ψ x=0,t =ψ x=l,t =0 ψ x,t = X x T t X x t t =X x 2 Something more about. Standing Waves Standing Waves Wave Function Differential Wave Equation X=L

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1 Something more about. Standing Waves Wave Function Differential Wave Equation Standing Waves Boundary Conditions: ψ x=0,t =ψ x=l,t =0 Separation of variables: X=0 sin(πx/l) sin(πx/l) sin(3πx/l) X=L Y Axis Title 1,0 ψ x,t = X x T t Wave Function: 0,5 0,0-0,5-1,0 0 X Axis Title L ψ X x =T t x x ψ T t =X x t t

2 Space: f(x) ψ 1 ψ = x v t TIme: f(t) 1 X x 1 T t = =constant X x x v T t t 1 d X x 1 d T t ω = = =constant X x dx v T t dt v Equivalent to two ordinary (not partial) differential equations: d X x ω = X x dx v d T t = ω T t dt ψ x, t = X x T t =sin Space: X(x) nπx A n sin ω n t B n cos ω n t L Time: T(t)

3 λ n =L Eigenvalue Condition: Eigenfunctions: ψ n x, t = X x T t =sin n=0, ±1, ±, ±3 nπx A n sin ω n t B n cos ω n t L General solution: Principle of superposition ψ = ψ n x, t = sin n=0 n=0 Since any linear Combination of the Eigenfunctions would also be a solution nπx A n sin ω n t B n cos ω n t L Fourier Series

4 Fourier Series Y Axis Title Any arbitrary function f(x) of period Λ can be expressed as a Fourier Series X Axis Title f x = f x Λ = n=0 π nx π nx An sin B n cos Λ Λ f x = f x Λ = n= π nx C n exp i Λ REAL Fourier Series COMPLEX Fourier Series

5 Wave Phenomena Reflexion Refraction Diffraction is the bending of a wave around an obstacle or through an opening. θi θi = θr Interference Diffraction n1 Wavelenght dependence Diffraction at Slits θt n n1 sin (θi) = n sin (θt) w θ p=w sin(θ)=mλ bright fringes Diffraction at a lattice θ p=w sin(θ)=mλ d The path difference must be a multiple of a wavelength to insure constructive interference. θ p=d sin(θ)=mλ bright fringes

6 Intensity pattern that shows the combined effects of both diffraction and interference when light passes through multiple slits. Interference and Diffraction: Huygens construction m= m=1 m=0

7 Black-Body Radiation A blackbody is a hypothetical object that absorbs all incident electromagnetic radiation while maintaining thermal equilibrium. E f df = 4πV f U f df 3 c U f =kt

8 Black-Body Radiation: classical theory L L λ= n= n λ dn L = 1D dλ λ Radiation as Electromagnetic Waves v λ= ; f dn 4πV = 4 dλ λ dλ v = df f dn dλ dn 4πV = = 3 f dλ df df v 3D Since there are many more permissible high frequencies than low frequencies, and since by Statistical Thermodynamics all frequencies have the same average Energy, it follows that the Intensity I of balck-body radiation should rise continuously with increasing frequency. Breakdown of classical mechanical principles when applied to radiation!!!ultraviolet Catastrophe!!!

9 The Quantum of Energy The Planck Distribution Law Physics is a closed subject in which new discoveries of any importance could scarcely expected. However He changed the World of Physics Nature does not make a Jump Matter Discrete Energy Continuous Classical Mechanics Max Planck 3 8πhν dν Energy Continuous E ν dν = Planck: Quanta dν 3 hν /kt c e 1 8 πν hν << kt, E ν dν= 3 kt dν h= 6.66 x Joule.sec E = hν c An oscillator could adquire Energy only in discrete units called Quanta!Nomenclature change!: ν f

10 Photoelectric Effect: Einstein The radiation itself is quantized Fluxe 1 Fluxe Metal ν>νο νο ν Ι Below a certain cutoff frequency νο of incident light, no photoelectrons are ejected, no matter how intense the light. Above the cutoff frequency the number of photoelectrons is directly proportional to the intensity of the light. 1 As the frequency of the incident light is increased, the maximum velocity of the photoelectrons increases. In cases where the radiation intensity is extremely low (but ν>νο) photoelectrons are emited from the metal without any time lag.

11 Photon E = hν Energy of light: Kinetic Energy = Energy of light Energy needed to escape surface (Work Function): ½ mev = hν hνο Φο : It depends on the Nature of the Metal Increasing the intensity of the light would correspond to increasing the number of photons. Increasing the frequency of the light would correspond to increasing the Energy of photons and the maximal velocity of the electrons.

12 Light as a stream of Photons? E = hν discrete Zero rest mass!! Light as Electromagnetic Waves? E = εο Eelec = (1/µο) Bmag continuous The square of the electromagnetic wave at some point can be taken as the Probability Density for finding a Photon in the volume element around that point. Energy having a definite and smoothly varying distribution. (Classical) Albert Einstein A smoothly varying Probability Density for finding an atomistic packet of Energy. (Quantical)

13 The Wave Nature of Matter All material particles are associated with Waves ( Matter waves De Broglie A central concept of Quantics: wave particle duality is the concept that all matter and energy exhibits both wave -like and particle -like properties. E = hν mc = hν = hc/λ E = mc or: mc = h/λ A normal particle with nonzero rest mass m travelling at velocity v Then, every particle with nonzero rest mass m travelling at velocity v has an related wave λ mv = p = h/λ λ = h/ mv 1. The particle property is caused by their mass.. The wave property is related with particles' electrical charges. 3. Particle-wave duality is the combination of classical mechanics and electromagnetic field theory.

14 Electron Diffraction Electron Diffraction Crystalline Material Expected Source Experimental Source Amorphous Material Conclusion: Under certain circunstances an electron behaves also as a Wave!

15 The Waves and the Incertainty Principle of Heisenberger The measurement of particle position leads to loss of knowledge about particle momentum and visceversa. y v y p θ m p sin (θ) = py=pλ/ y sin (θ) = ±λ/ y py. y pλ = h x The momentum of the incoming beam is all in the x direction. But as a result of diffraction at the slit, the diffracted beam has momentum p with components on both x and y directions.

16 Schrödinger's cat It is a Gedanken (thought experiment) often described as a paradox I don t like it and I regret that I got involved in it. Superposition of two States: Broadly stated, a quantum superposition is the combination of all the possible states of a system. ψalife+ ψdead Schrödinger Miau! ψalife ψdead

17 Schrödinger wrote: One can even set up quite ridiculous cases. A cat is penned up in a steel chamber, along with the following device (which must be secured against direct interference by the cat): in a Geiger counter there is a tiny bit of radioactive substance, so small, that perhaps in the course of the hour one of the atoms decays, but also, with equal probability, perhaps none; if it happens, the counter tube discharges and through a relay releases a hammer which shatters a small flask of hydrocyanic acid. If one has left this entire system to itself for an hour, one would say that the cat still lives if meanwhile no atom has decayed. The Ψ function of the entire system would express this by having in it the living and dead cat (pardon the expression) mixed or smeared out in equal parts. It is typical of these cases that an indeterminacy originally restricted to the atomic domain becomes transformed into macroscopic indeterminacy, which can then be resolved by direct observation. That prevents us from so naively accepting as valid a "blurred model" for representing reality. In itself it would not embody anything unclear or contradictory. There is a difference between a shaky or outof-focus photograph and a snapshot of clouds and fog banks.

18 Schrödinger's famous tought experiment poses the question: when does a quantum system stop existing as a mixture of states and become one or the other? (More technically, when does the actual quantum state stop being a linear combination of states, each of which resemble different classical states, and instead begin to have a unique classical description?) If the cat survives, it remembers only being alive. But explanations of experiments that are consistent with standard microscopic quantum mechanics require that macroscopic objects, such as cats and notebooks, do not always have unique classical descriptions. The purpose of the thought experiment is to illustrate this apparent paradox: our intuition says that no observer can be in a mixture of states, yet it seems cats, for example, can be such a mixture. Are cats required to be observers, or does their existence in a single well-defined classical state require another external observer? An interpretation of quantum mechanics. A key feature of quantum mechanics is that the state of every particle is described by a wavefunction, which is a mathematical representation used to calculate the probability for it to be found in a location, or state of motion. In effect, the act of measurement causes the calculated set of probabilities to "collapse" to the value defined by the measurement. This feature of the mathematical representations is known as wave function collapse.