Physics 2 formula sheet, Cheat Sheet of Physics

Download Physics 2 formula sheet and more Cheat Sheet Physics in PDF only on Docsity! Particle Physics Formula Sheet Special Relativity Spacetime Coordinates: xµ = (c t, ~x) xµ = (c t,−~x) ~x = (x1, x2, x3) = (x, y, z) 4-Momentum: pµ = (E/c, ~p) pµ = (E/c,−~p) ~p = (p1, p2, p3) = (px, py, pz) Derivatives: ∂µ = ∂ ∂xµ = ( 1 c ∂ ∂ t , ~∇ ) ∂µ = ( 1 c ∂ ∂ t ,−~∇ ) ~∇ = ( ∂ ∂x1 , ∂ ∂x2 , ∂ ∂x3 ) = ( ∂ ∂x , ∂ ∂y , ∂ ∂z ) Metric: ηµν = ηµν =  1 0 0 0 0 −1 0 0 0 0 −1 0 0 0 0 −1  ηµν = δµν =  1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1  Raising and Lowering Indices: aµ = ηµν aν aµ = ηµν a ν Einstein Summation Convention: aµ bµ = a0 b0 + a1 b1 + a2 b2 + a3 b3 = a 0 b 0 − ~a ·~b Lorentz Transformations: a′µ = Λµ ν aν a′µ = Λµν a ν ηµν = ΛµαΛνβ η αβ ηµν = Λµ αΛν β ηαβ Properties of Lorentz Transformations: 1. ηµν = ΛµαΛνβ η αβ 2. | det Λµν | = 1 3. (Λ1)µα (Λ2)αν = (Λ12)µν 4. |Λ0 0| ≥ 1 Boost Along the z-Axis: Λµν =  γ 0 0 −β γ 0 1 0 0 0 0 1 0 −β γ 0 0 γ  with β = v c γ = 1√ 1− β2 x′µ = Λµν x ν = ( γ (c t− β z), x, y, γ (z − β c t) ) p′µ = Λµν p ν = ( γ (E/c− β pz), px, py, γ (pz − β E/c) ) Rotation About the y-Axis: Λµν =  1 0 0 0 0 cos θ 0 − sin θ 0 0 1 0 0 sin θ 0 cos θ  x′µ = Λµν x ν = ( c t, x cos θ − z sin θ, y, z cos θ + x sin θ ) p′µ = Λµν p ν = ( E/c, px cos θ − pz sin θ, py, pz cos θ + px sin θ ) Relativistic Kinematics Conservation of 4-Momentum in Any Process:∑ i=initial pµi = ∑ j=final pµj Massive Particles: pµ pµ = E2 c2 − ~p · ~p = m2 c2 Massless Particles: pµ pµ = 0 pµ = |~p| ( 1, ~ε ) with ~ε · ~ε = 1 Important Matrices 2× 2 Matrices are written out explicitly, and 4× 4 matrices are written out in terms of 2× 2 blocks. The 2× 2 identity matrix is 1 = ( 1 0 0 1 ) Pauli Matrices: σ1 = ( 0 1 1 0 ) σ2 = ( 0 −i i 0 ) σ3 = ( 1 0 0 −1 ) σi σj = δij 1 + i εijk σk [σi, σj ] = 2 i εijk σk {σi, σj } = 2 δij Dirac Matrices: γ0 = ( 1 0 0 −1 ) γi = ( 0 σi −σi 0 ) γ5 = i γ0 γ1 γ2 γ3 = ( 0 1 1 0 ) { γµ, γν } = 2 ηµν /a = aµ γ µ = a0 γ0 − a1 γ1 − a2 γ2 − a3 γ3 2 Quarks (Spin 1/2) Generation Flavor Charge Mass Other Quantum Numbers First d -1/3 7 0 u 2/3 3 0 Second s -1/3 120 Strangeness = -1 c 2/3 1200 Charm = +1 Third b -1/3 4300 Bottomness = -1 t 2/3 174000 Topness = +1 (Note: Quark masses are approximate.) Baryons (Spin 1/2) Baryon Quark Content Charge Mass Lifetime p uud 1 938.272 ∞ n udd 0 939.565 885.7 Λ uds 0 1115.68 2.63× 10−10 Σ+ uus 1 1189.37 8.02× 10−11 Σ0 uds 0 1192.64 7.4× 10−20 Σ− dds -1 1197.45 1.48× 10−10 Ξ0 uss 0 1314.8 2.90× 10−10 Ξ− dss -1 1321.3 1.64× 10−10 Λ+ c udc 1 2286.5 2.00× 10−13 Baryons (Spin 3/2) Baryon Quark Content Charge Mass Lifetime ∆ uuu, uud, udd, ddd 2, 1, 0, -1 1232 5.6× 10−24 Σ∗ uus, uds, dds 1, 0, -1 1385 1.8× 10−23 Ξ∗ uss, dss 0, -1 1533 6.9× 10−23 Ω− sss -1 1672 8.2× 10−11 Vector Mesons (Spin 1) Meson Quark Content Charge Mass Lifetime ρ u d̄, (u ū− d d̄ )/ √ 2, d ū 1, 0, 0, -1 775.5 4× 10−24 K∗ u s̄, d s̄, d d̄, s ū 1, 0, 0, -1 894 1× 10−23 ω (u ū+ d d̄ )/ √ 2 0 782.6 8× 10−23 ψ c c̄ 0 3097 7× 10−21 D∗ c d̄, c ū, u c̄, d c̄ 1, 0, 0, -1 2008 3× 10−21 Υ b b̄ 0 9460 1× 10−20 5 Pseudoscalar Mesons (Spin 0) Meson Quark Content Charge Mass Lifetime π± u d̄, d ū ±1 139.570 2.6× 10−8 π0 (u ū− d d̄ )/ √ 2 0 134.977 8.4× 10−17 K± u s̄, s ū ±1 493.68 1.24× 10−8 K0, K̄ 0 d s̄, s d̄ 0 497.65 K0 S : 8.95× 10−11 K0 L : 5.11× 10−8 η (u ū+ d d̄− 2 s s̄ )/ √ 6 0 547.51 5.1× 10−19 η′ (u ū+ d d̄+ s s̄)/ √ 3 0 957.78 3.2× 10−21 D± c d̄, d c̄ ±1 1869.3 1.04× 10−12 D0, D̄ 0 c ū, u c̄ 0 1864.5 4.1× 10−13 D±s c s̄, s c̄ ±1 1968.2 5.0× 10−13 B± u b̄, b ū ±1 5279.0 1.6× 10−12 B0, B̄ 0 d b̄, b d̄ 0 5279.4 1.5× 10−12 Baryons, Mesons, and the Eightfold Way Light Baryons (` = 0) n p Σ− Σ 0 Σ + Λ Ξ− Ξ 0 The Baryon Octet udd−dud√ 2 udu−duu√ 2 dsd−sdd√ 2 usd−sud+dsu−sdu 2 usu−suu√ 2 2uds−2dus+usd−sud−dsu+sdu√ 12 dss−sds√ 2 uss−sus√ 2 χ12 : Antisymmetric in 1 and 2 dud−ddu√ 2 uud−udu√ 2 dds−dsd√ 2 dus−dsu+uds−usd 2 uus−usu√ 2 2sud−2sdu+dus−dsu−uds+usd√ 12 sds−ssd√ 2 sus−ssu√ 2 χ23 : Antisymmetric in 2 and 3 udd−ddu√ 2 uud−duu√ 2 dds−sdd√ 2 uds−sdu+dus−sud 2 uus−suu√ 2 2usd−2dsu+uds−sdu−dus+sud√ 12 dss−ssd√ 2 uss−ssu√ 2 χ13 : Antisymmetric in 1 and 3 6 ∆− ∆0 ∆+ ∆++ Σ∗− Σ∗0 Σ∗+ Ξ∗− Ξ∗0 Ω The Baryon Decuplet ddd ddu+dud+udd√ 3 uud+udu+duu√ 3 uuu dds+dsd+sdd√ 3 uds+usd+dus+dsu+sud+sdu√ 6 uus+usu+suu√ 3 dss+sds+ssd√ 3 uss+sus+ssu√ 3 sss χS : Completely Symmetric uds−usd+dsu−dus+sud−sdu√ 6 χA : Completely Antisymmetric Light Mesons (` = 0) K0 K+ π− π 0 π+ η η′ K− K̄0 Pseudoscalar Meson Nonet (Octet plus Singlet η′) K∗0 K∗+ ρ− ρ 0 ρ+ φ ω K∗− K̄∗0 Vector Meson Nonet (Octet plus Singlet ω) 7