PHYS-505 Parity and other Discrete Symmetries Lecture-7!

Transcription

1 PHYS-505 Parity ad other Discrete Symmetries Lecture-7! 1

2 Discrete Symmetries So far we have cosidered cotiuous symmetry operators that is, operatios that ca be obtaied by applyig successively ifiitesimal symmetry operatios. Not all symmetry operatios useful i quatum mechaics are ecessarily of this for. Parity, lattice traslatio ad time reversal are examples of these.! 2

3 Space Reversal Symmetry ad the Cocept of Coservatio! Cosider ow a case where the potetial is costat uder space reversal i.e V(r) = V(-r), or i a mathematical laguage the potetial is a eve fuctio of r.! I the above case the Hamiltoia has a space reversal symmetry. As we have see symmetries are always coected to coservatios laws. Is there ay coserved quatity which correspods to the space reversal symmetry?! The aswer is yes ad this quatity is called the parity.! 3

4 Parity ad Space Reversal! I quatum mechaics there is aother physical quatity which is coserved while it does ot have a classical couterpart. This quatity is parity.! Defiitio: We call space reversal (or iversio) the geometric actio r --> -r which maps every space poit at is diametric with respect to the origi of the coordiate system.! x r z -r It is more coveiet tha a simple reflectio because you eed specify oly the poit which is at the ceter of symmetry.! y 4

5 Parity ad Space Reversal! " Ğ;Ğ.. Ğ Ğ 1 ".Ğ " Ğ The space reversal is # # equivalet to a chage of a right-haded (RH) system ito a left-haded system (LH) system as show i figure.! Ƅ ɡė ŞėƋÞÈÞˋĆƄŻ*ˋɁ Þˋ ÈK NJėƋÞÈÞˋĆȪŻ*ˋ ÈƦ ˋ Let s focus o trasformatio & 0 *ˋ of kets ad ot system of ˋ axis. We itroduce the parity & *ˋ operator P.! ˋ ˋ ˋ ˋ ˋ ˋ ˋ Nࠕ ˋ ˋ: ˋ ˋ,ˋ ࠕ ˋ Nࠕ ˋ áǹ g ǯǹ ǹ ǹ ðǹ ǧ ǧ ǹ ǹ ðǹ ǹ g ǹ ǹ Ë ǿ Ë ȤȥȦȧȨȩ ėϧ $ ɰ g á ǹ TYVea H3> & ľg» ˋ D ˋ ˋ ˋ ˋ ˋ ˋ ˋ ˋ ˋ ˋ ±ˋ7 ˋ ˋ ˋ Nࠕ«ࠕ - -ɱࠕ <ࠕ Y& /*ˋ ˋ -ϕࠕ ˋ ˋ ˋ ˋ ˋ ˋ ˋ )ˋ " ۦ Eˋ ˋ ˋ ˋ ˋ 5 &-J*ˋ N ࠕ - N ࠕ - ȏ N ࠕ ) ˋ ¼ ˋ ˋ ˋNࠕŸ ࠕ ˋ ˋ ˋ ˋ ࠕ ˋ ˋ <ࠕ ˋ ˋ ˋ

6 Parity ad Space Reversal! " Ğ;Ğ.. Ğ Ğ 1 ".Ğ " Ğ B4@ø The basic requiremet is the expectatio value of a r take with respect to the iverted state to be opposite i sig:! a P rp a = a r a + # áǹ This is accomplished if! P+rP = r {P,r} = 0 g ǯǹ ǹ ǹ ðǹ ǧ ǧ ǹ ǹ ðǹ ǹ g ǹ ǹ Ë ǿ Ë ȤȥȦȧȨȩ ėϧ $ ɰ g á ǹ TYVea H3> # Ƅ ɡė ŞėƋÞÈÞˋĆƄŻ*ˋɁ Þˋ ÈK NJėƋÞÈÞˋĆȪŻ*ˋ ÈƦ ˋ Aticommutatio! ˋ & - - & 0 *ˋ & *ˋ Nࠕ«ࠕ - -ɱࠕ <ࠕ Y& /*ˋ ˋ ˋ ˋ ˋ ˋ ˋ ˋ Nࠕ ˋ ˋ: ˋ ˋ,ˋ ࠕ ˋ Nࠕ ˋ ľg» ˋ D ˋ ˋ ˋ ˋ ˋ ˋ ˋ ˋ ˋ ˋ ±ˋ7 ˋ ˋ ˋ ˋ -ϕࠕ ˋ ˋ ˋ ˋ ˋ ˋ ˋ )ˋ " ۦ Eˋ ˋ ˋ ˋ ˋ 6 &-J*ˋ N ࠕ - N ࠕ - ȏ N ࠕ ) ˋ ¼ ˋ ˋ ˋNࠕŸ ࠕ ˋ ˋ ˋ ˋ ࠕ ˋ ˋ <ࠕ ˋ ˋ ˋ

7 Parity ad Space Reversal! The parity operator P is a hermitia operator which has the followig properties:! Pψ ( r) = ψ ( r) P 2 = P P = 1 The eigevalues of this operator ca be show to be +1, -1. The eigefuctios which correspod to the positive value +1 are eve fuctios (or fuctios with positive parity), while the eigefuctios which correspod to the egative value -1 are odd fuctios (or fuctios with egative parity).! 7

8 Parity ad Symmetry Classically, if a state is symmetric uder a iversio, the operatio gives back the same state. I quatum mechaics, however, there are the two possibilities: we get the same state or mius the same state. Whe we get the same state, we say that the state has eve parity. Whe the sig is reversed we say that the state has odd parity.! There are, of course, states which are ot symmetric uder the operatio P; these are states with o defiite parity. For istace, i the H + 2 system the state Ι = ( ) / 2 has eve parity, the state Ι = 1 2 has odd parity, ad the state has o defiite parity.! ( ) / 2 Ι = 1 8

9 Parity ad Symmetry May of the laws of physics but ot all are uchaged by a reflectio or a iversio of the coordiates. They are symmetric with respect to a iversio. The laws of electrodyamics, for istace, are uchaged if we chage x to x, y to y, ad z to z i all the equatios. The same is true for the laws of gravity, ad for the strog iteractios of uclear physics.! 9

10 Parity ad Symmetry Oly the weak iteractios resposible for β-decay do ot have this symmetry. We will for ow leave out ay cosideratio of the β-decays. The i ay physical system where β-decays are ot expected to produce ay appreciable effect a example would be the emissio of light by a atom the Hamiltoia H ad the operator P will commute.! 10

11 Parity ad Symmetry Uder these circumstaces we have the followig propositio. If a state origially has eve parity, ad if you look at the physical situatio at some later time, it will agai have eve parity. For istace, suppose a atom about to emit a photo is i a state kow to have eve parity. You look at the whole thig icludig the photo after the emissio; it will agai have eve parity (likewise if you start with odd parity). This priciple is called the coservatio of parity. 11

12 Parity ad Symmetry You ca see why the words coservatio of parity ad reflectio symmetry are closely itertwied i the quatum mechaics. Although util a few years ago it was thought that ature always coserved parity, it is ow kow that this is ot true. It has bee discovered to be false because the β-decay reactio does ot have the iversio symmetry which is foud i the other laws of physics.! 12

13 Parity ad Symmetry Now we ca prove a iterestig theorem (which is true so log as we ca disregard weak iteractios):! Ay state of defiite eergy which is ot degeerate must have a defiite parity. It must have either eve parity or odd parity.! Remember that we have sometimes see systems i which several states have the same eergy we say that such states are degeerate. Our theorem will ot apply to them.! 13

14 Parity ad other Vectors! The parity operator P has the followig effect o the mometum operator:! { P,p} = 0 P + pp = p The parity operator P has the followig effect o the agular mometum operator:!!p,j# " $ = 0 P+ JP = J Traslatio followed by parity! ad vice versa! 14

15 Parity ad other Vectors! Uder rotatios, r ad J trasform i the same way, so they are both vectors, or spherical tesors of rak 1.! However r (or p) is odd uder parity, whereas J is eve uder parity. Vectors that are odd uder parity are called polar vectors, ad vectors that are eve uder parity are called axial vectors, or pseudovectors.! 15

16 Parity ad other Vectors! For the wavefuctio of a spiless particle we have the followig relatio:! ψ ( r ' ) = ±ψ r ' ( ) " $ # eve parity %$ odd parity Not all wavefuctios of physical iterest have defiite parities as above. Cosider, for istace, the mometum eigeket. The mometum operator aticommutes with the parity operator, so the mometum eigeket is ot expected to be parity eigeket. (E.g. a plae wave)! 16

17 ψ+(x)" ψ-(x)" x" Eve fuctio! Odd fuctio! (parity +1)! (parity -1)! x" 17

18 Parity! Coclusio a: If a system has space reversal symmetry, the parity is a coserved quatity.! Coclusio b: If the wavefuctio of a physical system, which has space reversal symmetry, is at a give istat eve or odd, the it will remai eve or odd respectively! 18

19 Parity ad Selectio Rules! Parity plays a importat ad fudametal role i the character of the states coected through a iteractio.! We ca show that the parity-odd operator r coects oly states of opposite parity. This has a very importat cosequece o radiative trasitios betwee atomic states.! 19

20 Lattice Traslatio as a Discrete Symmetry! Cosider a periodic potetial i oe dimesio, where:! V (x ± a) =V (x) A realistic example is the motio of a electro i a chai of regularly spaced positive ios.! 20

21 Lattice Traslatio as a Discrete Symmetry! If the traslatio is represeted by a operator τ(a), the we have:! τ + ( a)v (x)τ ( a) =V (x ± a) =V (x) Because the kietic-eergy part of the Hamiltoia is ivariat uder the traslatio with ay displacemet! τ + a ( ) Hτ ( a) = H Sice τ(a) is uitary we have! H,τ a ( ) = 0 21

22 Lattice Traslatio as a Discrete Symmetry! The previous relatio shows that the Hamiltoia ad the traslatio operator may have commo eigestates.! We may show that i the case of ifiite potetial barriers these eigestates are give from:! θ e iθ, π θ π = 22

23 Lattice Traslatio as a Discrete Symmetry! I the case of ot-ifiite barrier agai we may cosider the eigestates:! θ e iθ, π θ π = But ow we eed a assumptio about states of differet sites:! ' H 0 oly if ' = or ' = +1!! This, i the solid-state physics, is the assumptio kow as the tight-bidig approximatio.! 23

24 Lattice Traslatio as a Discrete Symmetry! I this case the states θ are Hamiltoia eigestates with eigevalues give by:! E = ( E 0 2Δcosθ), with Δ=- ±1 H This, i the solid-state physics, is the assumptio kow as the tight-bidig approximatio.! 24

25 Lattice Traslatio as a Discrete Symmetry! It ca be show that for the wavefuctio of θ, give as x θ we have:!!! x θ = e ikx u k ( x), θ = ka This the famous Bloch Theorem i solid state physics. Which states: The eigestates of a periodic potetial ca be always writte as a product of a expoetial fuctio with a fuctio which gas the periodicity of the potetial.! 25

26 Lattice Traslatio as a Discrete Symmetry! These eigestates look similar to the plae wave eigefuctios ψ k (x) = Ae ikx but with a amplitude havig the periodicity of the potetial.! The most fasciatig property is that there are o terms proportioal to e ikx. This meas there is o possibility for the particle to be reflected i the opposite directio! The particle moves iside the potetial with costat mometum. The presece of the potetial imposes just a spatial periodic modulatio of the of the wave amplitude.! 26

27 Lattice Traslatio as a Discrete Symmetry! C Ğ;Ğ The oly thig that Ğ 5 Ğ 1 Ğ 5 Ğ;Ğ Ğ 5 Ğ 1 Ğ 5 Ğ chages C ø is the 4 N h 4 N h dispersio relatio from E(k) =! 2 k 2 / 2M of a free particle to! E(k) = E0 2Acos(ka)! D@:Q H=:Q Q D@:Q H=:Q Q for the particle i the TZVfa º ģʫ T ɽ~ÂÉ º í{ ÂÜ~í ºÂíV íü~âé Éí íí ~íqâ ««º íçº ~ís t í ʮʯϦ periodic potetial.! TZVfa º ģʫ T ɽ~ÂÉ º í{ ÂÜ~í ºÂíV íü~âé Éí íí ~íqâ ««º íçº ~ís t í ʮʯϦÊ ā İ Ê ā İ ˋ ˋLˋ ˋ ˋ ˋVˋ ˋ Áˋ ˋ ˋLˋ ˋ ˋ ˋVˋ ˋ Áˋ L V"ˋȏ L R &İˋ Vľ ˋ 0 +W"ˋ L R &İˋ ˋ < Vľ ˋ X ˋ ˋ L V"ˋȏ ˋ ˋ ˋ ˋ ˋ ˋ 27 ˋ 0 +W"ˋ ˋ ˋ ˋ ˋ ˋ ˋ ˋ ˋ ˋ 8 ˋ % ˋ ˋ ˋX ˋ ˋ X ˋ ˋ ˋ ˋ ˋ < ˋ ˋ ˋ ˋ 0 ˋ +H"ˋ ˋ ˋ ˋ ˋ jˋ ˋ ˋ ˋ ˋ ˋ[ ˋˋ ˋĔÉ 1ࠕ ˋ ˋ ˋ

28 Lattice Traslatio as a Discrete Symmetry! What is the physical meaig of θ? With the way it is defied the quatity k ca cosidered as the mometum of the particle iside the periodic potetial.! This is ot rigorously correct. Oly the plae waves have a sharp mometum. The quatity k is ot exteded i the regio - < k < but oly i the regio %/a k +%/a, the so called Brilloui zoe.! 28

29 Lattice Traslatio as a Discrete Symmetry! I other words the motio of the particle feels o resistace at all! A exteral field ca accelerate the particle up to a mometum at the limits of the relevat zoe where reflectios occur ad the start the so called Bloch oscillatios.! For small mometa the particles move freely i the potetial. The mea free path ad the coductivity are ifiite!! Oly deviatios from the perfect periodicity (like impurities ad defects) ca delay the motio. This is what happes i a real coductor.! 29