{"id":100009,"date":"2020-03-13T05:17:39","date_gmt":"2020-03-13T05:17:39","guid":{"rendered":"https:\/\/martinphysics.club\/?p=100009"},"modified":"2026-08-20T15:51:38","modified_gmt":"2026-08-20T07:51:38","slug":"solution-to-one-dimensional-infinite-depth-potential-extraction","status":"publish","type":"post","link":"https:\/\/martinphysics.club\/?p=100009","title":{"rendered":"\u89e3\u4e00\u7ef4\u65e0\u9650\u6df1\u52bf\u9631(\u89e3\u5b9a\u6001\u859b\u5b9a\u8c14\u65b9\u7a0b)"},"content":{"rendered":"<h6>\u5b9a\u6001\u859b\u5b9a\u8c14\u65b9\u7a0b\u6c42\u89e3\uff1a<\/h6>\n<p>\u5047\u8bbe\u7c92\u5b50\u5728\u4e00\u7ef4\u52bf\u573a\u4e2d\u8fd0\u52a8\uff0c\u52bf\u573a\u6ee1\u8db3\uff1a<\/p>\n<p style=\"text-align: center;\"><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.image?U(x)=\\left\\{\\begin{matrix}0,0\\leq&amp;space;x\\leq&amp;space;a\\\\&amp;plus;\\infty,x&gt;a,x&lt;0\\end{matrix}\\right.\" border=\"0\" alt=\"\" width=\"206\" height=\"43\" loading=\"lazy\"><\/p>\n<p>\u89e3\uff1a<\/p>\n<p>\u8bbe\u7c92\u5b50\u7684\u6ce2\u51fd\u6570\u4e3a<img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\psi&amp;space;(x)\" border=\"0\" alt=\"\\psi (x)\" align=\"absmiddle\" loading=\"lazy\">, \u7531\u5b9a\u6001\u859b\u5b9a\u8c14\u65b9\u7a0b\u5f97\uff1a<\/p>\n<p style=\"text-align: center;\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\hat{H}\\psi&amp;space;(x)=E\\psi&amp;space;(x)\" border=\"0\" alt=\"\\hat{H}\\psi (x)=E\\psi (x)\" align=\"absmiddle\" loading=\"lazy\"><\/p>\n<p>\u5176\u4e2d<img decoding=\"async\" class=\"alignnone\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\hat{H}\" border=\"0\" alt=\"\\hat{H}\" align=\"absmiddle\" loading=\"lazy\">\u4e3a\u7c92\u5b50\u7684\u54c8\u5bc6\u987f\u91cf\uff0c\u4e3a\u7c92\u5b50\u52a8\u80fd\u4e0e\u52bf\u80fd\u7684\u603b\u548c\uff0cE\u4e3a\u7c92\u5b50\u7684\u80fd\u91cf\u3002<\/p>\n<p style=\"text-align: left;\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\hat{H}=\\frac{\\vec{p}^{2}}{2m}+U(\\vec{r},t)\" border=\"0\" alt=\"\\hat{H}=\\frac{\\vec{p}^{2}}{2m}+U(\\vec{r},t)\" align=\"absmiddle\" loading=\"lazy\">\u540c\u65f6\u5728\u6570\u5b66\u4e0a\uff0c\u5bf9\u6ce2\u51fd\u6570\u6709\u4ee5\u4e0b\u7b97\u6cd5\u5bf9\u5e94\u5173\u7cfb\uff1a<img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\vec{p}\\rightarrow&amp;space;-i\\hbar\\triangledown\" border=\"0\" alt=\"\\vec{p}\\rightarrow -i\\hbar\\triangledown\" align=\"absmiddle\" loading=\"lazy\"><\/p>\n<p>\u5f53x&lt;0\u548cx&gt;a\u65f6\uff0c\u5728\u52bf\u9631\u5916\u3002<\/p>\n<p>\u5f530&lt;x&lt;a\u65f6\uff0cU(x)=0\uff0c\u5728\u52bf\u9631\u5185\uff1a<\/p>\n<p>\u7531\u4e0a\u8ff0\u859b\u5b9a\u8c14\u65b9\u7a0b\u51fa\u53d1\uff0c\u7efc\u5408\u4e0a\u8ff0\u5f97\uff1a<\/p>\n<p style=\"text-align: center;\"><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?[\\frac{\\hbar^{2}}{2m}\\bigtriangledown^{2}&amp;space;+U(x)]\\psi&amp;space;(x)=E\\psi&amp;space;(x)\" border=\"0\" alt=\"[\\frac{\\hbar^{2}}{2m}\\bigtriangledown^{2} +U(x)]\\psi (x)=E\\psi (x)\" align=\"absmiddle\" loading=\"lazy\"><\/p>\n<p>\u7531\u4e8e\u7814\u7a76\u7684\u95ee\u9898\u662f\u4e00\u7ef4\u3001\u5b9a\u6001(\u4e0d\u542b\u65f6)\uff0c\u4e0a\u8ff0\u65b9\u7a0b\u6539\u5199\u4e3a\uff1a<\/p>\n<p style=\"text-align: center;\"><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\frac{\\hbar^{2}}{2m}\\frac{d^{2}}{dx^{2}}\\psi&amp;space;(x)&amp;space;+E\\psi&amp;space;(x)=0\" border=\"0\" alt=\"\\frac{\\hbar^{2}}{2m}\\frac{d^{2}}{dx^{2}}\\psi (x) +E\\psi (x)=0\" align=\"absmiddle\" loading=\"lazy\"><\/p>\n<p>\u4e3a\u89e3\u65b9\u7a0b\u65b9\u4fbf\uff0c\u8bbe\u6709\u67d0\u7cfb\u6570k:<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center;\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/latex.codecogs.com\/gif.latex?k=\\sqrt{\\frac{2mE}{&amp;space;\\hbar&amp;space;^{2}}}\" border=\"0\" alt=\"k=\\sqrt{\\frac{2mE}{ \\hbar ^{2}}}\" align=\"absmiddle\" loading=\"lazy\"><\/p>\n<p>\u539f\u65b9\u7a0b\u53d8\u5f62\u4e3a\uff1a<\/p>\n<p style=\"text-align: center;\"><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\frac{d^{2}}{dx^{2}}\\psi&amp;space;(x)+k^{2}\\psi&amp;space;(x)=0\" border=\"0\" alt=\"\\frac{d^{2}}{dx^{2}}\\psi (x)+k^{2}\\psi (x)=0\" align=\"absmiddle\" loading=\"lazy\"><\/p>\n<p>\u7531\u6570\u5b66\u7269\u7406\u65b9\u6cd5\u5f97\uff0c\u4e0a\u5f0f\u901a\u89e3\u4e3a\uff1a<\/p>\n<p style=\"text-align: center;\"><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\psi&amp;space;(x)=C_{1}\\cos&amp;space;kx+C_{2}\\sin&amp;space;kx\" border=\"0\" alt=\"\\psi (x)=C_{1}\\cos kx+C_{2}\\sin kx\" align=\"absmiddle\" loading=\"lazy\"><\/p>\n<p>\u73b0\u5728\u4ece\u8fb9\u754c\u6761\u4ef6\u5165\u624b\u903c\u51fa\u7279\u89e3\uff1a<\/p>\n<p style=\"text-align: center;\"><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\because&amp;space;\\psi&amp;space;(0)=0\\therefore&amp;space;C_{1}=0\" border=\"0\" alt=\"\\because \\psi (0)=0\\therefore C_{1}=0\" align=\"absmiddle\" loading=\"lazy\"><\/p>\n<p>\u539f\u5f0f\u53d8\u4e3a&nbsp;<img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\psi&amp;space;(x)=C_{2}\\sin&amp;space;kx\" border=\"0\" alt=\"\\psi (x)=C_{2}\\sin kx\" align=\"absmiddle\" loading=\"lazy\"><\/p>\n<p style=\"text-align: center;\">\u53c8<img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\because&amp;space;\\psi&amp;space;(a)=0,\\therefore&amp;space;C_{2}\\sin&amp;space;ka=0,ak=n\\pi&amp;space;(n=1,2,3\\cdots&amp;space;)\" border=\"0\" alt=\"\\because \\psi (a)=0,\\therefore C_{2}\\sin ka=0,ak=n\\pi (n=1,2,3\\cdots )\" align=\"absmiddle\" loading=\"lazy\"><\/p>\n<p>\u6240\u4ee5\uff0c\u672c\u5f81\u6ce2\u51fd\u6570\u4e3a<img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\psi&amp;space;(x)=C_{2}\\sin&amp;space;\\frac{n\\pi&amp;space;}{a}x,(n=1,2,3\\cdots&amp;space;)\" border=\"0\" alt=\"\\psi (x)=C_{2}\\sin \\frac{n\\pi }{a}x,(n=1,2,3\\cdots )\" align=\"absmiddle\" loading=\"lazy\"><\/p>\n<p>\u8fdb\u884c\u5f52\u4e00\u5316\u5904\u7406\uff1a<\/p>\n<p style=\"text-align: center;\"><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\int_{-\\infty&amp;space;}^{+\\infty&amp;space;}\\left&amp;space;|\\psi&amp;space;(x)&amp;space;\\right&amp;space;|^{2}d\\tau&amp;space;=\\int_{0}^{a}\\left&amp;space;|&amp;space;C_{2}&amp;space;\\right&amp;space;|^{2}\\sin&amp;space;^{2}(\\frac{n\\pi&amp;space;x}{a})dx=1\" border=\"0\" alt=\"\\int_{-\\infty }^{+\\infty }\\left |\\psi (x) \\right |^{2}d\\tau =\\int_{0}^{a}\\left | C_{2} \\right |^{2}\\sin ^{2}(\\frac{n\\pi x}{a})dx=1\" align=\"absmiddle\" loading=\"lazy\"><\/p>\n<p>\u5229\u7528\u500d\u89d2\u516c\u5f0f<img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\cos&amp;space;2\\alpha&amp;space;=1-2\\sin&amp;space;^{2}\\alpha\" border=\"0\" alt=\"\\cos 2\\alpha =1-2\\sin ^{2}\\alpha\" align=\"absmiddle\" loading=\"lazy\">\uff0c\u5316\u7b80\u5f97\uff1a<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center;\"><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\int_{0}^{a}\\left&amp;space;|&amp;space;C_{2}&amp;space;\\right&amp;space;|^{2}\\frac{1-\\cos&amp;space;2(\\frac{n\\pi&amp;space;x}{a})}{2}dx=1,(n=1,2,3\\cdots&amp;space;)\" border=\"0\" alt=\"\\int_{0}^{a}\\left | C_{2} \\right |^{2}\\frac{1-\\cos 2(\\frac{n\\pi x}{a})}{2}dx=1,(n=1,2,3\\cdots )\" align=\"absmiddle\" loading=\"lazy\"><\/p>\n<p>\u62c6\u6563\u79ef\u5206\u9879\u4ee5\u7b80\u6613\u8ba1\u7b97\uff1a<\/p>\n<p style=\"text-align: center;\"><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\int_{0}^{a}\\frac{1}{2}\\left&amp;space;|&amp;space;C_{2}&amp;space;\\right&amp;space;|^{2}dx-\\int_{0}^{a}\\frac{1}{2}\\cos&amp;space;2\\frac{n\\pi&amp;space;x}{a}dx=1,n=(1,2,3\\cdots&amp;space;)\" border=\"0\" alt=\"\\int_{0}^{a}\\frac{1}{2}\\left | C_{2} \\right |^{2}dx-\\int_{0}^{a}\\frac{1}{2}\\cos 2\\frac{n\\pi x}{a}dx=1,n=(1,2,3\\cdots )\" align=\"absmiddle\" loading=\"lazy\"><\/p>\n<p style=\"text-align: center;\"><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/gif.latex?=\\int_{0}^{a}\\frac{1}{2}\\left&amp;space;|&amp;space;C_{2}&amp;space;\\right&amp;space;|^{2}dx-\\frac{a}{4n\\pi&amp;space;}\\int_{0}^{2\\pi&amp;space;}cos&amp;space;udu=1,u=\\frac{2n\\pi&amp;space;x}{a},n=(1,2,3\\cdots&amp;space;)\" border=\"0\" alt=\"=\\int_{0}^{a}\\frac{1}{2}\\left | C_{2} \\right |^{2}dx-\\frac{a}{4n\\pi }\\int_{0}^{2\\pi }cos udu=1,u=\\frac{2n\\pi x}{a},n=(1,2,3\\cdots )\" align=\"absmiddle\" loading=\"lazy\"><\/p>\n<p>\u53ef\u4ee5\u89c2\u6d4b\u5230\uff0c\u7b2c\u4e8c\u9879\u79ef\u5206\u7ed3\u679c\u4e3a0\uff0c\u6240\u4ee5\uff1a<\/p>\n<p style=\"text-align: center;\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\int_{0}^{a}\\frac{1}{2}\\left&amp;space;|&amp;space;C_{2}&amp;space;\\right&amp;space;|^{2}dx=1\\Rightarrow&amp;space;C_{2}=\\sqrt{\\frac{2}{a}}\" border=\"0\" alt=\"\\int_{0}^{a}\\frac{1}{2}\\left | C_{2} \\right |^{2}dx=1\\Rightarrow C_{2}=\\sqrt{\\frac{2}{a}}\" align=\"absmiddle\" loading=\"lazy\"><\/p>\n<p>&nbsp;\u6240\u4ee5\u8be5\u7c92\u5b50\u7684\u6ce2\u51fd\u6570\u4e3a\uff1a<\/p>\n<p style=\"text-align: center;\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/latex.codecogs.com\/gif.latex?\\psi&amp;space;(x)=\\sqrt{\\frac{2}{a}}\\sin&amp;space;\\frac{n\\pi&amp;space;}{a}x,n=(1,2,3\\cdots&amp;space;)\" border=\"0\" alt=\"\\psi (x)=\\sqrt{\\frac{2}{a}}\\sin \\frac{n\\pi }{a}x,n=(1,2,3\\cdots )\" align=\"absmiddle\" loading=\"lazy\"><\/p>\n<p>\u540c\u65f6\u5229\u7528\u4e0a\u8ff0\u8bbe\u5b9a\u7684\u7cfb\u6570k\u6c42\u5f97\u4f53\u7cfb\u7684\u672c\u5f81\u80fd\u7ea7\u4e3a\uff1a<\/p>\n<p style=\"text-align: center;\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/latex.codecogs.com\/gif.latex?E=\\frac{n^{2}&amp;space;\\pi^{2}&amp;space;\\hbar^{2}}{2ma^{2}},n=(1,2,3\\cdots&amp;space;)\" border=\"0\" alt=\"E=\\frac{n^{2} \\pi^{2} \\hbar^{2}}{2ma^{2}},n=(1,2,3\\cdots )\" align=\"absmiddle\" loading=\"lazy\"><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u4e00\u7ef4\u52bf\u573a\u5b9a\u6001\u859b\u5b9a\u8c14\u65b9\u7a0b\u6c42\u89e3<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"wpai_generated_summary":"","footnotes":""},"categories":[1],"tags":[],"class_list":["post-100009","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"blocksy_meta":{"styles_descriptor":{"styles":{"desktop":"","tablet":"","mobile":""},"google_fonts":[],"version":7}},"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v28.3 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>\u89e3\u4e00\u7ef4\u65e0\u9650\u6df1\u52bf\u9631(\u89e3\u5b9a\u6001\u859b\u5b9a\u8c14\u65b9\u7a0b) - martinphysics.club<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" 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