\u88ab\u79ef\u51fd\u6570\ndef integrand(r, A1, alpha1, A2, alpha2, d):\n return Bessel1(r, A1, alpha1) * Bessel2(r, A2, alpha2, d)\n\n# \u8bbe\u7f6e\u53c2\u6570\nA1 = 1.0\nalpha1 = 0.5\nA2 = 1.0\nalpha2 = 0.5\n\n# \u8ba1\u7b97\u4e0d\u540c\u8ddd\u79bb\u4e0b\u7684\u91cd\u53e0\u79ef\u5206\ndistances = np.linspace(0, 5, 100)\noverlaps = []\n\nfor d in distances:\n result, error = quad(integrand, 0, 10, args=(A1, alpha1, A2, alpha2, d)) # \u5c06\u79ef\u5206\u533a\u95f4\u9650\u5236\u5728 [0, 10]\n overlaps.append(result)\n\n# \u53ef\u89c6\u5316\u7ed3\u679c\nplt.figure(figsize=(10, 6))\nplt.plot(distances, overlaps, label='Overlap Integral')\nplt.xlabel('Distance (d)')\nplt.ylabel('Overlap Integral')\nplt.title('Overlap Integral between Two Radial Bessel Distributions')\nplt.legend()\nplt.grid(True)\nplt.show()\n<\/code><\/pre>\n![](\"http:\/\/www.gnn.club\/wp-content\/uploads\/2024\/07\/output_29_0-2.png\")
\n<\/p>\nimport numpy as np\nimport matplotlib.pyplot as plt\nfrom scipy.integrate import quad\nfrom scipy.special import jv # Importing the Bessel function of the first kind\n\n# \u5b9a\u4e49\u4e24\u4e2a\u8d1d\u585e\u5c14\u51fd\u6570\u5f84\u5411\u5206\u5e03\ndef Bessel1(r, A1, alpha1):\n return A1 * jv(0, alpha1 * r)\n\ndef Bessel2(r, A2, alpha2, d):\n return A2 * jv(0, alpha2 * np.abs(r - d))\n\n# \u5b9a\u4e49\u91cd\u53e0\u79ef\u5206\u7684\u88ab\u79ef\u51fd\u6570\ndef integrand(r, A1, alpha1, A2, alpha2, d):\n return Bessel1(r, A1, alpha1) * Bessel2(r, A2, alpha2, d)\n\n# \u8bbe\u7f6e\u53c2\u6570\nA1 = 1.0\nalpha1 = 2.0 # \u9002\u5f53\u8c03\u6574\u4ee5\u4f53\u73b0\u9707\u8361\u6027\u8d28\nA2 = 1.0\nalpha2 = 2.0 # \u9002\u5f53\u8c03\u6574\u4ee5\u4f53\u73b0\u9707\u8361\u6027\u8d28\n\n# \u8ba1\u7b97\u4e0d\u540c\u8ddd\u79bb\u4e0b\u7684\u91cd\u53e0\u79ef\u5206\ndistances = np.linspace(0, 20, 200)\noverlaps = []\n\nfor d in distances:\n result, error = quad(integrand, 0, 10, args=(A1, alpha1, A2, alpha2, d)) # \u5c06\u79ef\u5206\u533a\u95f4\u9650\u5236\u5728 [0, 10]\n overlaps.append(result)\n\n# \u53ef\u89c6\u5316\u7ed3\u679c\nplt.figure(figsize=(10, 6))\nplt.plot(distances, overlaps, label='Overlap Integral')\nplt.xlabel('Distance (d)')\nplt.ylabel('Overlap Integral')\nplt.title('Overlap Integral between Two Radial Bessel Distributions')\nplt.legend()\nplt.grid(True)\nplt.show()\n<\/code><\/pre>\n![](\"http:\/\/www.gnn.club\/wp-content\/uploads\/2024\/07\/output_31_0.png\")
\n<\/p>\n\n
\n
\n$$
\n\\Phi(x, c)=\\Phi(|x-c|)
\n$$<\/p>\n
\n\u5bf9\u4e8e\u6b27\u51e0\u91cc\u5f97\u7a7a\u95f4\u4e2d\u6700\u5e38\u7528\u7684\u5f84\u5411\u57fa\u51fd\u6570\u2014\u2014\u9ad8\u65af\u6838\u51fd\u6570, \u5176\u5f62\u5f0f\u53ef\u4ee5\u7279\u522b\u8868\u793a\u4e3a:
\n$$
\nk(|x-c|)=\\exp \\left(-\\frac{|x-c|^2}{2 \\sigma^2}\\right)
\n$$<\/p>\n8. \u7403\u8c10\u51fd\u6570\u4e0e\u5f84\u5411\u6a21\u578b\u5728DL\u4e2d\u7684\u7ed3\u5408<\/h2>\n
\n