1. Trang chủ
  2. » Kỹ Thuật - Công Nghệ

Planck, atomic physics, einstein AB coeffecients

55 197 0

Đang tải... (xem toàn văn)

Tài liệu hạn chế xem trước, để xem đầy đủ mời bạn chọn Tải xuống

THÔNG TIN TÀI LIỆU

Thông tin cơ bản

Định dạng
Số trang 55
Dung lượng 264,41 KB

Nội dung

ECE 275B © P.J. Burke, Winter 2003 Last modified 1/2/2003 1:58 AM Lecture 4, Slide # 1 Lecture 4: Photons and atoms • Electromagnetic modes in a box • Blackbody radiation; photons, Planck law • Photoelectric effect • Energy spectrum of hydrogen • Einstein A/B coefficients •Three-level laser • Reading: Ch. 7 of Verdeyen ECE 275C © P.J. Burke, Winter 2003 Last modified 1/2/2003 1:58 AM Lecture 4, Slide # 2 Boundary conditions: L L L ∧ z ∧ y ∧ x 0=⋅∇ →→ E If we implement we find two things: ti zyxz ezkykxkEtzyxE ω )cos()sin()sin(),,,( 3 ⋅⋅= ti zyxx ezkykxkEtzyxE ω )sin()sin()cos(),,,( 1 ⋅⋅= ti zyxy ezkykxkEtzyxE ω )sin()cos()sin(),,,( 2 ⋅⋅= First: Second: 0 321 =++ zyx kEkEkE () 222 2 222 2 zyxzyx nnn L kkk c ++       =++=       πω n 1 ,n 2 ,n 3 integers; at least two must be non-zero. Each combination of n 1 ,n 2 ,n 3 is a “mode”. Discuss superposition. The magnitude of E 1 , E 2 , E 3 still can be any value! (Subject to constraints above.) ECE 275C © P.J. Burke, Winter 2003 Last modified 1/2/2003 1:58 AM Lecture 4, Slide # 3 How much energy is in the box?               +       = →→→→→ 2 0 2 0 ),(),( 2 1 ),( trHtrEtru µε Instantaneous energy per unit volume: Total energy in box: dVtruU box total ∫∫∫ → = ),( It can be shown that for the box: ()( ) 2 3 2 2 2 1 222 2 1 EEEnnnU zyxtotal ++++= So, the amount of energy in the box can have any value. We will show that this leads to a problem and must be wrong. The energy in the box must be quantized: these are photons. ECE 275C © P.J. Burke, Winter 2003 Last modified 1/2/2003 1:58 AM Lecture 4, Slide # 4 Concrete example of a mode: ()( ) 2 3 2 3 2 2 2 1 222 2 1 2 1 EEEEnnnU zyxtotal =++++= 0,,0;1;1 ===⇒=== zyxzyx k L k L knnn π π 00 21321 ==⇒=++ EEkEkEkE zyx ti z e L y L x EtzyxE ω ππ ⋅       ⋅       = sinsin),,,( 3 0),,,( =tzyxE x 0),,,( =tzyxE y The total energy in this mode can have a continuum of values, depending only on E 3 . The same is true for all other modes. (Discuss superposition). ECE 275C © P.J. Burke, Winter 2003 Last modified 1/2/2003 1:58 AM Lecture 4, Slide # 5 Blackbody radiation Consider E-M field in thermal equilibrium with matter at some temperature T. If one is inside a box, do the walls glow? Yes. ECE 275C © P.J. Burke, Winter 2003 Last modified 1/2/2003 1:58 AM Lecture 4, Slide # 6 How is energy in the box related to temperature? According to the equipartition theorem from thermodynamics, every mode of the system has an average energy <U>=(1/2)k B T. Note: This is already a problem. Energy infinite. What is the energy per frequency, then we will integrate over frequencies? There are many modes per unit frequency. Each has energy k B T. ECE 275C © P.J. Burke, Winter 2003 Last modified 1/2/2003 1:58 AM Lecture 4, Slide # 7 Modes per frequency ννννε dNkTd )( 2 1 )( ⋅= • ε(ν)dν is the energy between ν and ν+dν. • (This is the spectrum of the blackbody radiation.) •N(ν)dν is the number of modes between ν and ν+dν. ECE 275C © P.J. Burke, Winter 2003 Last modified 1/2/2003 1:58 AM Lecture 4, Slide # 8 Modes per frequency ν ν dN )( 32 3 )8( 1 )( L c dN νπνν = ν ν ν ν ε dNkTd )()( ⋅= 32 3 8 )( LkT c d ⋅⋅⋅= ⇒ ν π ννε ∞=⇒ ∫ ∞ 0 )( ννε d Rayleigh-Jeans law. Experiments confirm at low frequencies only. ECE 275C © P.J. Burke, Winter 2003 Last modified 1/2/2003 1:58 AM Lecture 4, Slide # 9 Recall Boltzmann factor P(ε): “The probability for a physical system to be in a state with energy ε is proportional to .” Equipartition: Tk B e / ε − (This is fundamentally linked to the concept of temperature. Take it as an absolute truth for the whole class.) ECE 275C © P.J. Burke, Winter 2003 Last modified 1/2/2003 1:58 AM Lecture 4, Slide # 10 Recall Boltzmann factor P(ε): “The probability for a physical system to be in a state with energy ε is proportional to .” In order to get p(ε) to be between 0, 1 we need to normalize it: Equipartition: Tk B e / ε − 1)( = ∑ i i p ε [...]... Note the length of the box did not really matter ECE 275C © P.J Burke, Winter 2003 Last modified 1/2/2003 1:58 AM Lecture 4, Slide # 22 Photoelectric effect Einstein s explanation: • Electrons bound to metal by work function W • If one photon is absorbed, energy of electron after being liberated is hν−W=h(ν−νc) • eV0 is the “stopping potential = h(ν-νc) • Slope of V0 vs ν is h/e Vacuum tube A B Current... R=107 m-1 (Rydberg) n>m m=1 Lyman series; m=2 Balmer series, etc ECE 275C © P.J Burke, Winter 2003 Last modified 1/2/2003 1:58 AM Lecture 4, Slide # 25 Energy spectrum of hydrogen: Absorption Certain wavelengths are strongly absorbed These are the Lyman series and at elevated temperatures the Balmer series What does it mean? For photons, ε = n h ν Hydrogen energy levels are quantized ∆ε n , m 1  1 =...Boltzmann distribution: Recall Boltzmann factor P(ε): “The probability for a physical system to be in a state with energy ε is proportional to e −ε / k T ” B In order to get p(ε) to be between 0, 1 we need to normalize it: ∑ p(ε i ) = 1 i p(ε ) = e − ε / k BT ∑e −ε . probability for a physical system to be in a state with energy ε is proportional to .” Equipartition: Tk B e / ε − (This is fundamentally linked to the concept of temperature. Take it as an absolute. superposition. The magnitude of E 1 , E 2 , E 3 still can be any value! (Subject to constraints above.) ECE 275C © P.J. Burke, Winter 2003 Last modified 1/2/2003 1:58 AM Lecture 4, Slide # 3 How. Blackbody radiation; photons, Planck law • Photoelectric effect • Energy spectrum of hydrogen • Einstein A/B coefficients •Three-level laser • Reading: Ch. 7 of Verdeyen ECE 275C © P.J. Burke,

Ngày đăng: 27/03/2014, 11:27