FYSA5, Kvanttimekaniikka I, osa B, tentti..4 Tentin yhteispistemäärä on 48 pistettä. Kaavakokoelma ja CG-taulukko paperinipun lopussa.. Selitä lyhyesti (a) Larmorin prekessio [ pt] (b) Clebsch-Gordan kertoimet [ pt] (c) bosonien ja fermionien aaltofunktioitten symmetriaominaisuudet hiukkasten vaihdossa [ pt] (d) Paulin kieltosääntö [ pt] yht. pt. Tarkastellaan yhtä elektronia kytkemättömässä kvanttitilassa ` =,s=/,m` =,m s =/i u. Kytke pyörimismäärät (Ĵ = ˆL+Ŝ) ja laske mitä mahdollisia arvoja ja millä todennäkösyydellä systeemistä voidaan mitata kun siihen operoidaan samanaikaisesti operaattoreilla Ĵ ja Ĵz. yht. pt. Tarkastellaan systeemiä jonka Hamiltonin operaattori on määritelty normitetussa ominaiskannassaan seuraavasti Ĥ i = E () i, Ĥ i = E () i, Ĥ i = E () i, Ĥ 4i = E () Lisätään systeemiin pieni häiriö (g > ja g << E (), E() 4i (E() <E () <E () ), E() ) siten että g ˆV i = g ( i + i), gˆv i = g i, gˆv i = g i, gˆv 4i =g 4i (a) Muodosta häiriömatriisi gv Ĥ:n ominaiskannassa. [4 pt] (b) Laske. kertaluvun häiriöteoriaa käyttäen korjaukset energian ominaisarvoihin. [4 pt] (c) Piirrä energiatasokaavio josta näkyy kuinka em. energiat muuttuvat häiriön seurauksena. [4 pt] yht. pt 4. Tarkastellaan yksiulotteista kvanttimekaanista systeemiä jonka Hamiltonin operaattori on muotoa d Ĥ = ~ m dx + V (x) jossa V on deltapotentiaali V (x) = (x), >. Arvioi perustilan energia tässä systeemissä käyttäen variaatiomenetelmää ja gaussista aaltofunktioyritettä (x) =Ae bx jossa A on normitusvakio ja b > variaatioparametri (muista normittaa ensin). Vertaa saamaasi tulosta tarkkaan ratkaisuun E = m /(~ ). yht. pt
FYSA5, Quantum mechanics I, part B, exam..4 Total points 48. Useful formulas and the CG table attached at the end.. Explain briefly (a) Larmor precession [ pt] (b) Clebsch-Gordan coefficients [ pt] (c) The symmetry properties of bosonic and fermionic wave functions regarding exchange of identical particles [ pt] (d) Pauli exclusion principle [ pt] yht. pt. Let us consider one electron in an uncoupled quantum state ` =,s=/,m` =,m s =/i u. Make the coupling of angular momenta (Ĵ = ˆL + Ŝ) and determine what values and with what probabilities one can measure from the system when simultaneous measurements are done corresponding to operators Ĵ ja Ĵz. yht. pt. Let us consider a system whose Hamilton operator is defined in its egenbasis as Ĥ i = E () i, Ĥ i = E () i, Ĥ i = E () i, Ĥ 4i = E () 4i (E() <E () <E () ) The system is subjected to a weak perturbation (g > and g << E (), E(), E() ) such that g ˆV i = g ( i + i), gˆv i = g i, gˆv i = g i, gˆv 4i =g 4i (a) Write the perturbation matrix gv in the eigenbasis of Ĥ. [4pt] (b) Evaluate the corrections to the energy eigenvalues by using the st order perturbation theory. [4 pt] (c) Draw an energy level diagram that shows how the unperturbed energies are changed as the result of the perturbation. [4 pt] yht. pt
4. Let us consider a one-dimensional quantum system whose Hamilton operator is d Ĥ = ~ m dx + V (x) where V is a delta potential V (x) = (x), >. Estimate the ground state energy of this system by using the variational principle and a Gaussian trial wave function (x) =Ae bx where A is the normalization constant and b> variational parameter (remember to normalize). Compare your result to the known exact ground state energy E = m /(~ ). yht. pt
Trigonometry = cos (x)+sin (x), e ıx = cos(x)+ı sin(x), cos(x) = cos (x) sin (x) = cos (x)+= sin (x), sin(x + y) =sin(x) cos(y) + cos(x)sin(y), cos(x + y) = cos(x) cos(y) sin(x)sin(y), Second degree polynomial equation Integrals Z Commutator relations ax + bx + c =! x = b ± p b 4ac a Z dx x n e ax = Z dx e ax = r a (a >) (n)! r n! n+ a n+ (a>, n =,,,...) dx x n+ e ax = n! a n+ (a>, n =,,,...) Z dx x n e ax = n! a n+ (n =,,,...) [AB, C] =A[B,C]+[A, C]B. Spherical coordinates and spherical functions r = r @ @r (r @ @r ) ~ r ˆL d r = r drd =r drsin d d General angular momentum Z d =4 X [Ĵi, Ĵj] =ı~ ijk Ĵ k [Ĵ, Ĵi] = Ĵ j, mi = ~ j(j+) j, mi Ĵ z j, mi = ~m j, mi k= Ĵ ± = Ĵx ± ıĵy Ĵ ± j, mi = ~ p (j m)(j ± m + ) j, m ± i
6. Clebsch-Gordan coefficients 6. CLEBSCH-GORDAN COEFFICIENTS, SPHERICAL HARMONICS, AND d FUNCTIONS Note: A square-root sign is to be understood over every coefficient, e.g., for 8/5 read 8/5. / / +/ +/ + +/ / / / / +/ / / / + / / + / / Y m l / +/ / / + +/ +/ +/ + / / / / / +/ / / / / + + + + + + + + + / / + =( ) m Y m l / +/ + / / + / / + + + + + Y = 4π cos θ Y = sin θeiφ 8π Y = 5 4π ( cos θ 5 Y = 8π Y = 4 ) sin θ cos θeiφ 5 π sin θe iφ j j m m j j JM =( ) J j j j j m m j j JM d j m,m =( )m m d j m,m = d j / / m, m + d, =cosθ d/ /,/ =cosθ d, = +cosθ +/ +/ + + / 7/ +/ +/ / / d / /, / = sin θ d, = sin θ +7/ 7/ 5/ +/ +/ / / + + + + +/ +5/ +5/ +/ / /5 / / ++/ /7 4/7 7/ 5/ / +/ +/ /5 /5 d, = cos θ ++/ 4/7 /7 +/ +/ +/ / +/ /5 / / + / /7 6/5 /5 +/ / / /4 9/ /4 + +/ 4/7 /5 /5 7/ 5/ / / 4 +/ / 9/ /4 / /4 +/ /7 8/5 /5 +/ +/ +/ +/ +4 4 / +/ 9/ /4 / /4 + + + + + / /5 6/5 /5 /5 / +/ / /4 9/ /4 + / /5 5/4 / + + / / 4 + + / / + + + +/ 8/5 7/ 5/ +/ 4/5 7/7 / / / +/ / /5 / / /5 /5 /5 / / / /5 /5 /5 / / / / +/ /5 / / + /4 / /7 + / 4/5 7/7 /5 / + + 4/7 /7 4 / / / / / 8/5 /5 /5 /5 + /4 / /7 + + + + / / / / +/ /5 5/4 / 7/ 5/ / + /4 / /7 /5 +/ /5 6/5 /5 /5 / / / / / + /7 /5 /4 / / /7 8/5 /5 + /7 /5 /4 / 4 / 4/7 /5 /5 7/ 5/ + /4 / /7 /5 +/ /7 6/5 /5 5/ 5/ + /7 / /7 /5 /5 / 4/7 /7 7/ + 8/5 /5 /4 / /5 / /7 4/7 7/ 8/5 /7 /5 d / /,/ = +cosθ cos θ + 8/5 /5 /4 / /5 4 / + /7 / /7 /5 /5 /4 / /7 /5 d / /,/ = +cosθ sin θ d / /, / = cos θ cos θ d / /, / = cos θ sin θ d / /,/ = cosθ cos θ d / cosθ + = sin θ /, / / / / / / / /5 8/5 /5 / + / /6 / + /6 / / / / + /6 / / / / / / + /5 /5 + /5 + /5 / / ( +cosθ ) d, = d, = +cosθ d, = 6 4 sin θ sin θ d, = cos θ sin θ ( cos θ ) d, = d, = +cosθ ( cos θ ) d, = sin θ cos θ d, = cos θ 5/ +5/ 5/ / +/ +/ + / /5 /5 5/ / +/ /5 /5 / / / +/ + +/ +/ + + /5 /5 5/ / /5 /5 / / / +/ 4/5 /5 /5 4/5 +/ / /4 /4 +/ +/ /4 /4 5/ +/ / / / +5/ 5/ / +/ + +/ +/ / +/ / / +/ /5 /5 5/ / / / / /4 /4 +/ + /5 /5 +/ +/ +/ / +/ /4 /4 / / / / /5 / /5 / / 8/5 /6 / + /5 / /5 +/ / /5 / +/ /5 /5 / 5/ / / / + / 8/5 /6 / / / / + +/ 4π d l m, = l + Y l m e imφ + / + +/ / / +/ / / + / / / / + + /5 4/5 5/ / 4/5 /5 +/ +/ / /5 / 8/5 /5 /5 /6 / 5/ / / / / /7 /5 /4 / /7 /5 /4 / /4 / /7 /5 Notation: m m / / /5 /5 / / /5 /5 / / 5/ 5/ ( cos θ +) d, = ( cos θ J M J M ).. m m Coefficients 4 5/ 5/ /4 / /7 4/7 /7 4 /4 / /7 / / 4 / / 4 Figure 6.: ThesignconventionisthatofWigner(Group Theory, AcademicPress,NewYork,959),alsousedbyCondonandShortley(The Theory of Atomic Spectra, CambridgeUniv. Press,NewYork,95),Rose(Elementary Theory of Angular Momentum, Wiley,NewYork,957), and Cohen (Tables of the Clebsch-Gordan Coefficients,NorthAmericanRockwellScienceCenter,ThousandOaks,Calif.,974).