... black body radiation black body black body radiation Gustav Kirchhoff 859 895 W. Wien O.R. Lummer cavity radiation ν ν +dν f T (ν) f T (ν)dν = 8πν2 c 3 kt dν (Rayleigh Jeans) (.) f T (ν) spectral energy density k Boltzmann constant f T ν 0
2 Max Planck ν E = hν (.2) f ( ) T ν [ 0 8 Js/m 3 ] 20 5 0 5 T = 000 K Rayleigh-Jeans Planck h = 6.62676 0 34 J s (.3) Planck constant ν ν +dν f T (ν) 0 0 50 00 50 200 ν [ THz ].: f T (ν) f T (ν)dν = 8πhν3 c 3 ( hν exp kt ) dν (Planck) (.4) f T (ν) (.4) (.). (.4) hν ( ) hν exp kt + hν kt + ( ) hν 2 + 2 kt h (.) (.4) ν λ = c/ν λ λ +dλ u T (λ) T u T (λ)dλ = 8πhc λ 5 ( hc exp λkt ) dλ (Planck) (.5) u T (λ) SI [J m 4 ] λ [J m 3 ] u T (λ).2 T = 4000 K 5000 K 6000 K 7000 K
.. 3 λ λ+dλ da z c c z θ c cos θ u T (λ) ccos θ 4π dω = sin θ dθ dϕ u T (λ)dλda c cos θ dω 4π. θ 0 π/2 ϕ 0 2π π/2 0 cos θ sin θ dθ = 2, 2π 0 dϕ =2π u ( ) T λ [ kj/m 4 ] 3000 2500 2000 500 000 500 0 0 500 000 500 2000 λ [ nm ].2: u T (λ) (c/4) u T (λ)dλda T λ λ +dλ s T (λ)dλ = c 4 u T (λ)dλ. (.6) u T (λ) λ λ max λ max T = 2.897796 0 3 m K (.7) Wilhelm Wien Wien s displacement law u T (λ) (.5) λ du T (λ) = 6π2 h 2 ( c 2π hc e 2π hc/λk B T ) dλ λ 6 λk B T (e 2π hc/λk B T ) 5 2 e 2π hc/λk B T =0, x = 2π hc λk B T x 5=0 e x x
4 Stefan-Boltzmann law I T I = σt 4, σ = k 4 B 60π h 3 c 2 =5.67 0 8 Wm 2 K 4. (.8) σ Stefan-Boltzmann constant I = s T (λ)dλ x =2π hc/(λk B T ) I = c 4 8π (k B T )4 (2π hc) 3 0 x 3 e x dx = kb 4 0 60π h 3 c 2 T 4 I = σt 4 (.8) ( 0 x 3 ) π4 e x dx = 5..2 P.E.A. Lenard 905 ν hν h photon
.. 5 E E = hν (.9) hν.3 hν φ 2 mv2 = hν φ (.0).3: φ ev electron volt, φ work function ev ev =.6029 0 9 J. (.) R.A. Millikan.4 M M M e V [ V ] 3.0 2.5 2.0.5.0 V.4: 0.5 0.0 0 2 4 6 8 0 2 ν [ 0 4 s - ].5: Na M M V (.0) 2 mv2 = hν φ V 0 V V = hν φ. (.2) ν V
6 V =0 ν 0 φ φ = hν 0. (.3).5 ν V V =0 ν 0 =4.39 0 4 s φ =.8 ev.5 h h..3 A.H. Compton 99 923 X Compton scattering momentum, momenta E = hν, p = hν c. (.4) X E + mc 2 = E + E e p +0 = p + p e. (.5) E p E p E e p e mc 2 E 2 e = (E E + mc 2 ) 2 = c 2 p 2 + c 2 p 2 2c 2 pp +2mc 3 (p p )+m 2 c 4 p 2 e = (p p ) 2 = p 2 + p 2 2pp cos θ (.4) θ Ee 2 = c 2 pe 2 + m 2 c 4 Δλ = λ λ = h ( cos θ ) (.6) mc λ = h/p λ = h/p h mc = 2.43 0 2 m (.7) (.6) θ Δλ X
.2. 7.2.2. A.J. Angstrom 884 J.J. Balmer λ n = 364.56 n 2 n 2 4 nm ( n =3, 4, 5, 6 ) (.8) λ λ λ 2 λ 3 /λ /λ 2 /λ 3 C. Runge J.R. Rydberg λ mn = λ n R m 2 ( m = n +,n+2, ) (.9) n λ n n R W. Ritz m A m λ mn = A m A n. (.20) combination principle.6 λ n = R H n 2 ( n =, 2, 3, ) (.2) λ n R H Rydberg constant R H =.097 0 5 cm λ
8 0 2 /λ [ 0 4 cm - ] 4 6 8 0 2.6: ( = R λ H n n 2 ) n =, 2, 3, n 2 (.22) n = n +,n +2,n +3,.2.2 909 E. Rutherford H. Geiger E. Marsden 0 4 93 N. Bohr () stationary state
.2. 9 (2) transition E = hν +e e m r v F = ma = mv2 r = e 2 4πε 0 r 2 (.23) e 2 /(4πε 0 r) e2 E = 2 mv2 e2 4πε 0 r = 8πε 0 r (.24) (.23) (.2) n n = n =2 n = n i n = n f n i >n f E = hc ( λ = E i E f = e2 ). (.25) 8πε 0 r f r i n 2 (.22) n r n = a B n 2 ( n =, 2, 3, ) (.26) a B n = Bohr radius E n = e 2 8πε 0 a B n 2 ( n =, 2, 3, ) (.27) (.2) a B = e 2 8πε 0 hcr H (.28)
0 R H a B +e +Q λ = me2 Q 2 ( 8ε0 2 ) ch3 ni 2 (.29) +e Q =+2e n f =4 n i =5, 6, 7, n i =6, 8, 0, +e e Q =+2e He + (.25) n 2 f hν nm = E n E m (.30) E n E m r n v n (.23) v n = 4πε 0 e 2 mr n T n =2πr n /v n ν n = = v n = e 2 T n 2πr n 2π 4πε 0 mab 3 n 3. (.3) (.25) r f = r n = a B n 2 r i = r m = r n+l l n ν n = E n+l E n e 2 [ ] = h 8πε 0 ha B n 2 e 2 (n + l) 2 4πε 0 ha B n 3 (.32) (.3) /n 3 n correspondence principle (.32) (.3) e 2 = e 2 4πε 0 ha B 2π 4πε 0 mab 3
.2. a B = ε 0 h2 πme 2 = 4πε 0 h 2 me 2 = 0.529 0 0 m, (.33) e 2 ( ) 2 me 4 E = = 8πε 0 a B 4πε 0 2 h 2 = 2.795 0 8 J = 3.606 ev (.34) n r n = a B n 2 (.35) e v n = 2 4πε 0 ma B n = h (.36) ma B n r n p n = n h (.37) p n = mv n angular momentum 2π h (.37) (.37) Bohr s quantum condition.2.3 924 L. de Broglie E = hν E = mc 2 p λ = h p (.38)
2 λ E = cp E = hν E = cp = hν = hc λ (.38) E = cp hν 0 m 0 ν 0 hν 0 = m 0 c 2. (.39) f 0 sin 2πν 0 t 0 t 0 v t 0 t ( t 0 = t vx ) (v/c) 2 c 2. [ ( 2πν f(x, t) sin 0 (v/c) 2 ν = (.40) ν 0 (v/c) 2, λ = u ν = c2 v t vx c 2 ) ] [ ( = sin 2πν t x )] u u = c2 v (v/c) 2 hν hν = (.40) (.4) ν 0 (.42) hν 0 = m 0 c 2 = (v/c) 2 (v/c) 2 mc2 (.43) E = mc 2 λ = hc2 (v/c) 2 = hc2 v hν 0 v mc 2 = h mv = h p (.44) v u = c 2 /v v<c u>c v u
.2. 3.7.7: x x ( ) f (x, t) = a sin(kx ωt), f 2 (x, t) = a sin (k + Δk)x (ω + Δω)t. u = ω k, u 2 = ω + Δω k + Δk phase velocity Δω ω Δk k f(x, t) = f (x, t)+f 2 (x, t) = [( 2a sin k + Δk ) x 2 ( ω + Δω 2 ) ] ( Δk t cos 2 x Δω ) 2 t (.45) t = t.8 t = t.8: ( Δk A(x, t ) = 2a cos 2 x Δω ) 2 t Δk/2 A(x, t ) [( f(x, t ) = A(x, t ) sin k + Δk ) ( x ω + Δω ) ] t 2 2
4 k + Δk/2 A(x, t ) t x v g = Δω Δk, (.46) group velocity = ω k, dω = dk (.47) ω = 2πν = 2πν 0 (v/c) 2, ω = 2π λ = 2πν 0 v/c c (v/c) 2 (.48) ω k = 2πν 0 (v/c) 2 c (v/c) 2 2πν 0 v/c = c2 v (.4) β = v/c dω dβ = 2πν 0 β ( β 2 ) 3/2, dk dβ = 2πν 0 c ( β 2 ) 3/2 v g = dω/dβ dk/dβ = βc = v. (.49) v