n * λ = 2 * d * sin θ (Bragg’s Law) λ = h / p (de Broglie wavelength of a particle) p = h / λ = h-­‐bar * k E = p * c = h * f = h * c / λ = h-­‐bar * ω (the energy of a photon) Ψ ( x , t ) = A * sin [ ( 2 * π / λ ) * ( x – v * t ) ] (a simple, classical sinusoidal wave). Also = A * sin ( k * x -­‐ ω * t ) traveling in positive-­‐x or A * sin ( k * x + ω * t ) in negative x A * sin (k * x -­‐ ω * t + φ ) traveling in positive-­‐x but now with arbitrary phase φ δ2Ψ / δx2 = ( 1 / v ^ 2) * δ2Ψ / δx2 (classical wave equation) λ = v_phase * T, v_phase = λ ∗ f (wavelength, frequency, period and velocity relationships) k = ( 2 * π / λ ), wavenumber and ω = ( 2 * π / Τ ), angular frequency Ψ ( x , t ) = Ψ1 ( x , t ) + Ψ2 ( x , t ) + … (the principle of superposition of waves) Δk * Δx = Δω * Δt = 2 * π (classical, non-­‐Gaussian version of the uncertainty principle) Ψ ( x , 0 ) = Ψ ( x ) = A e – k^2 * x^2 * cos ( k0 * x ) (a Gaussian function wave packet) Δk * Δx = 1.4 approximately in this case, or ½ after a more precise calculation v_group = Δω / Δk OR dω / dk (differential form). v_phase = ω / k (equivalent to above) v_group = dE / dp (relativistic E^2 = p^2 * c^2 + m^2 * c^4) v_group = v_phase + k * dvph / dk (v_group also known as u_gr in text) Δp * Δx >= h-­‐bar / 2 and ΔE * Δt >= h-­‐bar / 2 (Heisenberg uncertainty principle) P ( x ) dx = | Ψ ( x , t ) | 2 dx (probability density) | Ψ | 2 = Ψ* Ψ Integral from –inf to inf of P( x ) dx = 1 (normalization) From x1 to x2 for prob. betw. x1, x2 E_n = n^2 * h^2 / ( 8 * m * l^2 ) (particle in a box energy levels) i * h-­‐bar * δΨ( x , t )/δt = -­‐ (h-­‐bar ^2 / 2m) * δ2Ψ( x , t )/δx2 + V * Ψ ( x , t ) (Schrodinger eq.) 3D version has δ2Ψ( x , t )/δx2 + δ2Ψ( y , t )/δy2 + δ2Ψ( z , t )/δz2 in place of δ2Ψ( x , t )/δx2 Time-­‐independent version has -­‐ (h-­‐bar ^2 / 2m) * d2Ψ( x )/dx2 + V ( x ) * Ψ ( x ) = E * Ψ ( x ) When V does not depend on time, then Ψ( x , t ) = Ψ( x ) * e -­‐i t If Ψ is real OR as above, then Ψ* Ψ is same as squaring. Complex conjugate (*) means i -­‐> -­‐i Expectation value (quantum mechanical average) of an operator O is <O> = Integral over desired bounds of Ψ* ( x , t ) O Ψ( x , t ) dx (NOT always commutative) Momentum operator p-­‐hat = -­‐ i * h-­‐bar * δ / δx. Energy operator E-­‐hat = i * h-­‐bar * δ / δt Ψ n ( x ) = sqrt ( 2 / L ) * sin ( n * π * x / L ) (infinite square well potential. n >= 1 integer) E n ( x ) = ( n^2 * π^2 * h-­‐bar^2 ) / ( 2 * m * L^2 ) (quantized energy levels) If differential equation has form δ2Ψ/δx2 = constant * Ψ: (constant traditionally squared) Negative constant means the answer is complex exponentials or sinusoids (sin’s and cos’s) Positive constant means that the answer is real exponentials The above information allows you to solve the Schrodinger equation for a finite square-­‐
well potential and for potential barriers ( E > V0 ) and for tunneling though them ( E < V0 ) Δ
ω
Ψ n ( x ) = H n ( x ) * e – * x^2 / 2 (quantum mechanical simple harmonic oscillator) where alpha = sqrt ( m * κ / h-­‐bar ^ 2 ), where κ is the “spring constant” E n = ( n + ½ ) * h-­‐bar * ω (energy levels, where n >= 0 is an integer). ω = sqrt ( κ / m ) Hn ( x ) are the Hermite polynomials. H0 ( x ) = 1, H1 ( x ) = x, H2 ( x ) = (2*α*x^2 -­‐ 1),… kI = kIII = sqrt ( 2 * m * E ) / h-­‐bar where V = 0 (regions I and III, with barrier in region II) kII = sqrt ( 2 * m * ( E – V0 ) ) / h-­‐bar where V = V0 (if E > V0) T = [ 1 + V0^2 * sin2 ( kII * L ) / ( 4 * E * ( E – V0 ) ) ] ^ -­‐1 (transmission probability if E > V0) κ = sqrt ( 2m * ( V0 – E ) ) / h-­‐bar (region II when E < V0) T = [ 1 + V0^2 * sinh2 ( κ * L ) / ( 4 * E * ( V0 – E ) ) ] ^ -­‐1 (transmission when E < V0) a0 = ( 4 * π * ε0 * h-­‐bar^2 ) / ( µ * e^2 ), where µ = me-­‐ mp+ / ( me-­‐ + mp+ ) Ground state energy = -­‐ h-­‐bar^2 / (2*µ∗a0^2) = -­‐ 13.6 eV. En = divide this by n^2 Ψnlm ( r , θ , φ ) = R ( r ) * f ( θ ) ∗ g ( φ ) = Rnl ( r ) * Ylm ( θ, φ ) Principal quantum # n = 1,2,3…. l = 0 to n-­‐1, ml = -­‐l to l (l=0 is s, l=1 is p, l=2 is d, l=3 is f and n=1 is K and n=2 is L and n=3 is M and n=4 is N) Tables 7.1 and 7.2 will be provided separately, as will the integral appendix Probability of being in between radius r1 and r2 is the integral from r1 to r2 of r^2 * | Rnl ( r ) | 2 * dr (where r^2 comes from volume element of r^2 * sin(θ) dr dθ dφ L = sqrt ( l * ( l + 1 ) ) * h-­‐bar (using orbital q#) and Lz = ml * h-­‐bar (using azimuthal q#) S = sqrt ( s * ( s + 1 ) ) * h-­‐bar and Sz = ms * h-­‐bar VB = -­‐ µz B = + µB ml B, where µB = e * h-­‐bar / (2me-­‐) is the Bohr magneton (Zeeman effect) VB = + 2 µB ms B (due to intrinsic electron angular momentum; above due to orbital L) Selection rules: Δn = anything, Δl = +/-­‐ 1, Δml = 0, +/-­‐1, (Δmj = 0, +/-­‐1, Δj = 0, +/-­‐1) J = L + S (vector sum where total angular momentum is sum of orbital & spin momenta) J = sqrt ( j * ( j + 1 ) ) * h-­‐bar and Jz = mj * h-­‐bar and j = l +/-­‐ s State notation nLj Constants and units 1 eV (unit of energy) = 1.602 x 10-­‐19 J 1 e charge of proton or electron = +/-­‐ (respectively) 1.602 x 10-­‐19 C 1 atomic mass unit (amu or u) = 1.66 x 10-­‐27 kg = 931.494 MeV / c2 1 light-­‐year is the distance light travels in one year (not a unit of time) Speed of light in a vacuum c = 299,792,458 m/s (defined, exact) E0 (proton) = 1.67 x 10-­‐27 kg times c^2 = 938.27 MeV E0 (neutron) = 939.57 MeV E0 (electron) = 9.109 x 10-­‐31 kg times c^2 = 0.511 MeV Avogadro’s number = 6.022 x 1023 particles per mole σ = 5.6705 x 10-­‐8 W / ( m2 * K4 ) k_Boltzmann = 1.38 x 10-­‐23 Joules per Kelvin h (Planck’s constant) = 6.6261 x 10-­‐34 J * s and h-­‐bar equals h over 2*pi Bohr radius a0 = 0.53 x 10-­‐10 m 1 / ( 4 * pi * ε0 ) = 9 x 109 Newton – meters^2 per Coulomb^2 α
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