{"id":1984,"date":"2023-03-15T01:18:51","date_gmt":"2023-03-14T17:18:51","guid":{"rendered":"http:\/\/ustb.cc\/zh\/?p=1984"},"modified":"2023-03-16T09:31:43","modified_gmt":"2023-03-16T01:31:43","slug":"time-dependent-perturbation","status":"publish","type":"post","link":"http:\/\/ustb.cc\/zh\/time-dependent-perturbation\/","title":{"rendered":"\u542b\u65f6\u5fae\u6270\u90e8\u5206\u516c\u5f0f\u63a8\u5bfc"},"content":{"rendered":"\n

1. Quantum dynamics\u91cf\u5b50\u52a8\u529b\u5b66<\/h2>\n\n\n\n

\u5230\u76ee\u524d\u4e3a\u6b62\uff0c\u6240\u6709\u7684\u52bf\u80fd\u51fd\u6570\u5747\u4e0e\u65f6\u95f4\u65e0\u5173\uff1a<\/p>\n\n\n\n

$${V{(r,t)}} = {V{(r)}}$$<\/p>\n\n\n\n

\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u859b\u5b9a\u8c14\u65b9\u7a0b\u53ef\u901a\u8fc7\u5206\u79bb\u53d8\u91cf\u6c42\u89e3\uff1a<\/p>\n\n\n\n

$$i\\hbar \\frac{{\\partial \\psi }}{{\\partial t}} = H\\psi $$ $$\\Psi (r,t) = \\psi (r){e^{ – iEt\/\\hbar }}$$<\/p>\n\n\n\n

\u5176\u4e2d \u03c8<\/em>(r<\/em>) \u6ee1\u8db3\u4e0d\u542b\u65f6\u859b\u5b9a\u8c14\u65b9\u7a0b<\/p>\n\n\n\n

\u65f6\u95f4\u76f8\u5173\u6027\u7531\u6307\u6570\u56e0\u5b50(e-iEt\/\u045b<\/sup>)<\/em>\u4f53\u73b0\uff0c\u4f46\u5728\u6784\u5efa |\u03a8|2<\/sup>\u65f6\u4e92\u76f8\u62b5\u6d88\uff0c\u5bfc\u81f4\u6240\u6709\u6982\u7387\u548c\u671f\u671b\u503c\u4e0d\u968f\u65f6\u95f4\u53d8\u5316<\/p>\n\n\n\n

\u5c06\u5b9a\u6001\u7ebf\u6027\u7ec4\u5408\uff0c\u53ef\u4ee5\u4f7f\u5f97\u6ce2\u51fd\u6570\u5177\u6709\u65f6\u95f4\u76f8\u5173\u6027,\u4f46\u6d4b\u91cf\u80fd\u91cf\u7684\u53ef\u80fd\u503c\u53ca\u5176\u51fa\u73b0\u6982\u7387\u4f9d\u7136\u662f\u56fa\u5b9a\u7684<\/p>\n\n\n\n

\u4e3a\u4e86\u4f7f\u4e00\u4e2a\u80fd\u7ea7\u53ef\u4ee5\u8dc3\u8fc1\u5230\u53e6\u4e00\u4e2a\u80fd\u7ea7\uff0c\u9700\u8981\u5f15\u5165\u542b\u65f6\u5fae\u6270\u9879<\/p>\n\n\n\n

\u5f53\u54c8\u5bc6\u987f\u91cf\u542b\u65f6\u90e8\u5206\u5c0f\u4e8e\u4e0d\u542b\u65f6\u90e8\u5206\uff0c\u5219\u53ef\u5c06\u5176\u5904\u7406\u4e3a\u5fae\u6270<\/p>\n\n\n\n

2. Two-level systems\u4e8c\u80fd\u7ea7\u4f53\u7cfb<\/h2>\n\n\n\n

\u5047\u8bbe\u672a\u5fae\u6270\u4f53\u7cfb\u4e2d\u4ec5\u5b58\u5728\u4e24\u4e2a\u6001, \u03c8a<\/sub><\/strong><\/em> \u548c \u03c8b<\/sub><\/strong><\/em>\uff0c\u90fd\u662f\u672a\u5fae\u6270\u54c8\u5bc6\u987f\u91cfH<\/strong><\/em>0<\/strong><\/sup>\u7684\u672c\u5f81\u6001<\/p>\n\n\n\n

$${H^0}{\\psi _a} = {E_a}{\\psi _a}$$ $${H^0}{\\psi _b} = {E_b}{\\psi _b}$$<\/p>\n\n\n\n

\u8fd9\u4e24\u4e2a\u6001\u6b63\u4ea4\u5f52\u4e00<\/p>\n\n\n\n

$$\\langle{\\psi_a}|{\\psi _b}\\rangle = {\\delta_{ab}}$$<\/p>\n\n\n\n

\u8be5\u4f53\u7cfb\u5185\u7684\u4efb\u610f\u6001\u90fd\u53ef\u8868\u793a\u4e3a\u8fd9\u4e24\u4e2a\u672c\u5f81\u6001\u7684\u7ebf\u6027\u7ec4\u5408<\/p>\n\n\n\n

$$\\Psi (0) = {c_a}{\\psi _a} + {c_b}{\\psi _b}$$<\/p>\n\n\n\n

$$\\Psi (t) = {c_a}{\\psi _a}{e^{ – i{E_a}t\/\\hbar }} + {c_b}{\\psi _b}{e^{ – i{E_b}t\/\\hbar }}$$<\/p>\n\n\n\n

$${\\left| {{c_a}} \\right|^2} + {\\left| {{c_b}} \\right|^2} = 1$$<\/p>\n\n\n\n

3. The perturbed system<\/h2>\n\n\n\n

\u5f15\u5165\u542b\u65f6\u5fae\u6270H<\/em>‘(t<\/em>)<\/p>\n\n\n\n

\u7531\u4e8e\u03c8<\/em>a<\/sub><\/em> <\/em>\u548c \u03c8<\/em>b<\/sub><\/em> <\/em>\u6784\u6210\u4e00\u4e2a\u5b8c\u5907\u96c6\uff0c\u6545\u53ef\u4ee5\u901a\u8fc7\u7ebf\u6027\u7ec4\u5408\u8868\u793a \u03a8(t<\/em>)<\/p>\n\n\n\n

\u6b64\u65f6\uff0c c<\/em>a<\/sub><\/em> \u548c c<\/em>b<\/sub><\/em> \u662f\u4e0e\u65f6\u95f4\u6709\u5173\u7684\u51fd\u6570<\/p>\n\n\n\n

$$\\Psi (t) = {c_a}(t){\\psi _a}{e^{ – i{E_a}t\/\\hbar }} + {c_b}(t){\\psi _b}{e^{ – i{E_b}t\/\\hbar }}$$<\/p>\n\n\n\n

\u95ee\u9898\u5f52\u7ed3\u4e3a\u6c42\u89e3ca<\/sub><\/em> \u548c cb<\/sub><\/em>\u968f\u65f6\u95f4\u7684\u53d8\u5316\u89c4\u5f8b<\/p>\n\n\n\n

\u8bbe\u7c92\u5b50\u7684\u521d\u59cb\u72b6\u6001\u4e3a\u03c8a<\/sub><\/em>(ca<\/sub><\/em>(0)=1, cb<\/sub><\/em>(0)=0), \u7ecf\u8fc7\u65f6\u95f4t<\/em>1<\/sub>\uff0c\u53d1\u73b0\u7cfb\u6570\u53d8\u4e3aca<\/sub><\/em>(t<\/em>1<\/sub>)=0, cb<\/sub><\/em>(t<\/em>1<\/sub>)=1, \u8fd9\u8bf4\u660e\u7cfb\u7edf\u7ecf\u5386\u4e86\u4ece\u03c8a<\/sub><\/em> \u5230 \u03c8b<\/sub><\/em>\u7684\u8dc3\u8fc1<\/p>\n\n\n\n

\u03a8(t<\/em>)\u9700\u6ee1\u8db3\u542b\u65f6\u859b\u5b9a\u8c14\u65b9\u7a0b<\/p>\n\n\n\n

$$\\Psi (t) = {c_a}(t){\\psi _a}{e^{ – i{E_a}t\/\\hbar }} + {c_b}(t){\\psi _b}{e^{ – i{E_b}t\/\\hbar }}$$<\/p>\n\n\n\n

<\/p>\n\n\n\n

$$i\\hbar \\frac{{\\partial \\psi }}{{\\partial t}} = H\\psi ,{\\text{ }}H = {H^0} + H'(t)$$<\/p>\n\n\n\n

$$i\\hbar \\frac{\\partial }{{\\partial t}}\\left[ {{c_a}(t){\\psi _a}{e^{ – i{E_a}t\/\\hbar }} + {c_b}(t){\\psi _b}{e^{ – i{E_b}t\/\\hbar }}} \\right]$$<\/p>\n\n\n\n

$$\\eqalign{
& i\\hbar \\left[ {{{\\dot c}_a}{\\psi _a}{e^{ – i{E_a}t\/\\hbar }} + {{\\dot c}_b}{\\psi _b}{e^{ – i{E_b}t\/\\hbar }} + {c_a}{\\psi _a}\\left( { – \\frac{{i{E_a}}}{\\hbar }} \\right){e^{ – i{E_a}t\/\\hbar }} + {c_b}{\\psi _b}\\left( { – \\frac{{i{E_b}}}{\\hbar }} \\right){e^{ – i{E_b}t\/\\hbar }}} \\right] \\cr
& = {c_a}[{H^{\\text{0}}}{\\psi _a}]{e^{ – i{E_a}t\/\\hbar }} + {c_b}[{H^{\\text{0}}}{\\psi _b}]{e^{ – i{E_b}t\/\\hbar }} + {c_a}[H'{\\psi _a}]{e^{ – i{E_a}t\/\\hbar }} + {c_b}[H'{\\psi _b}]{e^{ – i{E_b}t\/\\hbar }} \\cr} $$<\/p>\n\n\n\n

$$\\left( {{H^0} + H'(t)} \\right)\\left( {{c_a}(t){\\psi _a}{e^{ – i{E_a}t\/\\hbar }} + {c_b}(t){\\psi _b}{e^{ – i{E_b}t\/\\hbar }}} \\right)$$<\/p>\n\n\n\n

\u8003\u8651\u5230$${H^0}{\\psi _a} = {E_a}{\\psi _a}$$ $${H^0}{\\psi _b} = {E_b}{\\psi _b}$$<\/p>\n\n\n\n

\u4e0a\u5f0f\u5de6\u53f3\u4e24\u4fa7\u5404\u6709\u4e24\u9879\u62b5\u6d88<\/p>\n\n\n\n

$${c_a}[H'{\\psi _a}]{e^{ – i{E_a}t\/\\hbar }} + {c_b}[H'{\\psi _b}]{e^{ – i{E_b}t\/\\hbar }} = i\\hbar \\left[ {{{\\dot c}_a}{\\psi _a}{e^{ – i{E_a}t\/\\hbar }} + {{\\dot c}_b}{\\psi _b}{e^{ – i{E_b}t\/\\hbar }}} \\right]$$<\/p>\n\n\n\n

\u4e3a\u4e86\u5206\u79bb${\\dot c_a}$, \u5de6\u53f3\u4e24\u4fa7\u4e0e\u03c8a<\/sub><\/em>\u505a\u5185\u79ef\uff0c\u5e76\u5229\u7528\u03c8a<\/sub><\/em>\u548c \u03c8b<\/sub><\/em>\u7684\u6b63\u4ea4\u6027<\/p>\n\n\n\n

$${c_a}\\langle {\\psi _a}|H’|{\\psi _a}\\rangle {e^{ – i{E_a}t\/\\hbar }} + {c_b}\\langle {\\psi _a}|H’|{\\psi _b}\\rangle {e^{ – i{E_b}t\/\\hbar }} = i\\hbar {\\dot c_a}{e^{ – i{E_a}t\/\\hbar }}$$<\/p>\n\n\n\n

\u7b80\u5355\u8d77\u89c1\uff0c\u5b9a\u4e49<\/p>\n\n\n\n

$$H’_{{ij}} \\equiv \\langle {\\psi _i}|H’|{\\psi _j}\\rangle $$<\/p>\n\n\n\n

$$H’_{{{\\text{j}}i}} = {(H’_{{ij}})^*}$$<\/p>\n\n\n\n

$${\\dot c_a} = – \\frac{i}{\\hbar }\\left[ {{c_a}H’_{{aa}} + {c_b}H’_{{ab}}{e^{ – i({E_b} – {E_a})t\/\\hbar }}} \\right]$$<\/p>\n\n\n\n

\u7c7b\u4f3c\u5730, \u4e0e\u03c8b<\/sub><\/em>\u505a\u5185\u79ef\u80fd\u591f\u5f97\u5230${\\dot c_b}$<\/p>\n\n\n\n

$${c_a}\\langle {\\psi _b}|H’|{\\psi _a}\\rangle {e^{ – i{E_a}t\/\\hbar }} + {c_b}\\langle {\\psi _b}|H’|{\\psi _b}\\rangle {e^{ – i{E_b}t\/\\hbar }} = i\\hbar {\\dot c_b}{e^{ – i{E_b}t\/\\hbar }}$$<\/p>\n\n\n\n

$${\\dot c_b} = – \\frac{i}{\\hbar }\\left[ {{c_b}H’_{{bb}} + {c_a}H’_{{ba}}{e^{i({E_b} – {E_a})t\/\\hbar }}} \\right]$$<\/p>\n\n\n\n

\u4e00\u822c\u5730\uff0c H<\/em>‘ \u7684\u5bf9\u89d2\u5143\u4e3a0<\/p>\n\n\n\n

$$H’_{{aa}} = H’_{{bb}} = 0$$<\/p>\n\n\n\n

$${\\dot c_a} = – \\frac{i}{\\hbar }H’_{{ab}}{e^{ – i{\\omega _0}t}}{c_b},{\\text{ }}{\\dot c_b} = – \\frac{i}{\\hbar }H’_{{ba}}{e^{i{\\omega _0}t}}{c_a}$$<\/p>\n\n\n\n

\u5176\u4e2d ${\\omega _0} \\equiv \\frac{{{E_b} – {E_a}}}{\\hbar }$ <\/em><\/strong> (\u5047\u8bbeEb<\/sub><\/em>\u2265Ea<\/sub><\/em>, \u6545 \u03c9<\/em>0<\/sub> \u22650)<\/p>\n\n\n\n

4. Time-dependent perturbation theory \u542b\u65f6\u5fae\u6270\u7406\u8bba<\/h2>\n\n\n\n

\u5230\u76ee\u524d\u4e3a\u6b62\uff0c\u5e76\u6ca1\u6709\u9650\u5b9a\u5fae\u6270\u7684\u5c3a\u5ea6<\/p>\n\n\n\n

\u4f46\u5982\u679cH<\/em>‘ \u662f\u5c0f\u91cf, \u5219\u53ef\u7528\u8fde\u7eed\u8fd1\u4f3c\u5730\u65b9\u6cd5\u5904\u7406<\/p>\n\n\n\n

\u5047\u8bbe\u521d\u59cb\u65f6\u523b\u7c92\u5b50\u5904\u4e8e\u4f4e\u80fd\u6001<\/p>\n\n\n\n

$${c_a}(0) = 1,{\\text{ }}{c_b}(0) = 0$$<\/p>\n\n\n\n

\u5982\u679c\u6ca1\u6709\u6270\u52a8\uff0c\u5219\u7c92\u5b50\u6c38\u8fdc\u5904\u4e8e\u8fd9\u4e2a\u6001 \uff08\u4e3a\u4ec0\u4e48\uff1f\uff09<\/p>\n\n\n\n

Zeroth Order<\/em><\/strong><\/p>\n\n\n\n

$${c_a}^{(0)}(t) = 1,{\\text{ }}{c_b}^{(0)}(t) = 0$$<\/p>\n\n\n\n

\u8ba1\u7b97\u4e00\u7ea7\u8fd1\u4f3c\uff0c\u5c06\u96f6\u7ea7\u8fd1\u4f3c\u7684\u503c\u4ee3\u5165\u4e0b\u5f0f\u53f3\u4fa7<\/p>\n\n\n\n

H’<\/em>\u662f\u5c0f\u91cf<\/p>\n\n\n\n

$${\\dot c_a} = – \\frac{i}{\\hbar }H’_{{ab}}{e^{ – i{\\omega _0}t}}{c_b},{\\text{ }}{\\dot c_b} = – \\frac{i}{\\hbar }H’_{{ba}}{e^{i{\\omega _0}t}}{c_a}$$<\/p>\n\n\n\n

First Order<\/em><\/strong><\/p>\n\n\n\n

$$\\frac{{dc_a^{(1)}}}{{dt}} = 0 \\Rightarrow c_a^{(1)}(t) = 1$$<\/p>\n\n\n\n

\u6b64\u5904\u7684\u4e0a\u6807\u542b\u4e49\u4e0e\u524d\u9762\u4e0d\u540c\uff0c\u662f\u201c\u6574\u4f53\u201d\u800c\u975e\u201c\u5dee\u503c\u201d<\/p>\n\n\n\n

$$\\frac{{dc_b^{(1)}}}{{dt}} = – \\frac{i}{\\hbar }H’_{{ba}}{e^{i{\\omega _0}t}} \\Rightarrow c_b^{(1)} = – \\frac{i}{\\hbar }\\int_0^t {H’_{{ba}}(t’){e^{i{\\omega _0}t’}}} dt’$$<\/p>\n\n\n\n

\u540c\u7406\uff0c\u8ba1\u7b97\u4e8c\u7ea7\u8fd1\u4f3c\uff0c\u5c06\u4e00\u7ea7\u8fd1\u4f3c\u7684\u503c\u4ee3\u5165\u53f3\u4fa7<\/p>\n\n\n\n

Second Order<\/em><\/strong><\/p>\n\n\n\n

$$\\dot c_a^{(2)} = – \\frac{i}{\\hbar }H’_{{ab}}{e^{ – i{\\omega _0}t}}c_b^{(1)}$$<\/p>\n\n\n\n

$$c_b^{(1)} = – \\frac{i}{\\hbar }\\int_0^t {H'{_{ba}}(t’){e^{i{\\omega _0}t’}}} dt’$$<\/p>\n\n\n\n

$$\\eqalign{
& \\frac{{dc_a^{(2)}}}{{dt}} = – \\frac{i}{\\hbar }H’_{{ab}}{e^{ – i{\\omega _0}t}}\\left( { – \\frac{i}{\\hbar }} \\right)\\int_0^t {H’_{{ba}}(t’){e^{i{\\omega 0}t’}}} dt’ \\Rightarrow \\cr & c_a^{(2)}(t) = 1 – \\frac{1}{{{\\hbar ^2}}}\\int_0^t {H’_{{ab}}(t’){e^{ – i{\\omega 0}t’}}} \\left[ {\\int_0^{t’} {H’_{{ba}}(t^”){e^{i{\\omega _0}t^”}}} dt^”} \\right]dt’ \\cr} $$<\/p>\n\n\n\n

$${\\text{ }}\\dot c_b^{(2)} = – \\frac{i}{\\hbar }H'{_{ba}}{e^{i{\\omega _0}t}}c_a^{(1)}$$<\/p>\n\n\n\n

$$c_a^{(1)}(t) = 1$$<\/p>\n\n\n\n

$$c_b^{({\\text{2}})} = – \\frac{i}{\\hbar }\\int_0^t {H'{_{ba}}(t’){e^{i{\\omega _0}t’}}} dt’$$<\/p>\n\n\n\n

cb<\/sub><\/em> \u4e0d\u53d8, $c_b^{({\\text{2}})}({\\text{t}}) = c_b^{({\\text{1}})}({\\text{t}})$<\/strong><\/em><\/p>\n\n\n\n

ca<\/sub>(2)<\/sup> <\/sub>(t)<\/em> \u7684\u8868\u8fbe\u5f0f\u4e2d\u5305\u542b\u96f6\u7ea7\u9879\uff0c\u56e0\u6b64\u4e8c\u7ea7\u4fee\u6b63\u9879\u4ec5\u662f\u79ef\u5206\u90e8\u5206<\/p>\n\n\n\n

\u4e0a\u8ff0\u6b65\u9aa4\u53ef\u4e00\u76f4\u91cd\u590d\u4e0b\u53bb\uff1a\u5c06\u7b2cn<\/em>\u7ea7\u8fd1\u4f3c\u63d2\u5165\u53f3\u4fa7\uff0c\u89e3\u51fa\u7b2cn<\/em>+1\u7ea7\u8fd1\u4f3c<\/p>\n\n\n\n

$${c_a}^{(0)}(t) = 1,{\\text{ }}{c_b}^{(0)}(t) = 0$$<\/p>\n\n\n\n

$$c_a^{(1)}(t) = 1,c_b^{(1)} = – \\frac{i}{\\hbar }\\int_0^t {H'{_{ba}}(t’){e^{i{\\omega _0}t’}}} dt’$$<\/p>\n\n\n\n

$$\\eqalign{
& c_a^{(2)}(t) = 1 – \\frac{1}{{{\\hbar ^2}}}\\int_0^t {H’_{{ab}}(t’){e^{ – i{\\omega _0}t’}}} \\left[ {\\int_0^{t’} {H’_{{ba}}(t^”){e^{i{\\omega 0}t”}}} dt^”} \\right]dt’ \\cr & c_b^{({\\text{2}})} = – \\frac{i}{\\hbar }\\int_0^t {H’_{{ba}}(t’){e^{i{\\omega _0}t’}}} dt’ \\cr} $$<\/p>\n\n\n\n

\u96f6\u7ea7\u9879\u4e0d\u542bH<\/em>‘, \u4e00\u7ea7\u9879\u542b\u6709\u4e00\u4e2aH<\/em>‘, \u4e8c\u7ea7\u9879\u542b\u6709\u4e24\u4e2aH<\/em>‘, \u4f9d\u6b64\u7c7b\u63a8<\/p>\n\n\n\n

\u4e00\u7ea7\u8fd1\u4f3c\u7684\u8bef\u5dee\u53ef\u4ece\u4e0b\u5f0f\u770b\u51fa\uff1a<\/p>\n\n\n\n

$${\\left| {c_a^{(1)}(t)} \\right|^2} + {\\left| {c_b^{(1)}(t)} \\right|^2} \\ne 1$$<\/p>\n\n\n\n

\u7136\u800c\uff0c\u5982\u679c\u4ec5\u8003\u8651\u5230H’<\/em>\u7684\u4e00\u7ea7\u9879\uff0c<\/p>\n\n\n\n

$${\\left| {c_a^{(1)}(t)} \\right|^2} + {\\left| {c_b^{(1)}(t)} \\right|^2} = 1$$<\/p>\n\n\n\n

\u540c\u7406\u9002\u7528\u4e8e\u9ad8\u7ea7\u9879<\/p>\n\n\n\n

5. Sinusoidal Perturbations\u6b63\u5f26\u6270\u52a8<\/h2>\n\n\n\n

\u5047\u8bbe\u5fae\u6270\u4e0e\u65f6\u95f4\u5b58\u5728\u6b63\u5f26\u5173\u7cfb:<\/p>\n\n\n\n

$$H’_(r,t) = V(r)\\cos (\\omega t)$$<\/p>\n\n\n\n

$$H’_{{ab}} = {V_{ab}}\\cos (\\omega t)$$<\/p>\n\n\n\n

\u5176\u4e2d$${V_{ab}} \\equiv \\langle {\\psi _a}|V|{\\psi _b}\\rangle $$<\/p>\n\n\n\n

\u4e00\u7ea7\u8fd1\u4f3c\u4e0b\uff0c\u6709<\/p>\n\n\n\n

$$c_b^{(1)} = – \\frac{i}{\\hbar }\\int_0^t {H'{_{ba}}(t’){e^{i{\\omega _0}t’}}} dt’$$<\/p>\n\n\n\n

\u5c06H<\/em>‘ab<\/sub><\/em>\u4ee3\u5165\u4e0a\u5f0f<\/p>\n\n\n\n

$$\\eqalign{
& {c_b} \\cong – \\frac{i}{\\hbar }{V_{ba}}\\int_0^t {\\cos (\\omega t’){e^{i{\\omega 0}t’}}} dt’ = – \\frac{{i{V_{ba}}}}{{2\\hbar }}\\int_0^t {\\left[ {{e^{i({\\omega 0} + \\omega )t’}} + {e^{i({\\omega _0} – \\omega )t’}}} \\right]} dt’ \\cr & {\\text{ }} = – \\frac{{{V_{ba}}}}{{2\\hbar }}\\left[ {\\frac{{{e^{i({\\omega _0} + \\omega )t}} – 1}}{{{\\omega _0} + \\omega }} + \\frac{{{e^{i({\\omega _0} – \\omega )t}} – 1}}{{{\\omega _0} – \\omega }}} \\right] \\cr} $$<\/p>\n\n\n\n

\u5047\u8bbe\u9a71\u52a8\u9891\u7387(driving frequencies, \u03c9<\/em>) \u4e0e\u8dc3\u8fc1\u9891\u7387(transition frequency \u03c9<\/em>0<\/sub>), \u975e\u5e38\u63a5\u8fd1\uff0c\u5219\u4e0a\u5f0f\u4e2d\u7b2c\u4e8c\u9879\u5360\u4e3b\u5bfc\uff1b\u5373<\/p>\n\n\n\n

$${\\omega _0} + \\omega \\gg \\left| {{\\omega _0} – \\omega } \\right|$$<\/p>\n\n\n\n

\u4e0a\u5f0f\u4e0d\u5168\u662f\u9650\u5236\uff0c\u56e0\u4e3a\u5176\u4ed6\u9891\u7387\u7684\u5fae\u6270\u4e00\u822c\u5f88\u96be\u9020\u6210\u8dc3\u8fc1<\/p>\n\n\n\n

\u4e0b\u9762\u5c06\u8be5\u7406\u8bba\u5e94\u7528\u4e8e\u5149\uff0c\u5176\u9891\u7387\u03c9<\/em>~1015<\/sup> s-1<\/sup>, \u56e0\u6b64\u4e24\u9879\u7684\u5206\u6bcd\u90fd\u975e\u5e38\u5927\uff0c\u9664\u975e\u5728\u03c9<\/em>0 <\/sub>\u9644\u8fd1<\/p>\n\n\n\n

\u5ffd\u7565\u7b2c\u4e00\u9879\uff0c\u5f97\u5230<\/p>\n\n\n\n

$$\\eqalign{
& {c_b}(t) \\cong – \\frac{{{V_{ba}}}}{{2\\hbar }}\\frac{{{e^{i({\\omega 0} – \\omega )t\/2}}}}{{{\\omega _0} – \\omega }}\\left[ {{e^{i({\\omega _0} – \\omega )t\/2}} – {e^{ – i({\\omega _0} – \\omega )t\/2}}} \\right] \\cr & {\\text{ }} = – i\\frac{{{V_{ba}}}}{\\hbar }\\frac{{\\sin [({\\omega _0} – \\omega )t\/2]}}{{{\\omega _0} – \\omega }}{e^{i({\\omega _0} – \\omega )t\/2}} \\cr} $$<\/p>\n\n\n\n

\u8dc3\u8fc1\u6982\u7387(transition probability)\uff1a\u521d\u6001\u4e3a\u03c8a<\/sub><\/em>\u7684\u7c92\u5b50\u5728t<\/em>\u65f6\u523b\u65f6\u5904\u4e8e\u6001 \u03c8b<\/sub><\/em>\u7684\u6982\u7387<\/p>\n\n\n\n

$${P_{a \\to b}}(t) = {\\left| {{c_b}(t)} \\right|^2} \\cong \\frac{{{{\\left| {{V_{ba}}} \\right|}^2}}}{{{\\hbar ^2}}}\\frac{{{{\\sin }^2}[({\\omega _0} – \\omega )t\/2]}}{{{{({\\omega _0} – \\omega )}^2}}}$$<\/p>\n\n\n\n

\u8dc3\u8fc1\u6982\u7387\u4e5f\u968f\u65f6\u95f4\u6b63\u5f26\u632f\u8361<\/p>\n\n\n\n

\u8dc3\u8fc1\u6982\u7387\u5347\u9ad8\u81f3\u6700\u5927\u503c|Vab<\/sub><\/em>|2<\/sup>\/\u045b<\/em>2<\/sup>(\u03c9<\/em>0 <\/sub>– \u03c9<\/em>)2<\/sup>\u540e\uff0c\u91cd\u65b0\u964d\u4e3a0\u3002\u4f46\u8be5\u6700\u5927\u503c\u5e94\u8fdc\u5c0f\u4e8e1\uff0c\u5426\u5219\u5fae\u6270\u4e0d\u9002\u7528<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

\u5728\u65f6\u523bt<\/em>n<\/sub><\/em> = 2n<\/em>\u03c0<\/em>\/|\u03c9<\/em>0 <\/sub>– \u03c9<\/em>|, n<\/em> = 1, 2, 3, \u2026 , \u7c92\u5b50\u91cd\u65b0\u5904\u4e8e\u57fa\u6001<\/p>\n\n\n\n

\u5f53\u9a71\u52a8\u9891\u7387\u4e0e\u56fa\u6709\u9891\u7387\u03c9<\/em>0<\/sub>\u63a5\u8fd1\u65f6\uff0c\u8dc3\u8fc1\u6982\u7387\u63a5\u8fd1\u4e8e\u6700\u5927\u503c\uff0c\u9ad8\u5ea6\u4e3a (|V<\/em>ab<\/sub><\/em>|t<\/em>\/2\u045b<\/em>)2<\/sup> \uff0c\u5bbd\u5ea6\u4e3a4\u03c0 \/ t<\/em><\/p>\n\n\n\n

\u968f\u65f6\u95f4\u5ef6\u957f\uff0c\u5cf0\u503c\u8d8a\u9ad8\u8d8a\u5bbd\uff0c\u4f46\u63a5\u8fd1\u4e8e1\u65f6\u5fae\u6270\u4e0d\u518d\u9002\u7528\uff0c\u6545\u4ec5\u9002\u7528\u4e8e\u8f83\u77edt<\/em><\/p>\n","protected":false},"excerpt":{"rendered":"

1. Quantum dynamics\u91cf\u5b50\u52a8\u529b\u5b66 \u5230\u76ee\u524d\u4e3a\u6b62\uff0c\u6240\u6709\u7684\u52bf\u80fd\u51fd\u6570\u5747\u4e0e\u65f6\u95f4\u65e0\u5173\uff1a $${V{(r,t […]<\/p>\n","protected":false},"author":16,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[4],"tags":[],"class_list":["post-1984","post","type-post","status-publish","format-standard","hentry","category-lab_resources"],"_links":{"self":[{"href":"http:\/\/ustb.cc\/zh\/wp-json\/wp\/v2\/posts\/1984","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/ustb.cc\/zh\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/ustb.cc\/zh\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/ustb.cc\/zh\/wp-json\/wp\/v2\/users\/16"}],"replies":[{"embeddable":true,"href":"http:\/\/ustb.cc\/zh\/wp-json\/wp\/v2\/comments?post=1984"}],"version-history":[{"count":114,"href":"http:\/\/ustb.cc\/zh\/wp-json\/wp\/v2\/posts\/1984\/revisions"}],"predecessor-version":[{"id":2304,"href":"http:\/\/ustb.cc\/zh\/wp-json\/wp\/v2\/posts\/1984\/revisions\/2304"}],"wp:attachment":[{"href":"http:\/\/ustb.cc\/zh\/wp-json\/wp\/v2\/media?parent=1984"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/ustb.cc\/zh\/wp-json\/wp\/v2\/categories?post=1984"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/ustb.cc\/zh\/wp-json\/wp\/v2\/tags?post=1984"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}