N(v) We then have a the distribution of velocities N(v) or the number of molecules at each velocity in the cavity.

Transcription

1 CHAPTER 2 QUANTUM THEORY OF LIGHT The quantum theory of light was realized in the later 19 th and early 20 th century, Soon after Boltzman set the laws of dealing with large ensembles of atoms with so called statistical mechanics some inconsistencies were seen involving radiation from solids. Wedgewood notes in 1789 that all objects in his ovens, regardless of chemical nature, size, shape, became red at the same temperature. By the mid 1800 s it is know that glowing objects emit continous light spectra. Thermal equilibrium We now understand that that objects emit and absorb radiation continuously. If the emission and absorption is perfectly matched at all frequencies as in the case of thermal equilibrium above the ball I the glowing oven becomes invisible! MAXWELL s DISTRIBUTION of SPEEDS Maxwell considered atoms in a container which is heated to temperature T. The container has a volume V. Through a slit some atoms escape and their speeds are measured. T v N(v) Velocity Distribution <v> AVG v We then have a the distribution of velocities N(v) or the number of molecules at each velocity in the cavity. Integrating over all v N(V) dv = N total number of molecules measured. We can write a normalized velocity distribution P(v) = 1/N N(v) which is the probability that a molecule is at velocity v.

2 Maxwell worked the distribution of speeds out out from statistical physics. Letting the number of molecules per unit volume be n=n/v and with KE = 1/2 m v 2 the Maxwell distribution of speeds is (, k=boltzmann constant and T=temperature) n(v) = A v 2 KE / kt e CAVITY RADIATION Such a cavity if made hot enough would also emit light! Attempts were made to understand the frequency distribution of light emitted from a cavity. This is called the cavity radiation problem. Kirchoff, circa. 1860, showed that for any body in thermal equlilibrium the radiant power emitted e EMITTED (per unit area per unit frequency) is proportional to the power absorbed. A ABSORBED at each frequency (a law of detailed balance). He showed that if this were not true then the 2 nd law of thermodynamics would be violated! e(f,t) EMITTED = α A(f,T) ABSORBED The radiant spectrum of a black body (α = 1) was sought. The black body perfectly absorbs all radtiation incident upon its surface appearing perfectly black. Absorption is not perfect in general, due to reflective losses or lack of absortion at some frequencies. A cavity with a small hole would absorb all radition incident on the hole. (Wein s idea!) Thus the hole (disk) becomes a black body and the emission spectrum can be measured.. STEPHAN s LAW Josef Stephan circa 1879 establishes from experiment, that the total power radiated from a near black body follows a universal law, (α = 1 for a true black body radiator.) e TOTAL = α σ T 4 σ = 5.67 x 10 8 W/m 2 K 4 is the StephanBoltzmann constant.

3 How much power are you radiating into the room? Let α =0.5 T BODY = 37C+273K = 310K A = surface area of your body ~ 4/5 m 2 e TOTAL = ( 5.7 x 10 8 W/m 2 K 4 e TOTAL = 260W/m 2 P = e TOTAL (4/5 )= 200W T BODY WEIN s DISPLACEMENT LAW (see example 2.2) Wein measured the radiant spectra from black bodies and proposed a universal law called the Wein Displacement Law. It describes the position of the maximum wavelength λ MAX versus body temperature. T. λ MAX T = x 10 3 mk RAYLEIGHJEANS and WEIN s EQUATIONS u~a f 4 RayleighJeans Law based on classical E&M and thermodynamics. Worked at Low frequencies only. Lead to an UltraViolet catastrophie.: diverging at hi frequency. u~a f 4 Wein s Law for higher frequencies. u~ A f 3 e C f / kt f Lord Rayleigh and Jeans formulate a spectral energy density u (f,t) from classical considerations with radiant energy increasing as th 4 th power of frquency which worked well at low frequencied and diverged at higher requencies (UltraViolet catastrophie). Wein also proposed a later which solved the UV catastrophie by only worked at high frequency?? u(f.t) = A f 3 e C f / kt

4 Each assumed that the cavity (or solid) is made of vibrating charges (radiators). Classically the energy of the radiation is proportional to the vibrational frequency or velocities of the oscillators thus the connection to Maxwell s velocity distribution for molecules. radiation radiation E = 1/2 M dx(t)/dt 2 X(t) = A cos(2π f t ) Initially believed to be successes for the classical views of thermodynamics and Maxwell s Equations, both laws failed to accurately describe the Black Body radiant energy spectrum MAX PLANCK Planck attacks the problem. He can only find correspondence with data if he assumes that the light spectra are coming from a discrete radiation source. Planck has stumbled on the quantum state of light the photon. Light exists in quana called photons. E = 3 E = E = 2 PHOTONS h = x evs Planck s Constant u(f.t) = A f 3 1 / [ e h f / kt 1 ] Planck s Law solved the cavity radiation problem and introduced the idea that light is not a continuous wave phenomena but acts as a quanta of energy E =. The idea is so new that many still disbelieve.

5 PHOTOELECTRIC EFFECT It was long known that high frequency light could liberate charge from a metal s surface. The reason was not clear. This was observed by Hertz. Later (1899) J.J. Thompson showed that these were negative charges (electrons). (1) Electrons were only emiitted once light of a high enough frequency was used no matter how intense the light? (2) Once this frequency was reached the photoelectric current rose with intensity. (3) Different metal seem to have different frequency theshold values. Einstein proposes the solution involves the interaction of light with the charges in the metal, not as a wave phenomena but like billiard balls! It is not understood if Einstein yet knew of Planck s quantum revelation. Einstein was a big fan of Deductive Reasoning or minimal influence with datasymmetry, beauty, God, thought experiment s etc. Left for further reading! Of course this is contrary to the most scientist of the time who used Inductive Reasoning whose conclusion s (equations) are drawn from experimental observation! KEe KEe KEe W 1 W 2 W 3 The electrons s kinetic energy upon exiting the metal derives from the photon s energy E= minus he work necessary for the electron to escape the metal, W = work. KE = W > 0 Must be positive! Electrons with the shortest paths attain the maximum kinetc energy, KE MAX. KE MAX = φ W MIN φ =W MIN is called the work function of the metal

6 PHOTOELECTRIC EFFECT APPARATUS I We can observe the photoelectric effect in the apparatus. depicted here. Light of frequency f is shone on a metal surface. A photocurrent I is generated. Stopping voltage Vs is increased from zero until the photocurrent I is reduced to zero. At this point evs = KE MAX I The electric field generated has stopped the most energetic electrons from crossing. Then we have evs = φ This equation of a line can be solved for the slope and intercept Vs Vs Vs = (h/e) f φ Slope = h/e QUANTUM WELL KE φ electrons Intercept = φ Conservation of energy φ = KE E=0 f E = φ e

7 PHOTOELECTRIC EFFECT A PARTICLE VIEW In the photoelectric effect all the photon s energy is transferred to the atomic electron. E= e E = KE + m e c 2 E = W + m e c 2 KE = 0 U = W e From conservation of energy W + m e c 2 = E = T + m e c 2 T = W LIGHT AS A QUANTA E = = hc/λ 0 E(eV) = 1240/λ(nm) E 2 = P 2 c 2 + m 2 c 4 E = Pc (photon) Because the photon is thought to have zero mass then in the relativistic view we have a simple relationship between momentum and energy! E = Pc XRAYS Xrays are light quanta on energy in the KeV range. Xrays were first produced by Roentgen (1895) by accident. He discovered that mysterious and very penetrating rays could be generated by smashing a beam of electrons into a metal. Ee

8 Xray Spectrum Intensity DISCRETE PEAKS λ MIN CONTINUOUS λ MIN λ (KeV) The Xray spectrum seem to have 2 components which needed explanation. (1) A continuous spectrum which rose and fell. No Xrays are observed below a minimum wavelenght or equivalently a maximum energy Emax.= hc/ λ MIN. (2) Intensity spikes occur at intermediate energies corresponding to atomic transitions?? CONTINUOUS SPECTRA. =EoE E E =E E Eo E E = Eo E = (KEo+mc 2 ) (KE + mc 2 ) = KEoKE λ MIN When the electron looses all of its energy in one collision E MAX = MAX = hc/ λ MIN occurs. =E E The continuous spectra is caused by the incident electron, energy Eo. undergoing a series of collisions with atoms Each time the electron is slowed and it s energy is degraded Eo>E >E >E A photon is emitted on each exchange to balance energy. This process is called Brehmstrahlung ( breaking radiation).

9 DISCRETE SPECTRA Eo Incident electron E2 E1 =E2E1 E Any incident electron at any stage of deacceleration can excite an orbital electron in to a higher state if E > E=E2E1. The orbital electron deexcites by emitting a photon of energy = E2E1. Incident electron of energy E wanders on with E = Eo, always conserving energy. This was not all perfectly understood at the time! COMPTON EFFECT Compton realized that he could test the quantum theory of light be scattering Xrays from Carbon atoms. And measureing the scattered energy. Classically light did not loose energy upon scattering in a medium. The wavelength did change λ n = λ/n. (Remember the λ n = λ/n is the wavelength change in a medium of index of refraction n) Compton suggests that a high energy photon (Xray) can scatter from an atomic electron and loose a fraction of it s energy, transferring the balance to the electron as Einstein suggests. =hc/λ y =hc/λ E=mc 2 φ E θ x Energy and Momentum Balance Equations in 2D + E = + E energy + mc 2 = cos(θ) + E cos(φ) xmomentum 0 = P γ sin(θ) + E sin(φ) ymomentum λ λ = (h/mc) ( 1cos(θ) ) λ C = h/mc = nm is called the Compton wavelength of an electron.

10 BRAGG SCATTERING LAW Bragg scattered Xrays from a crystal lattice and found interference phenomena which supported the wave picture of light. This was a confusing time. Was light a wave or particle phenomena? n λ = 2dsin θ λ θ θ dsin θ θ d GRAVITATIONAL REDSHIFT Does a photon have mass?? E =m c 2 = or m = / c 2 E= + mgy = mgy = ( / c 2 ) gy f = fgy/c 2 f/f = gy/ c 2 E = earth