1Foundations of Quantum Mechanics Wave-Particle Duality Diffraction of Waves Single-slit diffraction Light is a wave The wave relationship between frequency and wavelength is true for light. c=νλ 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 diffraction When 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. Direction of propagation screenconstructive interference destructive interference2Comments about diffraction Maximum intensity is directly behind barrier between two slits. Intensity pattern comes from interference of two wavefronts. Diffraction confirms that light is a wave. Photoelectric Effect Light 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. Eh=ν - 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. Conclusion Light 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?3DeBroglie Waves Einstein (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=λ Louis DeBroglie, in 1924, proposed that all matter has wave-like properties. (He did this in 2 pages article!) Nobel - 1929 Thus all moving particle have a wavelength (deBroglie wavelength) hpλ= 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 Principle In 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. ()()xp2∆∆≥ 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  (say h-bar) is the reduced Planck’s constanth2≡π **A consequence of the uncertainty principle is that measuring the state of a microscopic system changes the system.4Consider measurement in the macroscopic world To 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. position of particle at time 1position of particle at time 2apparent path of particleposition of particle at time 1position of particle at time 2‘actual’ path of particle - 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. ()()Et2∆∆≥ The 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/Wavefunctions A 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, ()rrx,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,tψ=ψ - 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 - 19335Born’s Interpretation of the Wavefunction ψ has no physical meaning. Only (ψ* ψ) has meaning as a probability density. - ψ* is complex conjugate of ψ. - 2*ψψ=ψ Probability density has physical meaning when associated with a region in space. - infinitesimal regions such as dx, dA, dτ, etc… Probability that electron is between x1 and x2 is21xxprob. * dx=ψψ∫ The probability that the electron is somewhere must be one. Thus prob. * dx 1∞−∞=ψψ =∫ - This requirement implies that the wavefunction must be normalized. Operators and Expectation Values Information is extracted from a wavefunction with an operator. Every observable (measurable quantity) has an operator. Two fundamental operators Position ˆxx= Momentum xˆpix∂=∂ - caret, ^, is used to denote operator. The information available in a wavefunction is extracted by using the operator to calculate an