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,