HW 2: 1. What are the main subdisciplines of nuclear science? Which one(s) do you find most intriguing? Why?

Transcription

1 HW 2: 1. What are the main subdisciplines of nuclear science? Which one(s) do you find most intriguing? Why? 2. Based on answer the following queseons: a) What is the neutron number of 160 Yb? b) What are pracecal applicaeons of c) What is the current temperature of the Universe (in Fahrenheit)? Explain your answers.

2 Symmetry: the secret of nature Wikipedia: A symmetry of a physical system is a physical or mathemaecal feature of the system (observed or intrinsic) that is preserved or remains unchanged under some transformaeon. The role of symmetry in Fundamental physics : David Gross h@p:// Einstein s great advance in 1905 was to put symmetry first, to regard the symmetry principle as the primary feature of nature that constrains the allowable dynamical laws. Thus the transformaeon properees of the electromagneec field were not to be derived from Maxwell s equaeons, as Lorentz did, but rather were consequences of relaevisec invariance, and indeed largely dictate the form of Maxwell s equaeons. With the development of quantum mechanics in the 1920s symmetry principles came to play an even more fundamental role. In the la@er half of the 20th century symmetry has been the most dominant concept in the exploraeon and formulaeon of the fundamental laws of physics. Today it serves as a guiding principle in the search for further unificaeon and progress.

3 The role of symmetries Provide conservaeon laws and quantum numbers. (In 1918 Emmy Noether proved her famous theorem relaeng symmetry and conservaeon laws.) Summarize the regulariees of the laws that are independent of the specific dynamics/boundary condieons. The theory of representaeons of conenuous and discrete groups plays an important role deducing the consequences of symmetry in quantum mechanics. Any quantum state can be as a sum of states transforming according to irreducible representa/ons of the symmetry group. These special states can be used to classify all the states of a system possessing symmetries and play a fundamental role in the analysis of such systems through quantum numbers and associated seleceon rules. Much of the texture of the world is due to mechanisms of symmetry breaking. o Explicit symmetry breaking (the symmetry violaeon as a small correceon and approximate conservaeon laws are present) o Spontaneous symmetry breaking (the laws of physics are symmetric but the state of the system is not). Example: crystals (translaeonal invariance), magneesm (rotaeonal invariance), supeconducevity (parecle number). Thus for every broken global symmetry there exist fluctuaeons with very low energy. These appear as massless parecles (Goldstone bosons).

4 Symmetries of the nuclear Hamiltonian (exact or almost exact) 1. Translational invariance 2. Galilean invariance (or Lorentz invariance) 3. Rotational invariance 4. Time reversal 5. Parity (space reflection) 6. Charge independence and isobaric symmetry 7. Baryon and lepton number symmetry 8. Permutation between the two nucleons (imposed by the exclusion principle) ConEnuous transformaeons (appear to be universally valid) Dynamical symmetries Apply in certain cases, provide useful coupling schemes 1. Chiral symmetry (broken by a quark condensate; valid for massless quarks) 2. SU(4) symmetry (Wigner supermultiplet) 3. SU(2) symmetry (seniority) Local symmetries (important for gauge theories) Different transformations at different points of spacetime

5 Symmetries in quantum mechanics (see "Symmetry in Physics", J.P. and P.G. Dawber, The Macmillan Press, London) Wave equaeon for the Hamiltonian operator: Group of transformaeons G whose elements G commute with H: GĤG 1 = Ĥ We say that H is invariant under G or totally symmetric with respect to the elements of G What are the properees of? RepresentaEon of the group (represents group elements as matrices so that the group operaeon can be represented by matrix muleplicaeon) dimension of the representaeon matrix representaeon of the group basis If all matrices D can be put into a block-diagonal form, the representaeon is irreducible

6 Classification of eigenstates with respect to symmetry group Wave funceons for different energy levels E k transform as basis funceons of irreducible representa/ons of the group G If we know the properees of G, we can classify the wave funceons Further, the same group-theoreecal structure will tell us about the spectroscopy of the system Point groups: geometric symmetries that keep at least one point fixed D 1 : (dihedral) refleceon group (2 element group: idenety and single refleceon) C n : cyclic n-fold rotaeon (C 1 is a trivial group containing idenety operaeon) Lie groups: conenuous transformaeon groups. TransformaEons generated by physical operators. The set of commutators between generators is closed. Its Casimir operator commuters with all the generators. SO(3): group of rotaeons in 3D (isomorphic with SU(2)) Poincare group (TranslaEons, Lorentz transformaeons)

7 Vector spaces: Scalars, Vectors, Tensors Orthogonal transformations U preserve lengths of vectors and angles between them map orthonormal bases to orthonormal bases Orthogonal transformations in two- or three-dimensional Euclidean space are stiff rotations, reflections, or combinations of a rotation and a reflection (also known as improper rotations). det(u)=1 usual (stiff) rotations (scalars, vectors, ) det(u)=-1 improper rotations (pseudo-scalars, axial vectors, ) Improper rotaeon operaeon S 4 in CH 4

8 Translational Invariance r k ' = r k a =U r k U 1 p k ' = p k, s k ' = s k U( a) " = exp# i $ " a P % & ' Total momentum (nucleons, mesons, photons, leptons, etc.) TransformaEon generator For [X,Y] central, i.e., commueng with both X and Y: Time displacement U(t 0 ) = exp " # i t 0Ĥ $ % & Unitary transformaeons: U + =U -1 Under unitary transformaeon U, an operator A transforms as A =UAU -1 e X Ye X = Y +[X, Y ]+ 1 2 [X, [X, Y ]] + 1 [X, [X, [X, Y ]]] + 3

9 Rotations in 3D (space isotropy) R ( χ) = exp{ i χ J } [J x, J y ] = ij z (+ cycl.) χ a set of three angles (a vector) represeneng rotaeons along x,y,z Total angular momentum TransformaEon generator SO(3) or SU(2) group RotaEonal states of the system labeled by the total angular momentum quantum numbers JM see examples of spectra at h@p://

10 Galilean (Lorentz) Invariance In atomic nucleus v 2 /c 2 <0.1, i.e., kinematics is nonrelativistic r k ' = r k v k ' = v k u, s k ' = s k U( u) = exp" i #" k M = m k, H = H int + P 2 2M p k ' = p k m k u um R c.m. $ % & R c.m. = 1 M m k r k, k Such a separaeon can be done for Galilean-invariant interaceons Depends only on relaeve coordinates and velociees

11 i [H, " R c.m. ] = 1 M " P no new conservaeon laws and quantum numbers Relativistic generalization ( ) 1/2 2 H = H int r + c 2 P 2 Center-of-mass coordinate cannot be introduced in a relaevisecally covariant manner All powers of c.m. momentum are present Unitary transformaeon contains gradient terms and spin-dependent pieces