{"id":173,"date":"2026-03-04T17:51:23","date_gmt":"2026-03-04T09:51:23","guid":{"rendered":"http:\/\/photocaloric.com\/?p=173"},"modified":"2026-03-04T17:51:24","modified_gmt":"2026-03-04T09:51:24","slug":"%e5%88%86%e6%95%b0%e9%87%8f%e5%ad%90%e9%9c%8d%e5%b0%94%e6%95%88%e5%ba%94%e4%b8%80%ef%bc%9a%e6%95%b4%e6%95%b0%e9%87%8f%e5%ad%90%e9%9c%8d%e5%b0%94%e6%95%88%e5%ba%94%e7%9a%84%e6%b3%a2%e5%87%bd%e6%95%b0","status":"publish","type":"post","link":"http:\/\/photocaloric.com\/index.php\/2026\/03\/04\/%e5%88%86%e6%95%b0%e9%87%8f%e5%ad%90%e9%9c%8d%e5%b0%94%e6%95%88%e5%ba%94%e4%b8%80%ef%bc%9a%e6%95%b4%e6%95%b0%e9%87%8f%e5%ad%90%e9%9c%8d%e5%b0%94%e6%95%88%e5%ba%94%e7%9a%84%e6%b3%a2%e5%87%bd%e6%95%b0\/","title":{"rendered":"\u5206\u6570\u91cf\u5b50\u970d\u5c14\u6548\u5e94(\u4e00\uff1a\u6574\u6570\u91cf\u5b50\u970d\u5c14\u6548\u5e94\u7684\u6ce2\u51fd\u6570)"},"content":{"rendered":"\n

\u6211\u4eec\u63d0\u5230\u91cf\u5b50\u970d\u5c14\u6548\u5e94\uff0c\u6211\u4eec\u5728\u8003\u8651\u7684\u662f\u4e00\u4e2a\u4e8c\u7ef4\u7684\u7535\u5b50\u4f53\u7cfb\u3002\u5728\u975e\u76f8\u4e92\u4f5c\u7528\u7684\u4e8c\u7ef4\u7535\u5b50\u4f53\u7cfb\uff08\u5373\uff1a\u4e8c\u7ef4\u7535\u5b50\u6c14\uff09<\/strong>\u4e2d\uff0c\u8fd9\u8003\u8651\u7684\u5c31\u662f\u5982\u4e0b\u54c8\u5bc6\u987f\u91cf\uff1a<\/p>\n\n\n\n

H<\/mi>=<\/mo>p<\/mi>2<\/mn><\/msup>2<\/mn>m<\/mi>e<\/mi><\/msub><\/mrow><\/mfrac>=<\/mo>p<\/mi>x<\/mi>2<\/mn><\/msubsup>+<\/mo>p<\/mi>y<\/mi>2<\/mn><\/msubsup><\/mrow>2<\/mn>m<\/mi>e<\/mi><\/msub><\/mrow><\/mfrac><\/mrow>H=\\frac{p^2}{2m_e}=\\frac{p_x^2+p_y^2}{2m_e}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

\u8fd9\u4e2a\u54c8\u5bc6\u987f\u91cf\u7684\u89e3\u662f\u975e\u5e38\u76f4\u89c2\u7684\uff0c\u5c31\u662f\u5c06x\u548cy\u65b9\u5411\u5206\u7acb\u53d8\u91cf\uff0c\u5f97\u5230\u672c\u5f81\u80fd\u529b\u548c\u6ce2\u51fd\u6570\u5206\u522b\u4e3a\uff1a<\/p>\n\n\n\n

E<\/mi>=<\/mo>\u210f<\/mi>2<\/mn><\/msup>2<\/mn>m<\/mi>e<\/mi><\/msub><\/mrow><\/mfrac>(<\/mo>k<\/mi>x<\/mi>2<\/mn><\/msubsup>+<\/mo>k<\/mi>y<\/mi>2<\/mn><\/msubsup>)<\/mo>,<\/mo><\/mspace>\u03c8<\/mi>(<\/mo>x<\/mi>,<\/mo>y<\/mi>)<\/mo>\u221d<\/mo>e<\/mi>i<\/mi>k<\/mi>x<\/mi><\/msub>x<\/mi><\/mrow><\/msup>e<\/mi>i<\/mi>k<\/mi>y<\/mi><\/msub>y<\/mi><\/mrow><\/msup><\/mrow>E=\\frac{\\hbar^2}{2m_e}(k_x^2+k_y^2)\n,\\quad\n\\psi(x,y)\\propto e^{ik_xx}e^{ik_yy}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

OK\uff0c\u73b0\u5728\u5982\u679c\u52a0\u4e0a\u78c1\u573a\u5462\uff1f\u6ce8\u610f\u5230\u78c1\u573a\u4e0d\u505a\u529f\uff0c\u56e0\u6b64\u5b83\u6539\u53d8\u7cfb\u7edf\u7684\u80fd\u91cf\uff08\u54c8\u5bc6\u987f\u91cf\uff09\u3002\u7cfb\u7edf\u7684\u54c8\u5bc6\u987f\u91cf\u4ecd\u65e7\u662fH<\/mi>=<\/mo>1<\/mn>2<\/mn><\/mfrac>m<\/mi>e<\/mi><\/msub>\ud835\udc99<\/mi>\u02d9<\/mo><\/mover>2<\/mn><\/msup><\/mrow>H=\\frac{1}{2}m_e\\dot{\\bm{x}}^2<\/annotation><\/semantics><\/math>\u3002\u7136\u800c\uff0c\u73b0\u5728\u552f\u4e00\u7684\u533a\u522b\u662f\uff0c\u4e0d\u518d\u6709\ud835\udc91<\/mi>=<\/mo>m<\/mi>e<\/mi><\/msub>\ud835\udc99<\/mi>\u02d9<\/mo><\/mover><\/mrow>\\bm{p}=m_e\\dot{\\bm{x}}<\/annotation><\/semantics><\/math>\u3002\u6b64\u65f6\u6211\u4eec\u5c06\u52a8\u91cfp\u66ff\u6362\u4e3a\ud835\udf45<\/mi>\\bm{\\pi}<\/annotation><\/semantics><\/math>\uff0c\u5b83\u53eb\u505a\u201c\u7269\u7406\u52a8\u91cf\u201d\uff0c\u800c\u4e0d\u662f\u54c8\u5bc6\u987f\u529b\u5b66\u4e2d\u53ef\u4ee5\u76f4\u63a5\u91cf\u5b50\u5316\u7684\u90a3\u4e2a\u201c\u6b63\u5219\u52a8\u91cf\u201d\u3002\u6b63\u5219\u52a8\u91cfp\u548c\u7269\u7406\u52a8\u91cf\u03c0<\/mi>\\pi<\/annotation><\/semantics><\/math>\u7684\u5173\u7cfb\u662f\uff1a<\/p>\n\n\n\n
\ud835\udf45<\/mi>=<\/mo>m<\/mi>e<\/mi><\/msub>\ud835\udc99<\/mi>\u02d9<\/mo><\/mover>=<\/mo>\ud835\udc91<\/mi>+<\/mo>e<\/mi>\ud835\udc68<\/mi><\/mrow>\\bm{\\pi}=m_e\\dot{\\bm{x}}=\\bm{p}+e\\bm{A}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

\u5176\u4e2d\u78c1\u573a\u7531\u78c1\u77e2\u52bf\ud835\udc68<\/mi>(<\/mo>x<\/mi>,<\/mo>y<\/mi>)<\/mo><\/mrow>\\bm{A}(x,y)<\/annotation><\/semantics><\/math>\u7ed9\u51fa\u3002\u6b63\u5219\u52a8\u91cf\ud835\udc91<\/mi>\\bm{p}<\/annotation><\/semantics><\/math>\u6ee1\u8db3\u548c\u4f4d\u7f6e\ud835\udc93<\/mi>\\bm{r}<\/annotation><\/semantics><\/math>\u7684\u91cf\u5b50\u529b\u5b66\u5bf9\u6613\u5173\u7cfb\uff1a[<\/mo>r<\/mi>i<\/mi><\/msub>,<\/mo>p<\/mi>j<\/mi><\/msub>]<\/mo>=<\/mo>i<\/mi>\u210f<\/mi>\u03b4<\/mi>i<\/mi>j<\/mi><\/mrow><\/msub><\/mrow>[r_i,p_j]=i\\hbar\\delta_{ij}<\/annotation><\/semantics><\/math>\u3002\u4e8e\u662f\u54c8\u5bc6\u987f\u91cf\u53d8\u4e3a<\/p>\n\n\n\n
H<\/mi>=<\/mo>\ud835\udf45<\/mi>2<\/mn><\/msup>2<\/mn>m<\/mi>e<\/mi><\/msub><\/mrow><\/mfrac>=<\/mo>(<\/mo>\ud835\udc91<\/mi>+<\/mo>e<\/mi>\ud835\udc68<\/mi>)<\/mo>2<\/mn><\/msup><\/mrow>2<\/mn>m<\/mi>e<\/mi><\/msub><\/mrow><\/mfrac><\/mrow>H=\\frac{\\bm{\\pi}^2}{2m_e}=\\frac{(\\bm{p}+e\\bm{A})^2}{2m_e}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

\uff08\u4e3a\u4ec0\u4e48\u662f\u8fd9\u6837\u7684\uff1f\u8fd9\u6d89\u53ca\u5230\u78c1\u573a\u7684\u6027\u8d28\u3002\u867d\u7136\u78c1\u573a\u4e0d\u6539\u53d8\u4f53\u7cfb\u7684\u80fd\u91cf\uff0c\u4f46\u7531\u4e8e\u7535\u5b50\u5e26\u7535\uff0c\u78c1\u573a\u4ecd\u65e7\u4f1a\u5bf9\u5176\u4e2d\u505a\u56de\u65cb\u8fd0\u52a8\u7684\u7535\u5b50\u65bd\u52a0\u4e00\u4e2a\u76f8\u4f4d\uff0c\u8fd9\u4f7f\u5f97\u7c92\u5b50\u7684\u5bf9\u6613\u5173\u7cfb\u2014\u2014\u5373\u4e24\u4e2a\u7c92\u5b50\u4ea4\u6362\u4f4d\u7f6e\uff0c\u53d7\u5230\u78c1\u573a\u7684\u5f71\u54cd\u3002\uff09\u8fdb\u4e00\u6b65\uff0c\u6211\u4eec\u5047\u8bbe\u78c1\u573a\u5782\u76f4\u4e8e\u4e8c\u7ef4\u5e73\u9762\u5411\u4e0a\ud835\udc69<\/mi>=<\/mo>B<\/mi>\ud835\udc9b<\/mi><\/mrow>\\bm{B}=B\\bm{z}<\/annotation><\/semantics><\/math>\u3002\u4e3a\u4e86\u5f97\u5230\u54c8\u5bc6\u987f\u91cf\u7684\u786e\u5207\u5f62\u5f0f\uff0c\u6211\u4eec\u5fc5\u987b\u786e\u5b9a\ud835\udc68<\/mi>(<\/mo>x<\/mi>,<\/mo>y<\/mi>)<\/mo><\/mrow>\\bm{A}(x,y)<\/annotation><\/semantics><\/math>\uff0c\u7136\u800c\u5b83\u662f\u4e0d\u786e\u5b9a\u7684\uff0c\u6211\u4eec\u53ea\u77e5\u9053\ud835\udc69<\/mi>=<\/mo>\u2207<\/mo>\u00d7<\/mo>\ud835\udc68<\/mi><\/mrow>\\bm{B}=\\nabla\\times\\bm{A}<\/annotation><\/semantics><\/math>\u3002\u4e00\u79cd\u7ea6\u5b9a\u662f\u6240\u8c13\u7684\u5bf9\u79f0\u89c4\u8303<\/strong>\uff1a<\/p>\n\n\n\n
\ud835\udc68<\/mi>(<\/mo>x<\/mi>,<\/mo>y<\/mi>)<\/mo>=<\/mo>\u2212<\/mo>1<\/mn>2<\/mn><\/mfrac>\ud835\udc93<\/mi>\u00d7<\/mo>\ud835\udc69<\/mi>=<\/mo>B<\/mi>2<\/mn><\/mfrac>(<\/mo>\u2212<\/mo>y<\/mi>,<\/mo>x<\/mi>)<\/mo><\/mrow>\\bm{A}(x,y)=-\\frac{1}{2}\\bm{r}\\times\\bm{B}=\\frac{B}{2}(-y,x)<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

\u5728\u8fd9\u79cd\u89c4\u8303\u4e0b\uff0c\u54c8\u5bc6\u987f\u91cf\u4e3a<\/p>\n\n\n\n

H<\/mi>=<\/mo>1<\/mn>2<\/mn>m<\/mi>e<\/mi><\/msub><\/mrow><\/mfrac>[<\/mo>(<\/mo>p<\/mi>x<\/mi><\/msub>\u2212<\/mo>e<\/mi>B<\/mi><\/mrow>2<\/mn><\/mfrac>y<\/mi>)<\/mo>2<\/mn><\/msup>+<\/mo>(<\/mo>p<\/mi>y<\/mi><\/msub>+<\/mo>e<\/mi>B<\/mi><\/mrow>2<\/mn><\/mfrac>x<\/mi>)<\/mo>2<\/mn><\/msup>]<\/mo><\/mrow>H=\\frac{1}{2m_e}[(p_x-\\frac{eB}{2}y)^2+(p_y+\\frac{eB}{2}x)^2]<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

\u8fd9\u5c31\u662f\u5728\u5782\u76f4\u78c1\u573a\u4e0b\u7684\u65e0\u76f8\u4e92\u4f5c\u7528\u7684\u4e8c\u7ef4\u7535\u5b50\u6c14<\/strong>\uff08\u5728\u5bf9\u79f0\u89c4\u8303\u4e0b\uff09\u7684\u54c8\u5bc6\u987f\u91cf\u7684\u5177\u4f53\u5f62\u5f0f\uff0c\u662f\u4e0d\u662f\u5f88\u7b80\u5355\uff1f\u89e3\u8d77\u6765\u5c31\u66f4\u7b80\u5355\u4e86\uff0c\u53ef\u4ee5\u5b9a\u4e49\u4ea7\u751f\/\u6e6e\u706d\u7b97\u7b26\uff1a<\/p>\n\n\n\n

a<\/mi>\u2020<\/mo><\/msup>=<\/mo>1<\/mn>2<\/mn>e<\/mi>\u210f<\/mi>B<\/mi><\/mrow><\/msqrt><\/mfrac>(<\/mo>\u03c0<\/mi>x<\/mi><\/msub>+<\/mo>i<\/mi>\u03c0<\/mi>y<\/mi><\/msub>)<\/mo><\/mspace>a<\/mi>=<\/mo>1<\/mn>2<\/mn>e<\/mi>\u210f<\/mi>B<\/mi><\/mrow><\/msqrt><\/mfrac>(<\/mo>\u03c0<\/mi>x<\/mi><\/msub>\u2212<\/mo>i<\/mi>\u03c0<\/mi>y<\/mi><\/msub>)<\/mo><\/mrow>a^\\dagger=\\frac{1}{\\sqrt{2e\\hbar B}}(\\pi_x+i\\pi_y)\\quad \na=\\frac{1}{\\sqrt{2e\\hbar B}}(\\pi_x-i\\pi_y)<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

\u4e8e\u662f\u6709<\/p>\n\n\n\n

H<\/mi>=<\/mo>\u210f<\/mi>\u03c9<\/mi>B<\/mi><\/msub>(<\/mo>a<\/mi>\u2020<\/mo><\/msup>a<\/mi>+<\/mo>1<\/mn>2<\/mn><\/mfrac>)<\/mo><\/mrow>H=\\hbar\\omega_B(a^\\dagger a + \\frac{1}{2})<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

\u5176\u4e2d\u03c9<\/mi>B<\/mi><\/msub>=<\/mo>e<\/mi>B<\/mi><\/mrow>m<\/mi>e<\/mi><\/msub><\/mfrac><\/mrow>\\omega_B=\\frac{eB}{m_e}<\/annotation><\/semantics><\/math>\u79f0\u4e3a\u56de\u65cb\u5171\u632f\u9891\u7387\u3002\u63a8\u5bfc\u8fc7\u7a0b\u4e2d\u5229\u7528\u4e86\u78c1\u573a\u4e0b\u7269\u7406\u52a8\u91cf\u7684\u5bf9\u6613\u5173\u7cfb[<\/mo>\u03c0<\/mi>i<\/mi><\/msub>,<\/mo>\u03c0<\/mi>j<\/mi><\/msub>]<\/mo>=<\/mo>\u2212<\/mo>i<\/mi>\u210f<\/mi>e<\/mi>\u03f5<\/mi>i<\/mi>j<\/mi>k<\/mi><\/mrow><\/msub>B<\/mi>k<\/mi><\/msub><\/mrow>[\\pi_i,\\pi_j]=-i\\hbar e\\epsilon_{ijk}B_k<\/annotation><\/semantics><\/math>\u3002<\/p>\n\n\n\n

\u6ce8\u610f\uff1a<\/strong>\u5f53\u78c1\u573aB<\/mi>\u2192<\/mo>0<\/mn><\/mrow>B\\to0<\/annotation><\/semantics><\/math>\u65f6\uff0c\u4f53\u7cfb\u56de\u5230\u6b63\u5e38\u7684\u8c10\u632f\u5b50\u5f62\u5f0f\u3002\u4f60\u53ef\u4ee5\u6ce8\u610f\u5230\uff0c\u8fd9\u4e00\u5207\u7684\u7f18\u7531\uff0c\u90fd\u662f\u56e0\u4e3a\u5728\u5b58\u5728\u78c1\u573a\u7684\u60c5\u51b5\u4e0b\uff0c\u7269\u7406\u52a8\u91cf\u7684\u5bf9\u6613\u5173\u7cfb[<\/mo>\u03c0<\/mi>i<\/mi><\/msub>,<\/mo>\u03c0<\/mi>j<\/mi><\/msub>]<\/mo>\u2260<\/mo>0<\/mn><\/mrow>[\\pi_i,\\pi_j]\\neq0<\/annotation><\/semantics><\/math>\uff0c\u8fd9\u5c31\u662f\u89c4\u8303\u7406\u8bba\u7684\u9b45\u529b\uff08\u78c1\u573a\u6539\u53d8\u4e86\u7a7a\u95f4\u7684\u8f9b\u7ed3\u6784\uff09\u3002\u4e4b\u540e\u4f60\u770b\u5230\u7684\u4e00\u5207\u7279\u522b\u7684\u6027\u8d28\u90fd\u7531\u5b83\u6240\u5bfc\u81f4\u3002<\/p>\n\n\n\n

\u6211\u4eec\u7ee7\u7eed\u89e3\u8fd9\u4e2a\u54c8\u5bc6\u987f\u91cf\u3002\u5b9e\u9645\u4e0a\u5176\u672c\u5f81\u80fd\u91cf\u5df2\u7ecf\u89e3\u51fa\u6765\u4e86\uff0c\u5c31\u662f\u7c7b\u4f3c\u8c10\u632f\u5b50\u7684\u5206\u7acb\u80fd\u7ea7\uff1a<\/p>\n\n\n\n

E<\/mi>=<\/mo>\u210f<\/mi>\u03c9<\/mi>B<\/mi><\/msub>(<\/mo>n<\/mi>+<\/mo>1<\/mn>2<\/mn><\/mfrac>)<\/mo><\/mrow>E=\\hbar\\omega_B(n+\\frac{1}{2})<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

\u6ce8\u610f\u548c\u4e0d\u5b58\u5728\u78c1\u573a\u65f6\u7684\u60c5\u51b5E<\/mi>=<\/mo>\u210f<\/mi>2<\/mn><\/msup>2<\/mn>m<\/mi>e<\/mi><\/msub><\/mrow><\/mfrac>(<\/mo>k<\/mi>x<\/mi>2<\/mn><\/msubsup>+<\/mo>k<\/mi>y<\/mi>2<\/mn><\/msubsup>)<\/mo><\/mrow>E=\\frac{\\hbar^2}{2m_e}(k_x^2+k_y^2)<\/annotation><\/semantics><\/math>\u4f5c\u6bd4\u8f83\uff0c\u540e\u8005\u662f\u8fde\u7eed\u8c31\uff0c\u800c\u524d\u8005\u5df2\u7ecf\u88ab\u79bb\u6563\u5316\u4e86\u3002\u80fd\u91cf\u4e0a\u7684\u79bb\u6563\u5316\u5f80\u5f80\u610f\u5473\u7740\u7a7a\u95f4\u4e0a\u7684\u5b9a\u57df\u6027\uff0c\u8fd9\u5f88\u597d\u7406\u89e3\uff1a\u5728\u5782\u76f4\u78c1\u573a\u4e0b\uff0c\u7535\u5b50\u503e\u5411\u4e8e\u7ed5\u7740\u67d0\u4e2a\u4e2d\u5fc3\u505a\u56de\u65cb\u8fd0\u52a8\uff0c\u800c\u4e0d\u662f\u5728\u4e8c\u7ef4\u7a7a\u95f4\u4e2d\u81ea\u7531\u8fd0\u52a8\u7684\u5e73\u9762\u6ce2\u3002\u8fd9\u4e5f\u89e3\u91ca\u4e86\u4e3a\u4ec0\u4e48\u03c9<\/mi>B<\/mi><\/msub>\\omega_B<\/annotation><\/semantics><\/math>\u88ab\u79f0\u4f5c\u56de\u65cb\u5171\u632f\u9891\u7387\u3002<\/p>\n\n\n\n

\u6211\u4eec\u5c06\u4e0a\u8ff0\u79bb\u6563\u80fd\u7ea7\u79f0\u4e3a\u6717\u9053\u80fd\u7ea7<\/strong>\u3002<\/p>\n\n\n\n

\u6211\u4eec\u8fdb\u4e00\u6b65\u6c42\u89e3\u8be5\u54c8\u5bc6\u987f\u91cf\u7684\u6ce2\u51fd\u6570\u3002\u5c06H<\/mi>=<\/mo>1<\/mn>2<\/mn>m<\/mi>e<\/mi><\/msub><\/mrow><\/mfrac>[<\/mo>(<\/mo>p<\/mi>x<\/mi><\/msub>\u2212<\/mo>e<\/mi>B<\/mi><\/mrow>2<\/mn><\/mfrac>y<\/mi>)<\/mo>2<\/mn><\/msup>+<\/mo>(<\/mo>p<\/mi>y<\/mi><\/msub>+<\/mo>e<\/mi>B<\/mi><\/mrow>2<\/mn><\/mfrac>x<\/mi>)<\/mo>2<\/mn><\/msup>]<\/mo><\/mrow>H=\\frac{1}{2m_e}[(p_x-\\frac{eB}{2}y)^2+(p_y+\\frac{eB}{2}x)^2]<\/annotation><\/semantics><\/math>\u62c6\u5f00\u5f97\u5230<\/p>\n\n\n\n
H<\/mi>=<\/mo>1<\/mn>2<\/mn>m<\/mi>e<\/mi><\/msub><\/mrow><\/mfrac>[<\/mo>\ud835\udc91<\/mi>2<\/mn><\/msup>+<\/mo>e<\/mi>2<\/mn><\/msup>B<\/mi>2<\/mn><\/msup><\/mrow>4<\/mn><\/mfrac>\ud835\udc93<\/mi>2<\/mn><\/msup>+<\/mo>e<\/mi>B<\/mi>L<\/mi>z<\/mi><\/msub>]<\/mo><\/mrow>H=\\frac{1}{2m_e}[\\bm{p}^2+\\frac{e^2B^2}{4}\\bm{r}^2+eBL_z]<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

\u5176\u4e2dL<\/mi>z<\/mi><\/msub>=<\/mo>x<\/mi>p<\/mi>y<\/mi><\/msub>\u2212<\/mo>y<\/mi>p<\/mi>x<\/mi><\/msub><\/mrow>L_z=xp_y-yp_x<\/annotation><\/semantics><\/math>\u662f\u6cbf\u7740z\u8f74\u65b9\u5411\u7684\u89d2\u52a8\u91cf\u3002\u8fd9\u5c31\u662f\u4e00\u4e2a\u6709\u5916\u52a0\u78c1\u573a\u7684\u4e8c\u7ef4\u8c10\u632f\u5b50\uff0c\u5176\u89e3\u7531\u4e3b\u91cf\u5b50\u6570n\uff08\u5b83\u4e0d\u662f\u6717\u9053\u80fd\u7ea7\u7684\u5e8f\u6570\uff0c\u6211\u4eec\u4e4b\u540e\u4e0d\u8ba8\u8bba\u5b83\u4e86\uff09\u548c\u78c1\u91cf\u5b50\u6570m\u51b3\u5b9a\u3002<\/p>\n\n\n\n

\u6c42\u89e3\u4e0a\u8ff0\u54c8\u5bc6\u987f\u91cf\u4e0d\u662f\u5fc5\u8981\u7684\uff08\u9664\u975e\u4f60\u5f15\u5165\u4e00\u4e2a\u70b9\u72b6\u6742\u8d28\uff0c\u53bb\u7814\u7a76\u7535\u5b50\u675f\u7f1a\u5728\u6742\u8d28\u9644\u8fd1\u7684\u884c\u4e3a\uff09\u3002\u6211\u4eec\u6709\u66f4\u52a0\u4f18\u96c5\u4e14\u4ee3\u6570\u7684\u65b9\u6cd5\u6765\u5904\u7406\u8fd9\u4e2a\u54c8\u5bc6\u987f\u91cf\u3002\u65e2\u7136\u5df2\u77e5\u8be5\u4f53\u7cfb\u5fc5\u7136\u75312\u4e2a\u91cf\u5b50\u6570\u51b3\u5b9a\uff0c\u90a3\u4e48\u6717\u9053\u80fd\u7ea7\u7684\u5e8f\u6570n\u5fc5\u7136\u662f\u7b80\u5e76\u7684\u3002\u6211\u4eec\u9700\u8981\u627e\u5230\u4e00\u4e2a\u7269\u7406\u91cf\uff0c\u5b83\u548c\u54c8\u5bc6\u987f\u91cf\u5bf9\u6613\uff0c\u6765\u53bb\u9664\u6717\u9053\u80fd\u7ea7\u7684\u7b80\u5e76\u3002\u6211\u4eec\u627e\u5230\u4e86\uff1a<\/p>\n\n\n\n

\u5b9a\u4e49\u201c\u52a8\u91cf\u201d\uff1a<\/p>\n\n\n\n

\ud835\udf45<\/mi>~<\/mo><\/mover>=<\/mo>\ud835\udc91<\/mi>\u2212<\/mo>e<\/mi>\ud835\udc68<\/mi><\/mrow>\\tilde{\\bm{\\pi}}=\\bm{p}-e\\bm{A}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

\u518d\u5b9a\u4e49\u4e0e\u4e4b\u642d\u914d\u7684\u4e00\u7ec4\u201c\u4ea7\u751f\/\u6e6e\u706d\u7b97\u7b26\u201d\uff1a<\/p>\n\n\n\n

b<\/mi>\u2020<\/mo><\/msup>=<\/mo>1<\/mn>2<\/mn>e<\/mi>\u210f<\/mi>B<\/mi><\/mrow><\/msqrt><\/mfrac>(<\/mo>\u03c0<\/mi>~<\/mo><\/mover>x<\/mi><\/msub>\u2212<\/mo>i<\/mi>\u03c0<\/mi>~<\/mo><\/mover>y<\/mi><\/msub>)<\/mo><\/mspace>b<\/mi>=<\/mo>1<\/mn>2<\/mn>e<\/mi>\u210f<\/mi>B<\/mi><\/mrow><\/msqrt><\/mfrac>(<\/mo>\u03c0<\/mi>~<\/mo><\/mover>x<\/mi><\/msub>+<\/mo>i<\/mi>\u03c0<\/mi>~<\/mo><\/mover>y<\/mi><\/msub>)<\/mo><\/mrow>b^\\dagger=\\frac{1}{\\sqrt{2e\\hbar B}}(\\tilde{\\pi}_x-i\\tilde{\\pi}_y)\\quad \nb=\\frac{1}{\\sqrt{2e\\hbar B}}(\\tilde{\\pi}_x+i\\tilde{\\pi}_y)<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

\u4e8e\u662f\u7b80\u5e76\u5c31\u88ab\u5b8c\u5168\u53bb\u9664\u4e86\uff08\u4e4b\u540e\u53ef\u4ee5\u9a8c\u8bc1\uff09\u3002\u7136\u540e\u4f53\u7cfb\u7684\u6ce2\u51fd\u6570\u5c31\u53ef\u4ee5\u7528\u4ea7\u751f\u7b97\u7b26\u5199\u4e3a\uff1a<\/p>\n\n\n\n

|<\/mi>n<\/mi>,<\/mo>m<\/mi><\/mrow>\u27e9<\/mo><\/mpadded>=<\/mo>a<\/mi>\u2020<\/mo><\/msup>n<\/mi><\/msup>b<\/mi>\u2020<\/mo><\/msup>m<\/mi><\/msup><\/mrow>n<\/mi>!<\/mo>m<\/mi>!<\/mo><\/mrow><\/msqrt><\/mfrac>|<\/mi>0,0<\/mn>\u27e9<\/mo><\/mpadded><\/mrow>\\ket{n,m}=\\frac{{a^\\dagger}^n {b^\\dagger}^m}{\\sqrt{n!m!}}\\ket{0,0}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

\u6211\u4eec\u9700\u8981\u77e5\u9053|<\/mi>0,0<\/mn>\u27e9<\/mo><\/mpadded>\\ket{0,0}<\/annotation><\/semantics><\/math>\u7684\u5f62\u5f0f\u3002\u7531\u6e6e\u706d\u7b97\u7b26\u7684\u6027\u8d28\uff0c\u5b83\u5e94\u5f53\u6ee1\u8db3\u5982\u4e0b\u7b49\u5f0f\uff1a<\/p>\n\n\n\n
a<\/mi>|<\/mi>0,0<\/mn>\u27e9<\/mo><\/mpadded>=<\/mo>b<\/mi>|<\/mi>0,0<\/mn>\u27e9<\/mo><\/mpadded>=<\/mo>0<\/mn><\/mrow>a\\ket{0,0}=b\\ket{0,0}=0<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

\u5b9a\u4e49z<\/mi>=<\/mo>x<\/mi>\u2212<\/mo>i<\/mi>y<\/mi>,<\/mo>z<\/mi>~<\/mo><\/mover>=<\/mo>x<\/mi>+<\/mo>i<\/mi>y<\/mi>,<\/mo>\u2202<\/mi>=<\/mo>1<\/mn>2<\/mn><\/mfrac>(<\/mo>\u2202<\/mi>\u2202<\/mi>x<\/mi><\/mrow><\/mfrac>+<\/mo>i<\/mi>\u2202<\/mi>\u2202<\/mi>y<\/mi><\/mrow><\/mfrac>)<\/mo>,<\/mo>\u2202<\/mi>~<\/mo><\/mover>=<\/mo>1<\/mn>2<\/mn><\/mfrac>(<\/mo>\u2202<\/mi>\u2202<\/mi>x<\/mi><\/mrow><\/mfrac>\u2212<\/mo>i<\/mi>\u2202<\/mi>\u2202<\/mi>y<\/mi><\/mrow><\/mfrac>)<\/mo><\/mrow>z=x-iy,\\tilde{z}=x+iy, \\partial=\\frac{1}{2}(\\frac{\\partial}{\\partial x}+i\\frac{\\partial}{\\partial y}), \\tilde{\\partial}=\\frac{1}{2}(\\frac{\\partial}{\\partial x}-i\\frac{\\partial}{\\partial y})<\/annotation><\/semantics><\/math>\uff08\u8fd9\u6837\u6ee1\u8db3\u2202<\/mi>z<\/mi>=<\/mo>\u2202<\/mi>~<\/mo><\/mover>z<\/mi>~<\/mo><\/mover>=<\/mo>1<\/mn>,<\/mo>\u2202<\/mi>~<\/mo><\/mover>z<\/mi>=<\/mo>\u2202<\/mi>z<\/mi>~<\/mo><\/mover>=<\/mo>0<\/mn><\/mrow>\\partial z=\\tilde{\\partial}\\tilde{z}=1, \\tilde{\\partial}z=\\partial\\tilde{z}=0<\/annotation><\/semantics><\/math>\uff09\uff0c\u4ea7\u751f\/\u6e6e\u706d\u7b97\u7b26\u5728\u5750\u6807\u8868\u8c61\u4e0b\u5e94\u5f53\u8bb0\u4e3a<\/p>\n\n\n\n
a<\/mi>\u2020<\/mo><\/msup>=<\/mo>\u2212<\/mo>i<\/mi>2<\/mn><\/msqrt>(<\/mo>l<\/mi>B<\/mi><\/msub>\u2202<\/mi>\u2212<\/mo>z<\/mi>~<\/mo><\/mover>4<\/mn>l<\/mi>B<\/mi><\/msub><\/mrow><\/mfrac>)<\/mo><\/mspace>a<\/mi>=<\/mo>\u2212<\/mo>i<\/mi>2<\/mn><\/msqrt>(<\/mo>l<\/mi>B<\/mi><\/msub>\u2202<\/mi>~<\/mo><\/mover>+<\/mo>z<\/mi>4<\/mn>l<\/mi>B<\/mi><\/msub><\/mrow><\/mfrac>)<\/mo><\/mrow>a^\\dagger=-i\\sqrt{2}(l_B\\partial-\\frac{\\tilde{z}}{4l_B})\n\\quad \na=-i\\sqrt{2}(l_B\\tilde{\\partial}+\\frac{z}{4l_B})<\/annotation><\/semantics><\/math><\/div>\n\n\n\n
b<\/mi>\u2020<\/mo><\/msup>=<\/mo>\u2212<\/mo>i<\/mi>2<\/mn><\/msqrt>(<\/mo>l<\/mi>B<\/mi><\/msub>\u2202<\/mi>~<\/mo><\/mover>\u2212<\/mo>z<\/mi>4<\/mn>l<\/mi>B<\/mi><\/msub><\/mrow><\/mfrac>)<\/mo><\/mspace>b<\/mi>=<\/mo>\u2212<\/mo>i<\/mi>2<\/mn><\/msqrt>(<\/mo>l<\/mi>B<\/mi><\/msub>\u2202<\/mi>+<\/mo>z<\/mi>~<\/mo><\/mover>4<\/mn>l<\/mi>B<\/mi><\/msub><\/mrow><\/mfrac>)<\/mo><\/mrow>b^\\dagger=-i\\sqrt{2}(l_B\\tilde{\\partial}-\\frac{z}{4l_B})\n\\quad \nb=-i\\sqrt{2}(l_B\\partial+\\frac{\\tilde{z}}{4l_B})<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

\u5176\u4e2dl<\/mi>B<\/mi><\/msub>=<\/mo>\u210f<\/mi>e<\/mi>B<\/mi><\/mrow><\/mfrac><\/msqrt><\/mrow>l_B=\\sqrt{\\frac{\\hbar}{eB}}<\/annotation><\/semantics><\/math>\u662f\u78c1\u957f\u5ea6\u3002\u4e8e\u662f\u65b9\u4fbf\u89e3\u5f97\u6ee1\u8db3\u6e6e\u706d\u7b97\u7b26\u6027\u8d28\u7684\u57fa\u6001\u4e3a<\/p>\n\n\n\n
|<\/mi>0,0<\/mn>\u27e9<\/mo><\/mpadded>\u221d<\/mo>e<\/mi>\u2212<\/mo>|<\/mi>z<\/mi>|<\/mi>2<\/mn><\/msup><\/mrow>4<\/mn>l<\/mi>B<\/mi>2<\/mn><\/msubsup><\/mrow><\/mfrac><\/mrow><\/msup><\/mrow>\\ket{0,0}\\propto e^{-\\frac{|z|^2}{4l_B^2}}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

\u5176\u5b83\u6001\u600e\u4e48\u5f97\u5230\uff1f\u5c31\u662f\u628a\u4ea7\u751f\u7b97\u7b26\u4e00\u4e2a\u4e00\u4e2a\u4f5c\u7528\u4e0a\u53bb\u5c31\u597d\u4e86\uff08a\u548cb\u8fd9\u4e24\u7ec4\u7b97\u7b26\u5b8c\u5168\u4e92\u76f8\u5bf9\u6613\uff0c\u4e92\u4e0d\u5f71\u54cd\uff09\u3002\u8fd9\u91cc\u6211\u4eec\u5173\u6ce8\u4e24\u79cd\u7c7b\u578b\u7684\u6ce2\u51fd\u6570\uff1a<\/p>\n\n\n\n

    \n
  • \u6700\u4f4e\u6717\u9053\u80fd\u7ea7\u4e2d\u7684\u6240\u6709\u6ce2\u51fd\u6570|<\/mi>0<\/mn>,<\/mo>m<\/mi><\/mrow>\u27e9<\/mo><\/mpadded>\\ket{0,m}<\/annotation><\/semantics><\/math>\u3002\u663e\u7136\u5b83\u7b49\u4e8e|<\/mi>0<\/mn>,<\/mo>m<\/mi><\/mrow>\u27e9<\/mo><\/mpadded>\u221d<\/mo>(<\/mo>b<\/mi>\u2020<\/mo><\/msup>)<\/mo>m<\/mi><\/msup>|<\/mi>0,0<\/mn>\u27e9<\/mo><\/mpadded>\u221d<\/mo>(<\/mo>z<\/mi>l<\/mi>B<\/mi><\/msub><\/mfrac>)<\/mo>m<\/mi><\/msup>e<\/mi>\u2212<\/mo>|<\/mi>z<\/mi>|<\/mi>2<\/mn><\/msup><\/mrow>4<\/mn>l<\/mi>B<\/mi>2<\/mn><\/msubsup><\/mrow><\/mfrac><\/mrow><\/msup><\/mrow>\\ket{0,m}\\propto (b^\\dagger)^m\\ket{0,0} \\propto (\\frac{z}{l_B})^m e^{-\\frac{|z|^2}{4l_B^2}}<\/annotation><\/semantics><\/math>\u3002\u5b83\u5bf9\u5206\u6570\u91cf\u5b50\u970d\u5c14\u6548\u5e94\u7684\u8ba8\u8bba\u6709\u57fa\u7840\u610f\u4e49\u3002<\/li>\n\n\n\n
  • m=0\u65f6\uff0c\u66f4\u9ad8\u6717\u9053\u80fd\u7ea7\u7684\u6ce2\u51fd\u6570|<\/mi>n<\/mi>,<\/mo>0<\/mn><\/mrow>\u27e9<\/mo><\/mpadded>\\ket{n,0}<\/annotation><\/semantics><\/math>\u3002\u663e\u7136\u5b83\u7b49\u4e8e|<\/mi>n<\/mi>,<\/mo>0<\/mn><\/mrow>\u27e9<\/mo><\/mpadded>\u221d<\/mo>(<\/mo>a<\/mi>\u2020<\/mo><\/msup>)<\/mo>n<\/mi><\/msup>|<\/mi>0,0<\/mn>\u27e9<\/mo><\/mpadded>\u221d<\/mo>(<\/mo>z<\/mi>~<\/mo><\/mover>l<\/mi>B<\/mi><\/msub><\/mfrac>)<\/mo>n<\/mi><\/msup>e<\/mi>\u2212<\/mo>|<\/mi>z<\/mi>|<\/mi>2<\/mn><\/msup><\/mrow>4<\/mn>l<\/mi>B<\/mi>2<\/mn><\/msubsup><\/mrow><\/mfrac><\/mrow><\/msup><\/mrow>\\ket{n,0}\\propto (a^\\dagger)^n \\ket{0,0} \\propto (\\frac{\\tilde{z}}{l_B})^n e^{-\\frac{|z|^2}{4l_B^2}}<\/annotation><\/semantics><\/math>\u3002\u5b83\u5bf9\u5149\u5b66\u8fc7\u7a0b\u7684\u8ba8\u8bba\u6709\u5e2e\u52a9\u3002<\/li>\n<\/ul>\n\n\n\n

    \u6ce8\u610f\u4e00\u4e2a\u6709\u8da3\u7684\u4e8b\u5b9e\uff1a\u5982\u679c\u6211\u4eec\u5b9a\u4e49\u89d2\u52a8\u91cf\u7b97\u7b26<\/p>\n\n\n\n

    L<\/mi>z<\/mi><\/msub>=<\/mo>x<\/mi>\u03c0<\/mi>y<\/mi><\/msub>\u2212<\/mo>y<\/mi>\u03c0<\/mi>x<\/mi><\/msub>=<\/mo>\u2212<\/mo>i<\/mi>\u210f<\/mi>(<\/mo>x<\/mi>\u2202<\/mi>y<\/mi><\/msub>\u2212<\/mo>y<\/mi>\u2202<\/mi>x<\/mi><\/msub>)<\/mo>=<\/mo>\u210f<\/mi>(<\/mo>z<\/mi>~<\/mo><\/mover>\u2202<\/mi>~<\/mo><\/mover>\u2212<\/mo>z<\/mi>\u2202<\/mi>)<\/mo>=<\/mo>\u210f<\/mi>(<\/mo>n<\/mi>\u2212<\/mo>m<\/mi>)<\/mo><\/mrow>L_z=x\\pi_y-y\\pi_x=-i\\hbar(x\\partial_y-y\\partial_x)=\\hbar(\\tilde{z}\\tilde{\\partial}-z\\partial)\n=\\hbar(n-m)<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

    \u5219\u5728\u6700\u4f4e\u6717\u9053\u80fd\u7ea7\uff08n=0\uff09\u4e2d\uff0cm\u5b9e\u9645\u4e0a\u5145\u5f53\u4e86\u89d2\u52a8\u91cf\u7684\u89d2\u8272\u3002\uff08\u6ce8\u610f\uff1aDavid Tong\u4e66\u4e2d\u5b9a\u4e49\u7684L<\/mi>z<\/mi><\/msub>=<\/mo>i<\/mi>\u210f<\/mi>(<\/mo>x<\/mi>\u2202<\/mi>y<\/mi><\/msub>\u2212<\/mo>y<\/mi>\u2202<\/mi>x<\/mi><\/msub>)<\/mo><\/mrow>L_z=i\\hbar(x\\partial_y-y\\partial_x)<\/annotation><\/semantics><\/math>\uff0c\u4e0e\u6211\u4eec\u8fd9\u91cc\u5dee\u4e00\u4e2a\u8d1f\u53f7\uff0c\u5b9e\u9645\u4e0a\u53ea\u662f\u5c06\u89d2\u52a8\u91cf\u91cd\u65b0\u5b9a\u4e49\u4e86\u4e00\u4e0b\u3002\u8fd9\u79cd\u5b9a\u4e49\u5728\u91cf\u5b50\u970d\u5c14\u4f53\u7cfb\u91cc\u4f3c\u4e4e\u6bd4\u8f83\u5e38\u89c1\uff09\u3002<\/p>\n\n\n\n

    \u8fd9\u4e00\u8282\u7684\u6700\u540e\uff0c\u6211\u4eec\u518d\u8ba8\u8bba\u4e00\u4e0b\u6717\u9053\u80fd\u7ea7\u7684\u7b80\u5e76\u5ea6\u3002\u6ce8\u610f\u5230\u5728\u6211\u4eec\u8ba8\u8bba\u7684\u60c5\u51b5\uff08\u5bf9\u79f0\u89c4\u8303\uff09\u4e0b\uff0c\u6ce2\u51fd\u6570\u90fd\u662f\u73af\u72b6\u7684\uff0c\u5728\u6700\u4f4e\u6717\u9053\u80fd\u7ea7\u4e2d\uff0c\u4ee5m\u4e3a\u91cf\u5b50\u6570\u7684\u6ce2\u51fd\u6570\u7684\u5cf0\u503c\u51fa\u73b0\u5728\u534a\u5f84\u4e3ar<\/mi>=<\/mo>2<\/mn>m<\/mi><\/mrow><\/msqrt>l<\/mi>B<\/mi><\/msub><\/mrow>r=\\sqrt{2m}l_B<\/annotation><\/semantics><\/math>\u7684\u5706\u73af\u9644\u8fd1\u3002\u4e5f\u5c31\u662f\u8bf4\uff0c\u5982\u679c\u6211\u4eec\u5047\u8bbe\u6709\u4e00\u4e2a\u4ee5R\u4e3a\u534a\u5f84\u7684\u5706\u5f62\u533a\u57df\uff0c\u90a3\u4e48\u5176\u4e2d\u5305\u542b\u7684\u91cf\u5b50\u6001\u7684\u6570\u91cf\u5927\u7ea6\u4e3aR<\/mi>=<\/mo>r<\/mi>(<\/mo>m<\/mi>)<\/mo>\u2192<\/mo>m<\/mi>=<\/mo>1<\/mn>2<\/mn><\/mfrac>(<\/mo>R<\/mi>l<\/mi>B<\/mi><\/msub><\/mfrac>)<\/mo>2<\/mn><\/msup><\/mrow>R=r(m)\\to m=\\frac{1}{2}(\\frac{R}{l_B})^2<\/annotation><\/semantics><\/math>\u4e2a\uff0c\u4e8e\u662f\u6001\u5bc6\u5ea6\u5c31\u4e3a<\/p>\n\n\n\n
    \ud835\udca9<\/mi>=<\/mo>1<\/mn>2<\/mn><\/mfrac>(<\/mo>R<\/mi>l<\/mi>B<\/mi><\/msub><\/mfrac>)<\/mo>2<\/mn><\/msup>\/<\/mi>\u03c0<\/mi>R<\/mi>2<\/mn><\/msup>=<\/mo>1<\/mn>2<\/mn>\u03c0<\/mi>l<\/mi>B<\/mi>2<\/mn><\/msubsup><\/mrow><\/mfrac><\/mrow>\\mathcal{N}=\\frac{1}{2}(\\frac{R}{l_B})^2\/\\pi R^2=\\frac{1}{2\\pi l_B^2}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

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