QMGreensite_merged
21.2. LINEARALGEBRAINBRA-KETNOTATION 315 Ofcourse,incallingthecomponentsofavectora”wavefunction”,weareantici- patingtheuseofeq. ...
316 CHAPTER21. QUANTUMMECHANICSASLINEARALGEBRA Sinceboth|v′>andM|ei>areketvectors,theymusthaveanexpansioninbasis vectorswh ...
21.2. LINEARALGEBRAINBRA-KETNOTATION 317 LikewiseLactsonbravectors<q|bytakinganinnerproductontheright <q|L = <q|(|u> ...
318 CHAPTER21. QUANTUMMECHANICSASLINEARALGEBRA WearenowreadytoexpressanylinearoperatorMintermsof|u><v|symbols. Wehave M = ...
21.2. LINEARALGEBRAINBRA-KETNOTATION 319 TheHermitianconjugateM†ofalinearoperatorM isdefinedtobethatoperator withtheproperty < ...
320 CHAPTER21. QUANTUMMECHANICSASLINEARALGEBRA Thisis theorm provedin the last sectionfor matrices. Below we simply run throught ...
21.3. HILBERTSPACE 321 Inparticular,thematrixelementsofM intheM-basisformadiagonalmatrix Mmn = <vm|M|vn> = λnδmn (21.109) ...
322 CHAPTER21. QUANTUMMECHANICSASLINEARALGEBRA Thex-representationreferstothebasisinHilbertSpacespannedbytheeigen- states ofthep ...
21.3. HILBERTSPACE 323 variousangular-momentumrepresentations. Theharmonic-oscillatorrepresentation is frequently usedinquantum ...
324 CHAPTER21. QUANTUMMECHANICSASLINEARALGEBRA Theotheroperators,{P, Hho, Hsq},wedefinebytheirmatrixelementsintheX- representati ...
21.3. HILBERTSPACE 325 asamatrixequationintheX-representation: <x|[X,P]|y> = <x|XP|y>−<x|PX|y> = <x|XIP|y&g ...
326 CHAPTER21. QUANTUMMECHANICSASLINEARALGEBRA Aningeneral,anystatewhoseeigenfunctionisψ(x)inthex-representationwillhave aneigen ...
21.3. HILBERTSPACE 327 where x ̃=i ̄h ∂ ∂p (21.148) Ingeneral,onecanshowthatinthep-representation <p 1 |Xn|φ 2 >= ( i ̄h ∂ ...
328 CHAPTER21. QUANTUMMECHANICSASLINEARALGEBRA WenextconsiderabasisforHilbertSpaceconsistingoftheeigenstates|φn>of theHarmoni ...
21.3. HILBERTSPACE 329 = [φ∗ 0 (x),φ∗ 1 (x),φ∗ 2 (x),...] φ 0 (y) φ 1 (y) φ 2 (y) . . . (21.160) ...
330 CHAPTER21. QUANTUMMECHANICSASLINEARALGEBRA or X = √ ̄h 2 mω 0 1 0... 1 0 √ 2... 0 √ 2 0 √ 3.. 0 0 √ 3 0 √ 4. .. ...
21.3. HILBERTSPACE 331 Theselookjust like the eigenstates of the harmonicoscillator Hamiltonian φn in theHO-representation,but o ...
332 CHAPTER21. QUANTUMMECHANICSASLINEARALGEBRA wherethistime En=n^2 ̄h^2 π^2 2 mL^2 (21.177) ThematrixrepresentationsXmnisgivenb ...
21.3. HILBERTSPACE 333 Writtenoutas∞×∞matrices,XandPare X = L 1 2 − 16 9 π^20... − 916 π 2 12 − 2548 π 2... 0 − ...
334 CHAPTER21. QUANTUMMECHANICSASLINEARALGEBRA oftheelectron.Inthatcasethe|θ,φ>basisisinadequate,andtheuseoftheangular moment ...
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