Revision #1470 → #1779 · back to history
modifiedPower series expansion in λ73ef2f716a6c
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| anchors | [{"section":"First order corrections","snippet":"they can be written as a (Maclaurin) power series in λ"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}E_{n}&=E_{n}^{(0)}+\\lambda E_{n}^{(1)}+\\lambda ^{2}E_{n}^{(2)}+\\cdots \\\\[1ex]|n\\rangle &=\\left|n^{(0)}\\right\\rangle +\\lambda \\left|n^{(1)}\\right\\rangle +\\lambda ^{2}\\left|n^{(2)}\\right\\rangle +\\cdots \\end{aligned}}}"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}E_{n}^{(k)}&={\\frac {1}{k!}}{\\frac {d^{k}E_{n}}{d\\lambda ^{k}}}{\\bigg |}_{\\lambda =0}\\\\[1ex]\\left|n^{(k)}\\right\\rangle &=\\left.{\\frac {1}{k!}}{\\frac {d^{k}|n\\rangle }{d\\lambda ^{k}}}\\right|_{\\lambda =0.}\\end{aligned}}}"}] | — |
| mathlib.decl | — | HasFPowerSeriesAt |
| mathlib.match_kind | — | generalization |
| mathlib.module | — | Mathlib.Analysis.Analytic.Basic |
| note | While Mathlib has general power-series infrastructure, the perturbative expansion of eigenstates/eigenvalues in λ is a physics ansatz not formalized. | Mathlib has general formal/analytic power-series infrastructure (`HasFPowerSeriesAt`, `FormalMultilinearSeries`), but the specific perturbative ansatz for eigenstates/eigenvalues in λ is not formalized. |
| status | not_formalized | partial |
modifiedZeroth-order equationfb094c3b872c
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| anchors | [{"section":"First order corrections","snippet":"The zeroth-order equation is simply the Schrödinger equation for the unperturbed system"},{"type":"math_alttext","value":"{\\displaystyle H_{0}\\left|n^{(0)}\\right\\rangle =E_{n}^{(0)}\\left|n^{(0)}\\right\\rangle .}"}] | — |
modifiedFirst-order equation9b796ad95b8b
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| anchors | [{"section":"First order corrections","snippet":"The first-order equation is"},{"type":"math_alttext","value":"{\\displaystyle H_{0}\\left|n^{(1)}\\right\\rangle +V\\left|n^{(0)}\\right\\rangle =E_{n}^{(0)}\\left|n^{(1)}\\right\\rangle +E_{n}^{(1)}\\left|n^{(0)}\\right\\rangle .}"}] | — |
modifiedFirst-order energy shiftc150ae296c83
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| anchors | [{"section":"First order corrections","snippet":"This leads to the first-order energy shift"},{"type":"math_alttext","value":"{\\displaystyle E_{n}^{(1)}=\\left\\langle n^{(0)}\\right|V\\left|n^{(0)}\\right\\rangle .}"}] | — |
modifiedEnergy correction via correlation functions7a2c3ef9825f
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| anchors | [{"section":"Second-order and higher-order corrections","snippet":"It is possible to relate the k -th order correction to the energy E n to the k -point connected correlation function"},{"type":"math_alttext","value":"{\\displaystyle \\langle n^{(0)}|V(\\tau )V(0)|n^{(0)}\\rangle -\\langle n^{(0)}|V|n^{(0)}\\rangle ^{2}=\\mathrel {\\mathop {:} } \\int _{\\mathbb {R} }\\!ds\\;\\rho _{n,2}(s)\\,e^{-(s-E_{n}^{(0)})\\tau }}"},{"type":"math_alttext","value":"{\\displaystyle E_{n}^{(2)}=-\\int _{\\mathbb {R} }\\!{\\frac {ds}{s-E_{n}^{(0)}}}\\,\\rho _{n,2}(s).}"}] | — |
modifiedFirst-order degenerate perturbation73c4d8eac612
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| anchors | [{"section":"Degenerate perturbation theory","snippet":"we need solve the perturbed Hamiltonian restricted to the degenerate subspace D"},{"type":"math_alttext","value":"{\\displaystyle V|k^{(0)}\\rangle =\\epsilon _{k}|k^{(0)}\\rangle +{\\text{small}}\\qquad \\forall |k^{(0)}\\rangle \\in D,}"},{"type":"math_alttext","value":"{\\displaystyle \\langle k^{(0)}|V|l^{(0)}\\rangle =V_{kl}\\qquad \\forall \\;|k^{(0)}\\rangle ,|l^{(0)}\\rangle \\in D.}"}] | — |
modifiedHigher-order degenerate corrections533343325af1
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| anchors | [{"section":"Degenerate perturbation theory","snippet":"Higher-order corrections due to other eigenstates outside D can be found in the same way as for the non-degenerate case"},{"type":"math_alttext","value":"{\\displaystyle \\left(E_{n}^{(0)}-H_{0}\\right)|n^{(1)}\\rangle =\\sum _{k\\not \\in D}\\left(\\langle k^{(0)}|V|n^{(0)}\\rangle \\right)|k^{(0)}\\rangle .}"}] | — |
modifiedEnergy and state corrections (lifted degeneracy)58375321dd07
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| anchors | [{"section":"Degeneracy lifted to first order","snippet":"we have the following formulae for the energy correction to the second order"},{"type":"math_alttext","value":"{\\displaystyle E_{nk}=E_{n}^{0}+\\lambda V_{nk,nk}+\\lambda ^{2}\\sum \\limits _{m\\neq n}{\\frac {\\left|V_{m,nk}\\right|^{2}}{E_{n}^{(0)}-E_{m}^{(0)}}}+{\\mathcal {O}}(\\lambda ^{3})\\,,}"},{"type":"math_alttext","value":"{\\displaystyle \\left|\\psi _{nk}\\right\\rangle =\\left|\\psi _{nk}^{(0)}\\right\\rangle +\\lambda \\sum \\limits _{m\\neq n}{\\frac {V_{m,nk}}{E_{m}^{(0)}-E_{n}^{(0)}}}\\left(-\\left|\\psi _{m}^{(0)}\\right\\rangle +\\sum \\limits _{l\\neq k}{\\frac {V_{nl,m}}{E_{nl}^{(1)}-E_{nk}^{(1)}}}\\left|\\psi _{nl}^{(0)}\\right\\rangle \\right)+{\\mathcal {O}}(\\lambda ^{2})\\,.}"}] | — |
modifiedLinearly parameterized Hamiltonianb1a8d0077975
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| anchors | [{"section":"Hamiltonian and force operator","snippet":"The following linearly parameterized Hamiltonian is frequently used"},{"type":"math_alttext","value":"{\\displaystyle H(x^{\\mu })=H(0)+x^{\\mu }F_{\\mu }.}"}] | — |
modifiedFirst Hellmann–Feynman theorem3973e6ba7c08
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| anchors | [{"section":"Hellmann–Feynman theorems","snippet":"The first Hellmann–Feynman theorem gives the derivative of the energy"},{"type":"math_alttext","value":"{\\displaystyle \\partial _{\\mu }E_{n}=\\langle n|\\partial _{\\mu }H|n\\rangle }"}] | — |
modifiedSecond Hellmann–Feynman theoremb1c2929ce347
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| anchors | [{"section":"Hellmann–Feynman theorems","snippet":"The second Hellmann–Feynman theorem gives the derivative of the state"},{"type":"math_alttext","value":"{\\displaystyle \\langle m|\\partial _{\\mu }n\\rangle ={\\frac {\\langle m|\\partial _{\\mu }H|n\\rangle }{E_{n}-E_{m}}},\\qquad \\langle \\partial _{\\mu }m|n\\rangle ={\\frac {\\langle m|\\partial _{\\mu }H|n\\rangle }{E_{m}-E_{n}}}.}"}] | — |
modifiedSecond-order state correction (geometric)4df2028d20cc
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| anchors | [{"section":"Correction of energy and state","snippet":"The same computational scheme is applicable for the correction of states"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}\\left|n\\left(x^{\\mu }\\right)\\right\\rangle =|n\\rangle &+\\sum _{m\\neq n}{\\frac {\\langle m|\\partial _{\\mu }H|n\\rangle }{E_{n}-E_{m}}}|m\\rangle x^{\\mu }\\\\&+\\left(\\sum _{m\\neq n}\\sum _{l\\neq n}{\\frac {\\langle m|\\partial _{\\mu }H|l\\rangle \\langle l|\\partial _{\\nu }H|n\\rangle }{(E_{n}-E_{m})(E_{n}-E_{l})}}|m\\rangle -\\sum _{m\\neq n}{\\frac {\\langle m|\\partial _{\\mu }H|n\\rangle \\langle n|\\partial _{\\nu }H|n\\rangle }{(E_{n}-E_{m})^{2}}}|m\\rangle -{\\frac {1}{2}}\\sum _{m\\neq n}{\\frac {\\langle n|\\partial _{\\mu }H|m\\rangle \\langle m|\\partial _{\\nu }H|n\\rangle }{(E_{n}-E_{m})^{2}}}|n\\rangle \\right)x^{\\mu }x^{\\nu }+\\cdots .\\end{aligned}}}"}] | — |
modifiedOccupation probability25e2820680b2
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| anchors | [{"section":"Method of variation of constants","snippet":"The square of the absolute amplitude c n ( t ) is the probability that the system is in state n at time t"},{"type":"math_alttext","value":"{\\displaystyle \\left|c_{n}(t)\\right|^{2}=\\left|\\langle n|\\psi (t)\\rangle \\right|^{2}~.}"}] | — |
modifiedCoupled equations for amplitudesa1a5b6cec948
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| anchors | [{"section":"Method of variation of constants","snippet":"this can be reduced to a set of coupled differential equations for the amplitudes"},{"type":"math_alttext","value":"{\\displaystyle {\\frac {dc_{n}}{dt}}={\\frac {-i}{\\hbar }}\\sum _{k}\\langle n|V(t)|k\\rangle \\,c_{k}(t)\\,e^{-i(E_{k}-E_{n})t/\\hbar }~.}"}] | — |
modifiedIterative (perturbative) solutionf18f6f12670d
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| anchors | [{"section":"Method of variation of constants","snippet":"Repeatedly substituting this expression for c n back into right hand side, yields an iterative solution"},{"type":"math_alttext","value":"{\\displaystyle c_{n}(t)=c_{n}^{(0)}+c_{n}^{(1)}+c_{n}^{(2)}+\\cdots }"},{"type":"math_alttext","value":"{\\displaystyle c_{n}^{(1)}(t)={\\frac {-i}{\\hbar }}\\sum _{k}\\int _{0}^{t}dt'\\;\\langle n|V(t')|k\\rangle \\,c_{k}^{(0)}\\,e^{-i(E_{k}-E_{n})t'/\\hbar }~.}"},{"type":"math_alttext","value":"{\\displaystyle c_{n}^{(1)}(t)={\\frac {-i}{\\hbar }}\\int _{0}^{t}dt'\\;\\langle n|V(t')|k\\rangle \\,e^{-i(E_{k}-E_{n})t'/\\hbar }~.}"}] | — |
modifiedDyson series1cca2456f26b
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| anchors | [{"section":"Method of Dyson series","snippet":"where T is the time ordering operator"},{"type":"math_alttext","value":"{\\displaystyle H(t)|\\psi (t)\\rangle =i\\hbar {\\frac {\\partial |\\psi (t)\\rangle }{\\partial t}}}"},{"type":"math_alttext","value":"{\\displaystyle |\\psi (t)\\rangle =T\\exp {\\left[-{\\frac {i}{\\hbar }}\\int _{t_{0}}^{t}dt'H(t')\\right]}|\\psi (t_{0})\\rangle ~,}"},{"type":"math_alttext","value":"{\\displaystyle TA(t_{1})A(t_{2})={\\begin{cases}A(t_{1})A(t_{2})&t_{1}>t_{2}\\\\A(t_{2})A(t_{1})&t_{2}>t_{1}\\end{cases}}~.}"},{"type":"math_alttext","value":"{\\displaystyle |\\psi (t)\\rangle =\\left[1-{\\frac {i}{\\hbar }}\\int _{t_{0}}^{t}dt_{1}H(t_{1})-{\\frac {1}{\\hbar ^{2}}}\\int _{t_{0}}^{t}dt_{1}\\int _{t_{0}}^{t_{1}}dt_{2}H(t_{1})H(t_{2})+\\ldots \\right]|\\psi (t_{0})\\rangle ~.}"}] | — |
modifiedFirst-order transition amplitudea15e6b246118
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| anchors | [{"section":"Method of Dyson series","snippet":"The corresponding transition probability amplitude to first order is"},{"type":"math_alttext","value":"{\\displaystyle A_{\\alpha \\beta }=-{\\frac {i\\lambda }{\\hbar }}\\int _{t_{0}}^{t}dt_{1}\\langle \\beta |V(t_{1})|\\alpha \\rangle e^{-{\\frac {i}{\\hbar }}(E_{\\alpha }-E_{\\beta })(t_{1}-t_{0})}~,}"}] | — |
modifiedTime-independent PT within Dyson seriese3e92c576625
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| anchors | [{"section":"Method of Dyson series","snippet":"time-independent perturbation theory is also organized inside this time-dependent perturbation theory Dyson series"},{"type":"math_alttext","value":"{\\displaystyle U(t)=1-{\\frac {i\\lambda }{\\hbar }}\\int _{t_{0}}^{t}dt_{1}e^{{\\frac {i}{\\hbar }}H_{0}(t_{1}-t_{0})}V(t_{1})e^{-{\\frac {i}{\\hbar }}H_{0}(t_{1}-t_{0})}-{\\frac {\\lambda ^{2}}{\\hbar ^{2}}}\\int _{t_{0}}^{t}dt_{1}\\int _{t_{0}}^{t_{1}}dt_{2}e^{{\\frac {i}{\\hbar }}H_{0}(t_{1}-t_{0})}V(t_{1})e^{-{\\frac {i}{\\hbar }}H_{0}(t_{1}-t_{0})}e^{{\\frac {i}{\\hbar }}H_{0}(t_{2}-t_{0})}V(t_{2})e^{-{\\frac {i}{\\hbar }}H_{0}(t_{2}-t_{0})}+\\cdots }"}] | — |
modifiedQuartic oscillator ground-state energya19eac67b77b
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| anchors | [{"section":"Example of first-order perturbation theory – ground-state energy of the quartic oscillator","snippet":"Consider the quantum harmonic oscillator with the quartic potential perturbation"},{"type":"math_alttext","value":"{\\displaystyle H=-{\\frac {\\hbar ^{2}}{2m}}{\\frac {\\partial ^{2}}{\\partial x^{2}}}+{\\frac {m\\omega ^{2}x^{2}}{2}}+\\lambda x^{4}.}"}] | — |
modifiedQuantum pendulum4feeda7f87c1
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| anchors | [{"section":"Example of first- and second-order perturbation theory – quantum pendulum","snippet":"Consider the quantum-mathematical pendulum with the Hamiltonian"},{"type":"math_alttext","value":"{\\displaystyle H=-{\\frac {\\hbar ^{2}}{2ma^{2}}}{\\frac {\\partial ^{2}}{\\partial \\phi ^{2}}}-\\lambda \\cos \\phi }"},{"type":"math_alttext","value":"{\\displaystyle V=-\\cos \\phi .}"}] | — |
modifiedFree particle with weak potentialff341df9278f
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| anchors | [{"section":"Potential energy as a perturbation","snippet":"When the unperturbed state is a free motion of a particle with kinetic energy"},{"type":"math_alttext","value":"{\\displaystyle \\nabla ^{2}\\psi ^{(0)}+k^{2}\\psi ^{(0)}=0}"},{"type":"math_alttext","value":"{\\displaystyle \\nabla ^{2}\\psi ^{(1)}+k^{2}\\psi ^{(1)}={\\frac {2mU}{\\hbar ^{2}}}\\psi ^{(0)},}"},{"type":"math_alttext","value":"{\\displaystyle \\psi ^{(1)}(x,y,z)=-{\\frac {m}{2\\pi \\hbar ^{2}}}\\int \\psi ^{(0)}U(x',y',z'){\\frac {e^{ikr}}{r}}\\,dx'dy'dz',}"},{"type":"math_alttext","value":"{\\displaystyle \\psi ^{(1)}(x,y)=-{\\frac {im}{2\\hbar ^{2}}}\\int \\psi ^{(0)}U(x',y')H_{0}^{(1)}(kr)\\,dx'dy',}"},{"type":"math_alttext","value":"{\\displaystyle \\psi ^{(1)}(x)=-{\\frac {im}{\\hbar ^{2}}}\\int \\psi ^{(0)}U(x'){\\frac {e^{ikr}}{k}}\\,dx',}"}] | — |
addedHilbert space9f85268fc1b0
addedHermitian operator2e2dd6d83be0
addedMaclaurin / Taylor series411c16987f97
addedVector bundle over parameter manifolda039d80d5317
addedCovariant derivative / connectionb0c70517e253
addedLie derivative57f491bcd6d7
addedInverse Laplace transformd9607b5108f4
addedOrthonormal basis of eigenstatesfdda06021452
addedResolution of the identity969eb8db9776
addedTime-ordering operatorc035d72c5d66
addedInteraction (Dirac) pictureee591efd07f6
addedSchrödinger picture1b8e2709da35
addedBra–ket notationcaec64e083cc
addedHankel function of the first kindf94660b5ad34
addedRabi cycle1266cfa2a6a4
addedMuon spin spectroscopya207295174f5
addedPerturbed angular correlationed3f5c8fff0b