Want to review modes of electromagnetic radiation in cavity. Start with Maxwell s equations in free space(si units)

Size: px

Start display at page:

Download "Want to review modes of electromagnetic radiation in cavity. Start with Maxwell s equations in free space(si units)"

Austen Barton
6 years ago
Views:

1 2 Quantization of Normal Modes 2.1 Wave equation Want to review modes of electromagnetic radiation in cavity. Start with Maxwell s equations in free space(si units) E = 0 (1) B = 0 (2) E + B = 0 (3) t E B µ 0 ɛ 0 = 0 (4) t where is gradient operator = î x + ĵ y + ˆk z (5) Will also need subsidiary condition, charge conservation: and Lorentz force: ρ t + j = 0 (6) F = q(e + v B) (7) Now take curl of Eq. (3), and use identity ( E = ( E) 2 E: 0 = ( E + B t ) (8) 1

2 = 2 E + t B }{{} (9) E µ 0 ɛ 0 t (10) = 2 2 E E + µ 0 ɛ 0 t 2 (11) i.e. a wave equation for each component of the electric field E for waves travelling with speed c µ 0 ɛ 0. Why is it a wave eqn.? Consider soln. E = E 0 f(x ct), with f any function. Check! Compare field profile f(x) at time t = 0 with profile time t later: simply translated to right distance ct = pulse f(x) moving to right with velocity c (vel. of light!) Check that 1. E î (direction of propagation) 2. B field î, E 2.2 Normal modes of cavity Example: transverse waves on string. displacement of string δ(x, t), obeys wave equation 2 2 δ t = 2 δ 2 c2 s (12) x 2

3 where c s velocity of sound. Boundary conditions: suppose δ(x = 0, t) = 0, (13) δ(x = L, t) = 0 (14) Normal modes of vibration: solutions of form δ = f(x)cos(ωt + φ) (15) consistent with boundary conditions. Substitute in (12) solution only if Note these are standing waves. ω 2 f(x) = c 2 d 2 f s dx2. (16) (16) called Eigenvalue problem (German: proper value) since eqn has solutions consistent with B.C. only for special values (eigenvalues) ω. General soln. to (16): f(x) = A sin ωx c s + B cos ωx c s (17) B.C. δ(x = 0) = 0 = B = 0, δ(x = L) = 0 = A = 0 or ω = nπc s /L, n = 1, 2, 3... (18) N.B. This purely classical result has connection with quantization as used in SHO problem. 3

4 Modes in cavity: wave eqn. for electric field. Assume walls perfect conductors, so B.C. are E = 0, B = 0 at walls. Search for solns of Maxwell eqns. of form consistent with these B.C. 2.3 Periodic boundary conditions E = cos(ωt + φ)f(r) (19) We can find solutions with fixed (standing wave) boundary conditions, but since it is sometimes convenient to use other B.C., let s look at periodic B.C., i.e. assume cavity is periodically repeated with period L along x, y, z x, y, z is same pt. as x + L, y, z, etc. This is ok since the shape of cavity, exact pos. of walls can t matter if we are calculating something which depends only on the density of modes in freq. range. such that L sizes of interest. Consider solutions E = Re E 0 e i(k r ωt) (20) 4

5 = E 0 cos(k r ωt) (21) E 0 = const. φ = const Re= real part of Remarks: Plug into wave eqn. = ω 2 = k 2 c 2 E = 0 = k E 0 = 0 transverse wave Periodic B.C. = k x L = 2πn x, n x = 0, ±1, ±2... k y L = 2πn y, n y = 0, ±1, ±2... (22) k z L = 2πn z, n z = 0, ±1, ±2... Note allowed k s separated by 2 larger interval compared to fixed b.c., but n x,y,z can be ± = same # per dν. note that for, e.g., n x = ±1, + and - give independent solutions check! This was not true for fixed B.C. Allowed frequencies are therefore 5

6 here ω = kc = 2πc L (n2 x + n 2 y + n 2 z) 1/2. (23) Maxwell s eqns linear lin. comb. of solns. is soln. General: E = Re p=1,2 n x,n y,n z E 0 (n, p)e i(k r) ωt (24) p is polarization of wave: 2 lin. ind. directions of E for given k n = n x, n y, n z k = 2πn/L Exercise: derive form for B from Eq. (24) and James Clerk s equations Counting modes Need # of solutions e i(k r ωt) with freqs in range ν to ν + δν. Look at k-space : 6

7 Allowed k form cubic lattice in k-space, spacing 2π/L volume occupied by each allowed k is (2π/L) 3 volume in k-space between k and k + δk is 4πk 2 δk therefore number of k s in this shell is δn = 4πk 2 δk (L/2π) 3 (25) Now relate number of k s in (k, k + δk) to number of allowed solutions with ν s in (ν, ν + δν) (recall ω = 2πν = kc!) : L 3 (2π δn = }{{} 2 4π }{{} ) 3 ν 2 δν 8π 3 } c {{} 2 polarizations θ, φ integration k = 2πν/c = 8πL 3 ν 2 δν/c (26) 2.5 Planck Law Planck idea (1900): each mode of radiation in cavity acts like SHO suppose it contributes mean energy hν/(exp hν/kt 1). Mean energy in (ν, ν + δν) is then δe = 8πL3 hν 3 δν c 3 (e hν/kt 1) (27) Energy per unit volume and bandwidth: u ν 1 L 3 δe δν = 8πhν 3 c 3 (e hν/kt 1) (28) 7

8 ( Aside: at high temperatures, hν/kt 1, expand exponential to find u ν = 8πkT ν 2 /c 3, Rayleigh-Jeans law. Note there s no h classical result.) Net energy density now u = 0 u ν dν = 8πh c 3 0 ν 3 dν (e hν/kt 1) (29) Make integral dimensionless, put α = hν/kt, dν = kt dα/h: u = 8πh (kt ) 4 x 3 dx c 3 h 0 e x 1 }{{} π 4 /15 (30) So Planck also recovered : and determined u = at 4 Stephan-Boltzmann law (31) a = 8π5 15 k 4 (hc) 3 (32) But if had let h 0 as originally intended, nonsense! Although made no sense classically, P. brave enough to allow h to stay. Stephan s const. pretty well known expt lly in 1900 = Planck determined h = erg sec, close to modern h = erg sec! 8

Physics 576 Stellar Astrophysics Prof. James Buckley. Lecture 2 Radiation

Physics 576 Stellar Astrophysics Prof. James Buckley Lecture 2 Radiation Reading/Homework Assignment Read chapter 1, sections 1.1, 1.2, 1.5 Homework will be assigned on Thursday. Radiation Radiation A