Download Formulas and Equations for First Midterm in Physics and more Cheat Sheet Physics in PDF only on Docsity!
Formula Sheet: First midterm 2203, Fall 2012
1) Relativity: space and time Transforms from S to S’: : γ = 1/(1 vo^2 /c^2 )1/2^ vo in along the x axis. x’ = γ(x – vot) v’x = (vx – vo)/(1 – vxvo/c^2 ) y’ = y v’y = vy/γ (1 – vxvo/c^2 ) z’ = z v’z = vz/γ (1 – vxvo/c^2 ) z’ = z t’ = γ (t – (vo/c^2 )x) Change sign to go from S’ to S: Time dilation Δt=γΔt 0 : Δt 0 Proper measurement of time. Length contraction Λ=ΔL 0 /γ : L 0 proper measurement of length. 2) Relativistic Mechanics E’ = γ [E – (vo/c)(pc)] p’ = γ [p – (vo/c^2 )/E] p = γ m v p c = γ mc^2 ( v /c) EK(kinetic energy) = mc^2 (γ – 1) F =d p /dt E (total energy) = γ mc^2 E^2 = p^2 c^2 + (mc^2 )^2 : E 2 − mc 2
2 E=pc for massless particle v = p c^2 /E E = EK + mc^2 Relativistic Doppler shift f’/f = γ [1 – (v/c)cosθ] or f’/f = [(1‐v/c)/(1+v/c)]1/2^ (for θ = 0o) 3) Atoms Notation € Z A ZN Where A=Z+N with Z being the number of protons and N the number of neutrons. An example is Carbon with 6 neutrons and 6 protons € 6 12 C 6 Kinetic theory pV=nRT or pV=NkBT, with kB the Boltzmann’s constant and R the universal gas constant. kB=R/NA NA is Avogadro’s number average kinetic energy €
m v^2 =
kB T 4 ) Quantization of light: Photon has energy hf
E = hf, E = ω , with ω = 2 π f E = hc/λ = 1240/λ eV nm
E=pc Photoelectic effect Compton Scattering Kmax = hf – eφ λ’ ‐ λ = λc(1 – cos θ ) eφ is work function λc = h/mec = 0.00246 nm Group velocity vg = d ω dk , Phase velocity v (^) p = c^2 v and v = vg
Bragg Scattering 2dsinθ=nλ 5 ) Quantization of Atomic Energy levels: Bohr Atom Rydberg series series
λ
= R [
n 2 f
n 2 i ], i is the initial state and f the final state: R is Rydberg constant. Bohr Model: The angular momentum is quantized L = mvr = n or €
2 π rn = n λ yielding the
following equations: rn = n^2 ^2 ke 2 m where the Bohr orbit radius is defined a 0 =
^2
ke 2 m = 0.05292 nm En = − m 2 n
ke^2
2 giving En = − 13.6 eV n 2 velocity: vn = ke^2 n For a real system m should be the center of mass m’ where € m ' = mM m + M = M 1 + M / m Then r ' n = n^2 ^2 ke 2 m ' = n 2 me m ' a 0 and E ' n =^ −^ m ' 2 n
2 (^
ke^2
2 = − m ' me
E 1
n 2 Hydrogen like ions: one electron bound to Ze nucleus: vn = ke^2 n , En = − mZ 2 2 n^2
ke 2
2 , rn = n^2 ^2 kZe 2 m 6) Particles as waves p = h/λ or λ = h/p ke 2 = 1.44 nm ( eV ) λ = hc 2 mc^2 K
1240 eV i nm 2 mc^2 K de Broglie wave length λ = h/p Uncertainty principle
Δ k Δ x ≥ 1 / 2 and Δ ωΔ t ≥ 1/ 2 which can also be written as
Δ p Δ x ≥ / 2 and Δ E Δ t ≥ / 2 7) Schrödinger Equation in one Dimension General properties of Schrödinger’s Equation: Quantum Mechanics Schrödinger Equation (time dependent) €
^2
2 m
∂^2 Ψ
∂ x^2
∂ t Standing wave € Ψ( x , t ) = Ψ( x ) e − i ω t Schrödinger Equation (time independent) −
2 2 m
2 ψ ∂ x 2 + U^ ψ^ =^ E^ ψ
Constants c = 2.998 x 10+8^ m/s h = 6.626 x 10‐^34 J.sec =4.136 x 10‐^15 eV.sec = 1.055 x 10 − 34 J. s = 6.582 x 10 − 16 eV. s k =
4 π ε 0 = 8.988 x 10 9 N. m 2 / C 2 me = 9.109 x x 10‐^31 kg mec^2 = 0.511 MeV mp = 1.673 x 10‐^27 kg mpc^2 = 938.28 MeV mn = 1.675 x 10‐^27 kg mnc^2 = 939.57 MeV mp= 1836me mn=1839me 1u = 931.5 MeV/ c^2 nm = 10 ‐^9 m e = 1.6 x 10‐^19 coul kB = 8.617 x 10‐^5 eV/K eV=1.6 x 10‐^19 J NA= 6.022x10^23 objects/mole binomial expansion: (1 + x)n^ = 1 + nx + n(n‐1)x^2 /2! + n(n‐1)(n‐2)x^3 /3! μm=10‐^6 m ,nm=10‐^9 m, pm=10‐^12 m, fm=10‐^15 m K = 10^3 , M = 10^6 , G = 10^9
Useful relationships:
hc = 1240 eV i nm c = 197 eV i nm ke 2 = 1.44 eV i nm ke^2 c
R=
m ( kee )
2 4 π c ^3
mc^2 ( k ee )
2
4 π ( c )
3 R = 0.011 nm −^1
