Physics 486. Classical Newton s laws Motion of bodies described in terms of initial conditions by specifying x(t), v(t).

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1 Physis 486 Tony M. Liss Leture 1 Why quantum mehanis? Quantum vs. lassial mehanis: Classial Newton s laws Motion of bodies desribed in terms of initial onditions by speifying x(t), v(t). Hugely suessful until the 0 th entury Any new theory must have Newton as limiting ase. Quantum mehanis Not all observables an be measured with arbitrary auray at the same time. The system under study is disturbed by the proess of measurement in an unpreditable way. ΔpΔx Probabilities enter as a fundamental part of the theory. Why is quantum mehanis needed? -Blakbody radiation Blakbody in thermal equilibrium with surroundings I vs. depends on T 1-1

2 peak given by Wien s displaement law: T = p onstant Total power radiated /area T 4 aording to Stefan-Boltzmann law: 4 ( ) = σt E T Rayleigh-Jeans law (lassial Blakbody) Cavity w/ onduting walls, heated to temperature T Cube of side a z y x Cube is filled w/ EM radiation due to eletron motion in the walls. Maxwell s equations tell us that the parallel omponent of the eletri field at the walls vanishes Wave propagation nodes at wall x, y, z = 0,a standing waves In one dimension we have E(x, t) = E0 ( π x ) Sin( πvt) v= 1-

3 With nodes at x = 0, a we require a = 1,,. whih implies that the allowed wavelengths are given by a = n = 1,,, n In terms of v =, we have n v = a Now, we are trying to alulate the energy density as a funtion of wavelength (or frequeny). The model is our ubial avity with onduting walls heated to a temperature T. We have established that for suh a body, there are only ertain allowed wavelengths that satisfy the boundary onditions. Eah allowed wavelength or frequeny (i.e. eah value of "n") is alled a mode. To find the energy density as a funtion of wavelength, or equivalently, frequeny, we need to ount the number of allowed modes in a narrow range of or ν. We will do the alulation for frequenies, the onversion to wavelength at the end is straightforward. How many modes between v & v+ dv? a 4a N ( v) dv = dv = dv where the fator of omes from the two allowed polarizations of the eletromagneti wave. This is the one dimensional ase, sine we derived the requirement for allowed frequenies above in one dimension. In three dimensions this beomes 8π a N( v) dv= Where does the fator of v ome from? Count -d modes v v+ dv v dv 1-

4 The volume element of sphere is 4π rdr, and given that our ube oupies one otant (1/8) of the -d retangular oordinate system above, we find that 1 n( v) dv = n( r) dr = 4 πr dr = πr dr 8 (again, the fator of two in the rd term is for the two polarizations) whih, given the relationship between r and ν from the figure above, we find a a n( v) dv π r dr π v dv 8 v dv π = = = This gives the number of standing waves (modes) in the frequeny interval v v+ dv To find the energy density as a funtion of frequeny, we have to evaluate the amount of energy assoiated with eah mode. Up to this point we have just used boundary onditions from Maxwell's E&M, and ounted modes. None of this is affeted by the quantum theory. This next step, however, is key. This is where the use of the lassial theory brings us to an inorret result. We assume the system to be in thermal equilibrium at temperature T, in whih ase the equipartition theorem tells us that the average kineti energy per degree of freedom is KE = KT the average total energy is twie the average kineti energy, or, independent of frequeny. Therefore, all we need to omplete our formula for energy density as a funtion of frequeny, is to multiply the number of modes per frequeny interval by the average energy of eah, or. This gives πν ρν b gdν = 8 dν And now, onverting bak to wavelength, we get the (lassial) Rayleigh-Jeans law π ρ b gd= 8 d 4 1-4

5 ρ() Rayleigh- Jeans Experiment Plank's quantum theory: Energy of waves proportional to frequeny E = Waves have disrete energies E = n (Note that from Maxwell's equations eah wave an have any energy). These assumptions require that the equipartition theorem, whih gave <E>= independent of ν, be modified. To understand what has to be done, let's first look at how the lassial result is derived: z E EPbEgdE e E =, where PE = z PEdE bg b g is the Boltzman distribution. This distribution is valid when the number of states is independent of energy. The integral in the denominator on the left is 1, sine the distribution is properly normalized. Therefore E E = z E e de = 0 whih is the equipartition theorem. If instead we have E=n, then we must replae the integral above by a sum over n: n E e n n= 0 = = e 1 n e 1 n= 0 whih is not independent of ν. Note two things about this result: b g E ν as 0 Ebνg 0 as whih is just what the dotor ordered. At long wavelength (low frequeny), the lassial theory is OK. At short wavelength (high frequeny), the average energy goes to zero, whih fixes the 'ultraviolet atastrophe'! Putting it all together, we find: 1-5

6 8πν ρ d e 1 Plank's blakbody spetrum ( ν ) dν = ν The remaining question is to find the arbitrary onstant 'h', known as Plank's onstant. This is done by fitting the funtional form above to the data, and the result is h=6.6 x 10-4 joule-se In terms of Plank's onstant, the Stephan-Boltzmann onstant σ from page above is given by 4 π k σ =, where = h π 60 The ultraviolet atastrophe was not the only experimental result that pointed towards the quantum nature of eletromagneti radiation. Two other results were the photoeletri effet, and Compton sattering. Photoeletri Effet - Eletrons are emitted from metal plates irradiated with eletromagneti radiation. Several features of this effet are A threshold in frequeny is observed, i.e. below a ertain (material dependent) frequeny, the emission of eletrons eases. The magnitude of the urrent of emitted eletrons is proportional to the intensity of the EM radiation. The energy of the emitted eletrons is independent of the intensity, but is proportional to the frequeny. These features an all be understood if one assumes that the impinging radiation is quantized aording to the Plank formula, and that the eletrons are held in the metal in a potential well of depth W, so that it takes at a photon with at least an energy of W, or frequeny ν=w/h, to ejet an eletron. The intensity determines the number of photons striking the surfae of the metal, while the energy of the eletron emitted is given by Einstein's formula E e = ( ) W 1-6