ConclusionHeisenberg’s Uncertainty PrincipleQuantum States/WavefunctionsBorn’s Interpretation of the WavefunctionOperators and Expectation ValuesEigenfunctions and operatorsCommutators and commuting variablesCommutator notationRequirements of the WavefunctionHilbert SpaceBasis SetsOrthonormality of WavefunctionsFoundations of Quantum MechanicsWave-Particle DualityDiffraction of WavesSingle-slit diffractionLight is a waveThe wave relationship between frequency and wavelength is true for light. c  More evidence that light is a wave exists since it can be reflected, refracted and diffracted.When a traveling wave hits a hole (slit) that is approximately the size of the wave’s wavelength, the wave “expands” as it goes through the hole.Double-slit diffractionWhen a wave is incident on two slits relatively close to each other, a diffraction pattern appears when the waves constructively and destructively interfere with each other.1Direction of propagationscreenconstructive interferencedestructive interferenceComments about diffractionMaximum intensity is directly behind barrier between two slits.Intensity pattern comes from interference of two wavefronts.Diffraction confirms that light is a wave.Photoelectric EffectLight shining on a metal surface may cause electrons to be ejected from the surface.e-Frequency of light needs to be above threshold frequency to induce photoelectric emission.Kinetic energy of electrons is proportional to frequency of incident radiation.Kinetic energy of electrons is independent of light intensity.- I.e., microwave laser will not induce photoelectron emission.This independence contradicts wave nature of light.- According to wave nature, energy of electrons should be proportional to the intensity.**Albert Einstein proposes in 1905 that light has particle properties.** Nobel - 1921- Each quanta of light (photon) has energy proportional to frequency.E h - h is Planck’s constant- h = 6.626  10-34 Js- Planck’s is a universal, fundamental constant of nature.If light is a particle as Einstein suggests, then the photoelectric effect is easily explained in terms of collisions between photons and electrons.ConclusionLight has been shown to be a particle and a wave.Waves by their nature are “spread out”Particles by their nature are “localized”, i.e., in one place.How can light be spread out and in one place at the same time? WHO KNOWS?2DeBroglie WavesEinstein (also in 1905!), in his special theory of relativity, demonstrated that the photon has momentum like a particle where the momentum is inversely proportional to the wavelength.hp Louis DeBroglie, in 1924, proposed that all matter has wave-like properties. (He did this in 2pages article!) Nobel - 1929Thus all moving particle have a wavelength (deBroglie wavelength)hp Clinton Davisson and Lester Germer in 1927 confirm deBroglie waves by discovering that electrons of the correct energy will diffract from the surface of Ni metal. Nobel - 1937- just like photons diffract off a grating.George P. Thomson in 1927 also confirms deBroglie waves by passing electron beam through thin sheet of Pt and finding diffraction rings. Nobel – 1937- G. P. Thomson’s father, J. J. Thomson won Nobel (1906) for discovering electron as a particle.Heisenberg’s Uncertainty PrincipleIn 1925, based on a relationship from classical wave theory and deBroglie’s hypothesis; Heisenberg formulated his Uncertainty Principle.Heisenberg’s uncertainty principle states that one cannot measure a particle’s position and momentum simultaneously with infinite precision.   x p2  h where x is the uncertainty in the measurement of the particle’s position, p is the uncertainty in the measurement of the particle’s momentum and h (say h-bar) is the reduced Planck’s constanth2h**A consequence of the uncertainty principle is that measuring the state of a microscopic system changes the system.3Consider measurement in the macroscopic worldTo measure position of a cart at a specific time, we need to measure with our eye or a photograph.- Either instrument uses photons reflected off the cart to image the cart.To measure momentum, we need two photographs separated by time.Consider measurement in the microscopic world.Since photons have momentum, photons reflected off electron will the momentum of the electron.Measuring the momentum using two photographs is impossible.p o s i t i o n o f p a r t i c l e a t t i m e 1p o s i t i o n o f p a r t i c l e a t t i m e 2a p p a r e n t p a t h o f p a r t i c l ep o s i t i o n o f p a r t i c l e a t t i m e 1p o s i t i o n o f p a r t i c l e a t t i m e 2‘ a c t u a l ’ p a t h o f p a r t i c l e- Using photons, we have no way knowing the actual path between position 1 and position 2.Uncertainty principle also relates uncertainty of time and energy.   E t2  hThe uncertainty principle is a consequence of wave-particle duality. It cannot be ‘fixed’ with more sophisticated technology. It is fundamental limitation on measurements.Quantum States/WavefunctionsA proper description of a state allows all the known properties of a system to be found.For a macroscopic system, the state is described by a trajectory,  r r x, t.- A particle’s momentum, energy, etc… can be found from the trajectory.For a thermodynamic system, the state is described by the equation of state.For a microscopic system, the state is described by a wavefunction,  x, t - Unlike trajectory, wavefunction has no independent reality- Wavefunction is with operators to find particle properties, but the value of the wavefunction itself has no physical meaning.Erwin Schrödinger in 1926 developed the idea of a wavefunction. – more very soon. - Nobel - 19334Born’s Interpretation of the Wavefunction has no physical meaning. Only (* ) has meaning as a probability density.- * is complex conjugate of .- 2*   Probability density has physical meaning when associated with a region in space.- infinitesimal regions such as dx, dA, d, etc…Probability that electron is between x1 and x2 is21xxprob. * dx  The probability that the electron is somewhere must be one. Thus prob. * dx 1    - This requirement implies that the wavefunction must be normalized.Operators and Expectation ValuesInformation