曾谨言《量子力学教程》（第3版）笔记和课后习题（含考研真题）详解

绗?绔犮€€娉㈠嚱鏁颁笌Schrödinger鏂圭▼1.1銆€澶嶄範绗旇鏈珷浠嬬粛浜嗛噺瀛愬姏瀛︿腑鏈€鍩烘湰鐨勬尝鍑芥暟姒傚康鍜岃〃杈惧舰寮忥紝骞朵笖浠嬬粛浜嗘尝鍑芥暟鐨勭粺璁¤癄閲婂拰閲忓瓙鎬佺殑鍙犲姞鍘熺悊锛岀浉搴旂殑寮曞叆鎻忚堪閲忓瓙鎬佸姩鍔涘杩囩▼鐨勮枦瀹氳皵鏂圭▼锛岀浉姣旇緝浜庣粡鍏哥墿鐞嗕腑鐨勭墰椤垮畾寰嬶紝鍦ㄩ噺瀛愬姏瀛︿腑锛岄潪鐩稿璁烘儏鍐典笅鐨勮枦瀹氳皵鏂圭▼鍚屾牱鍗犳嵁浜嗛噸瑕佸湴浣嶏紝瀵逛簬鏈夐檺鍑犵鍙В妯″瀷锛屽寘鎷娍闃便€佸娍鍨掑拰绾挎€ц皭鎸瓙锛屼互鍙婃阿鍘熷瓙鎯呭舰锛岃枦瀹氳皵鏂圭▼鍙互姹傝В鍑烘尝鍑芥暟鍜岃兘绾х殑鍏蜂綋褰㈠紡锛屽叾缁撴灉鍙互瀹岀編瑙ｉ噴鐗╃悊瀹為獙涓浉鍏崇幇璞″拰鏁版嵁锛岃€屽浜庝笉鍙簿纭眰鍑鸿В鏋愯В鐨勬ā鍨嬶紝閽堝鍏蜂綋涓嶅悓鐨勬ā鍨嬶紝鍦ㄧ墿鐞嗚儗鏅殑鑰冭檻涓嬪彲浠ヨ繘琛岀浉搴旂殑杩戜技澶勭悊锛岀敱钖涘畾璋旀柟绋嬪緱鍑虹殑杩戜技瑙ｄ篃鍙互姣旇緝婊℃剰鍦板瀹為獙鐜拌薄鍜岀粨鏋滃仛鍑鸿В閲婏紝鍚庨潰鐨勭珷鑺傚井鎵扮悊璁洪儴鍒嗕細瀵硅繎浼兼眰瑙ｈ缁嗛槓杩帮紝鍦ㄩ珮绛夐噺瀛愬姏瀛︿腑锛屽皢浼氫粙缁嶇浉瀵硅鎯呭舰涓嬬殑鐙勬媺鍏嬫柟绋嬨€?/p>涓€銆佹尝鍑芥暟鐨勭粺璁¤癄閲?/p>1瀹炵墿绮掑瓙鐨勬尝鍔ㄦ€?/p>deBroglie锛?923锛夋彁鍑轰簡瀹炵墿绮掑瓙锛堥潤璐ㄩ噺m鈮?鐨勭矑瀛愶紝濡傜數瀛愶級涔熷叿鏈夋尝绮掍簩璞℃€э紙wave-particleduality锛夌殑鍋囪锛屽嵆涓庡姩閲忎负p鍜岃兘閲忎负E鐨勭矑瀛愮浉搴旂殑娉㈢殑娉㈤暱位鍜岄鐜囄戒负位锛漢/p锛屛斤紳E/h骞剁О涔嬩负鐗╄川娉紙matterwave锛夈€?/p>2姒傜巼娉紝澶氱矑瀛愪綋绯荤殑娉㈠嚱鏁?/p>Born鎻愬嚭鐨勬尝鍑芥暟鐨勬鐜囪癄閲婏細|蠄锛?ruby>r(鈫?/rt>)锛墊2螖x螖y螖z琛ㄥ緛鍦╮鐐瑰鐨勪綋绉厓螖x螖y螖z涓壘鍒扮矑瀛愮殑姒傜巼锛屾牴鎹尝鍑芥暟鐨勭粺璁¤癄閲婏紝瑕佹眰娉㈠嚱鏁跋堬紙r锛夋弧瓒充笅鍒楁潯浠?/p>杩欑О涓烘尝鍑芥暟鐨勫綊涓€鍖栵紙normalization锛夋潯浠躲€?/p>褰掍竴鍖栨潯浠跺氨鍙互绠€鍗曡〃绀轰负锛埾堬紝蠄锛夛紳13鍔ㄩ噺鍒嗗竷姒傜巼锛堜笁缁存儏鍐典笅锛?/p>鍔ㄩ噺鍒嗗竷姒傜巼瀵嗗害鍗硘蠁锛坧锛墊2銆?/p>4涓嶇‘瀹氭€у師鐞嗕笌涓嶇‘瀹氬害鍏崇郴浣嶇疆锛堝潗鏍囷級鍜屽姩閲忕殑涓嶇‘瀹氭€у叧绯?/p>涓婂紡琛ㄦ槑绮掑瓙澶勪簬浠讳綍閲忓瓙鎬佷笅锛屽畠鐨勪綅缃紙鍧愭爣锛夊拰鍔ㄩ噺涓嶈兘鍚屾椂鍏锋湁瀹屽叏纭畾鐨勫€硷紝x鍜宲鏄竴瀵瑰鍋堕噺锛屽悓鐞嗭紝鑳介噺E鍜屾椂闂磘涔熷叿鏈夌浉鍚岀殑鍏崇郴寮忋€?/p>5鍔涘閲忕殑骞冲潎鍊间笌绠楃鐨勫紩杩?/p>p(鈭?/rt>)涓哄姩閲忕畻绗︼紝鍦ㄥ潗鏍囪〃璞′腑鍏跺叿浣撹〃杈惧舰寮忎负l(鈫?/rt>)锛?ruby>r(鈫?/rt>)脳p(鈭?/rt>)锛堣鍔ㄩ噺绠楃锛?/p>l鏄竴涓煝閲忕畻绗︼紝瀹冪殑涓変釜鍒嗛噺鍙互琛ㄧず涓?/p>A(鈭?/rt>)鏄笌鍔涘閲廇鐩稿簲鐨勭畻绗︼紝鍏跺钩鍧囧€间负濡傛尝鍑芥暟鏈綊涓€鍖栵紝鍒?/p>涓庣粡鍏窰amilton閲廐锛漈锛媀鐩稿簲鐨勭畻绗﹁〃绀轰负鍏朵腑锛孴涓虹矑瀛愮殑鍔ㄨ兘锛孷涓虹矑瀛愬湪鍔垮満涓殑鍔胯兘銆?/p>6缁熻璇犻噴瀵规尝鍑芥暟鎻愬嚭鐨勮姹?/p>鏍规嵁缁熻璇犻噴锛屽娉㈠嚱鏁跋堬紙r锛夌殑瑕佹眰涓?/p>锛?锛墊蠄锛坮锛墊2鍙栨湁闄愬€硷紝鍗宠姹傁堬紙r锛夊彇鏈夐檺鍊笺€?/p>锛?锛夋尝鍑芥暟闇€瑕佹弧瓒冲綊涓€鍖栨潯浠讹紙骞虫柟鍙Н锛?/p>浣嗘鐜囨弿杩颁腑瀹炶川鐨勯棶棰樻槸鐩稿姒傜巼锛屽洜姝わ紝鍦ㄩ噺瀛愬姏瀛︿腑骞朵笉鎺掗櫎浣跨敤鏌愪簺涓嶈兘褰掍竴鍖栫殑鐞嗘兂鐨勬尝鍑芥暟锛屾瘮濡傚钩闈㈡尝銆?/p>锛?锛墊蠄锛坮锛墊2鍗曞€硷紝浣喯堬紙r锛変笉涓€瀹氬彇鍗曞€笺€?/p>锛?锛夋尝鍑芥暟蠄锛坮锛夊強鍏跺悇闃跺井鍟嗙殑杩炵画鎬э紙涓€鑸儏褰㈠彲浠ヨ繖鏍疯€冭檻锛屼絾閬囧埌鐗规畩鎯呭喌瑕佽鍏蜂綋鐨勫娍鑳借€屽畾锛夈€?/p>7绱㈡湯鑿查噺瀛愬寲鏉′欢鐨勬帹骞?/p>8娉㈢矑浜岃薄鎬э紙瑙佽〃1-1-1锛?/p>琛?-1-1銆€娉㈢矑浜岃薄鎬х浉鍏虫蹇?/p>浜屻€丼chrödinger鏂圭▼1Schrödinger鏂圭▼鐨勫紩杩?/p>鍦ㄥ娍鍦篤锛坮锛変腑鐨勭矑瀛愮殑娉㈠嚱鏁版弧瓒崇殑寰垎鏂圭▼锛岀О涓篠chrödinger娉㈠姩鏂圭▼銆?/p>2Schrödinger鏂圭▼鐨勮璁?/p>瀹氬煙鐨勬鐜囧畧鎭?/p>瀵逛簬涓€涓矑瀛愶紙闈炵浉瀵硅鎯呭舰锛夋潵璇达紝鍦ㄥ叏绌洪棿涓壘鍒板畠鐨勬鐜囦箣鎬诲拰搴斾笉闅忔椂闂存敼鍙橈紝鍗?/p>銆€鈶?/span>鈶?/span>寮忎负姒傜巼瀹堟亽鐨勫井鍒嗚〃杈惧紡锛屽叾褰㈠紡涓庢祦浣撳姏瀛︿腑鐨勮繛缁€ф柟绋嬬浉鍚屻€?/p>3鑳介噺鏈緛鏂圭▼鍋囪鍔胯兘V涓嶆樉鍚玹锛屽垯娉㈠嚱鏁板彲浠ュ啓涓?/p>鍏朵腑蠄E锛坮锛夋弧瓒充笅鍒楁柟绋嬶細銆€鈶?/span>鍦ㄦ湁鐨勬潯浠朵笅锛岀壒鍒槸鏉熺細鎬佽竟鏉′欢锛屽彧鏈夋煇浜涚鏁ｇ殑E鍊兼墍瀵瑰簲鐨勮В鎵嶆槸鐗╃悊涓婂彲浠ユ帴鍙楃殑锛岃繖浜汦鍊肩О涓轰綋绯荤殑鑳介噺鏈緛鍊硷紙energyeigenvalue锛夛紝鑰岀浉搴旂殑瑙Ｏ堬紙r锛夌О涓鸿兘閲忔湰寰佸嚱鏁帮紙energyeigenfunction锛夛紝鏂圭▼鈶?/span>灏辨槸鍔垮満V锛坮锛変腑绮掑瓙鐨勮兘閲忔湰寰佹柟绋嬶紝涔熺О涓轰笉鍚椂锛坱ime-independent锛塖chrödinger鏂圭▼銆?/p>涓嶅悓鐨勮兘閲忔湰寰佸€肩浉搴旂殑鏈緛鍑芥暟鏄浜ゅ綊涓€鍖栫殑锛堣E鍙栫鏁ｅ€硷級锛屽嵆Schrödinger鏂圭▼鐨勬洿鏅亶鐨勮〃绀烘槸銆€鈶?/span>H(鈭?/rt>)鏄綋绯荤殑Hamilton绠楃锛屽綋H(鈭?/rt>)涓嶆樉鍚玹鏃讹紝浣撶郴鐨勮兘閲忔槸瀹堟亽閲忥紝鏂圭▼鈶?/span>鍙互鍒嗙鍙橀噺锛屾鏃讹紝涓嶅惈鏃禨chrödinger鏂圭▼锛屽嵆鑳介噺鏈緛鏂圭▼锛屼负H(鈭?/rt>)蠄锛滶蠄4瀹氭€佷笌闈炲畾鎬?/p>鑻ュ湪鍒濆鏃跺埢锛坱锛?锛変綋绯诲浜庢煇涓€涓兘閲忔湰寰佹€佅堬紙r锛?锛夛紳蠄E锛坮锛夛紝鍒?/p>銆€鈶?/span>褰㈠紡濡傚紡鈶?/span>鐨勬尝鍑芥暟鎵€鎻忚堪鐨勬€侊紝绉颁负瀹氭€侊紙stationarystate锛夛紝澶勪簬瀹氭€佷笅鐨?/p>绮掑瓙鍏锋湁濡備笅鐗瑰緛锛?/p>锛?锛夌矑瀛愬湪绌洪棿鐨勬鐜囧瘑搴ο侊紙r锛夛紳|蠄锛坮锛墊2浠ュ強姒傜巼娴佸瘑搴鏄剧劧涓嶉殢鏃堕棿鏀瑰彉锛?锛変换浣曪紙涓嶆樉鍚玹鐨勶級鍔涘閲忕殑骞冲潎鍊间笉闅忔椂闂存敼鍙樸€?/p>锛?锛変换浣曪紙涓嶆樉鍚玹鐨勶級鍔涘閲忕殑娴嬮噺姒傜巼鍒嗗竷涔熶笉闅忔椂闂存敼鍙樸€?/p>鐢辫嫢骞蹭釜鑳介噺涓嶅悓鐨勬湰寰佹€佺殑鍙犲姞鎵€褰㈡垚鐨勬€侊紝绉颁负闈炲畾鎬侊紙nonstationarystate锛夈€?/p>5澶氱矑瀛愪綋绯荤殑Schrödinger鏂圭▼璁句綋绯荤敱N涓矑瀛愮粍鎴愶紝绮掑瓙璐ㄩ噺鍒嗗埆涓簃i锛坕锛?锛?锛?锛屸€︼紝N锛夛紝浣撶郴鐨勬尝鍑芥暟琛ㄧず涓合堬紙r1锛屸€︼紝rN锛宼锛夛紝璁剧i涓矑瀛愬彈鍒扮殑澶栧娍鍦轰负Ui锛坮i锛夛紝绮掑瓙涔嬮棿鐩镐簰浣滅敤涓篤锛坮1锛屸€︼紝rN锛宼锛夛紝鍒橲chrödinger鏂圭▼琛ㄧず涓?/p>鍏朵腑鑰屼笉鍚椂Schrödinger鏂圭▼琛ㄧず涓?/p>E涓哄绮掑瓙浣撶郴鐨勬湰寰佽兘閲忋€?/p>涓夈€侀噺瀛愭€佸彔鍔犲師鐞?/p>1閲忓瓙鎬佸強鍏惰〃璞?/p>褰撓堬紙r锛夌粰瀹氬悗锛屼笁缁寸┖闂翠腑涓€涓矑瀛愭墍鏈夊姏瀛﹂噺鐨勬祴鍊兼鐜囧垎甯冨氨纭畾浜嗭紝浠庤繖涓剰涔変笂鏉ヨ锛屜堬紙r锛夊畬鍏ㄦ弿杩颁簡涓€涓笁缁寸┖闂翠腑绮掑瓙鐨勯噺瀛愭€侊紝鎵€浠ユ尝鍑芥暟涔熺О涓烘€佸嚱鏁般€?/p>2閲忓瓙鎬佸彔鍔犲師鐞嗭紝娴嬮噺涓庢尝鍑芥暟鍧嶇缉锛?锛夎浣撶郴澶勪簬蠄鎻忚堪鐨勬€佷笅锛屾祴閲忓姏瀛﹂噺A鎵€寰楃粨鏋滄槸涓€涓‘鍒囧€糰1锛埾?sub>1涔熺О涓篈鐨勬湰寰佹€侊紝A鐨勬湰寰佸€间负a1锛夛紝鍙堝亣璁惧湪蠄2鎬佷笅锛屾祴閲廇寰楃殑缁撴灉鏄彟涓€涓‘鍒囧€糰2锛埾?sub>2涔熸槸A鐨勪竴涓湰寰佹€侊紝鏈緛鍊间负a2锛夛紝鍒欏湪蠄锛漜1蠄1锛媍2蠄2鎵€鎻忚堪鐨勭姸鎬佷笅锛屾祴閲廇鎵€寰楃粨鏋滐紝鏃㈠彲鑳戒负a1锛屼篃鍙兘涓篴