n p + e e e e + Ne * Ne + N C + e e Pu U + Fundamental particle decays Nuclear decays Some observed decays The transition rate, W (the “Golden Rule”) of initial final is also invoked to understand a b+c (+ ) decays How do you calculate an “overlap” between ???
It almost seems a self-evident statement: Any decay that’s possible will happen! What makes it possible? What sort of conditions must be satisfied? Total charge q conserved. J conserved.
probability of surviving through at least time t mean lifetime = 1/ For any free particle (separation of space-time components) Such an expression CANNOT describe an unstable particle since Instead mathematically introduce the exponential factor:
then a decaying probability of surviving Note: = ħ Also notice: effectively introduces an imaginary part to E
Applying a Fourier transform: still complex! What’s this represent? E distribution of the unstable state
Breit-Wigner Resonance Curve Expect some constant
EoEo EE MAX = FWHM When SPIN of the resonant state is included:
130-eV neutron resonances scattering from 59 Co Transmission -ray yield for neutron radiative capture
+ p elastic scattering cross-section in the region of the Δ ++ resonance. The central mass is 1232 MeV with a width =120 MeV
Cross-section for the reaction e + e anything near the Z 0 resonance plotted against cms energy
Cross section for the reaction B 10 + N 14 * versus energy. The resonances indicate levels in the compound nucleus N 14 *. [Talbott and Heydenburg, Physical Review, 90, 186 (1953).]
Spectrum of protons scattered from Na 14 indicating its energy levels. [Bockelman et al., Physical Review, 92, 665 (1953).]
Resonances observed in the radiative proton capture by 23 Na. [P.W.m. Glaudemans and P.M. Endt, Nucl. Phys. 30, 30 (1962).]
In general: cross sections for free body decays (not resonances) are built exactly the same way as scattering cross sections. DECAYS (2-body example) (2-body) SCATTERING except for how the “flux” factor has to be defined in C.O.M. in Lab frame: enforces conservation of energy/momentum when integrating over final states Now the relativistic invariant phase space of both recoiling target and scattered projectile
Number scattered per unit time = (FLUX) × N × total (a rate) /cm 2 ·sec A concentration focused into a small spot and small time interval density of targets size of each target Notice: is a function of flux!
X Y Z Rotations Y´Y´ X´X´ = Z ´ Changes in frame of reference or point of view involve transformations of coordinate axes (or, more generally, basis set)
X Y Z Rotations Y´Y´ X´X´ = Z ´ x y x´ y´
R =R = cos sin 0 -sin cos v ´ = R v
X Z Y r a X´ Y´ Z´ r´r´ Translations parallel translation (no rotation) of axes r´ = r a Vectors (and functions) are translated in the “opposite direction” as the coordinate system. How can we possibly express an operator like this as a matrix?
The trick involves using to cast matrix operators as exponentials where H is an operator…or matrix the unit matrix ···
Taylor Series (in 1-dimension) and we’ll make that connection through …and this useful limit
For an infinitesimal translation f (x 0 +δx) f (x 0 ) + δx fxfx 3 i=1 Ok…but how can any matrix represent this? Imagine dividing the entire translation a into δa x = δa y = δa z = axNaxN ayNayN azNazN N f (r) x=x0x=x0 and applying this little step N times
making this a continuous smooth translation lim N∞N∞ iħiħ (-iħ )
For homework you will be asked to do the same thing for rotations i.e., show you can cast in the same form. R=R= cos sin 0 -sin cos You should start from: R=R= 1 0 - Later we will generalize this result to:
Rotation of coordinate axes by about any arbitrary axis ^ Rotation of the physical system within fixed coordinate axes Recall, even more fundamentally, the QM relation: Time evolution of an initial state, generated by the Hamiltonian
“Generator”Operator Amount of transformation Nature of the transformation p J H a t Translation: moving linearly through space rotating through space translation through time
The Silver Surfer, Marvel Comics Group, 1969