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Foundations for quantum mechanics Atomic Spectra A little history Roentgen: discovered x-rays in 1895 More history • X-rays were discovered to ionize air • Until then, only ions observed were in solution (e.g., Na+, Cl-) • Implications: Neutral atoms must possess small electric charges as constituents • Atom is a complex structure • Electrical charges enter into its make-up Thomson: identified electron in 1897 • Early experimenter with mass spectrometer • Determined mass-to-charge ratio of electron • Proposed ‘plum-pudding’ model of atom, in which electrons move throughout a solid atom with evenly-distributed positive charge. A little more history Rutherford conducts gold foil experiment in 1911 • Directed high-energy, positively-charged alpha particles (helium ions from radon) at gold foil surrounded by cylindrical detector • Thomson’s model = no massive concentration of positive charge = nothing to significantly deflect ions • Results: • Most ions passed through as if foil were empty space • Few deflected at very large angles • Conclusion: relatively small nucleus with positive charge around which electrons orbit But… Electrons are negatively-charged and relatively far away from positivelycharged nucleus. If electrons were stationary, they would quickly be attracted to the nucleus. Easy explanation: They must be orbiting! But… Since… • Electrons orbit stable nuclei. • Moving charges emit energy in the form of electromagnetic waves. • Energy is conserved. Therefore… • As electrons orbit the nucleus, their energy must decrease. • As their energy decreases, they spiral into the nucleus. • As they spirals inward, their speed increases and so would frequency of light emitted. • The process repeats. • The universe does not last very long. Oops… If our assumptions are true, we would expect to see light emitted from atoms shift from red (low-energy) to orange to yellow to green to blue (high-energy) relatively quickly. (and also, we would expect not to exist as the electrical potential in atoms would have evened out long ago, making life as we know it impossible.) Instead, we see this (Also, we exist.) Spectral lines We find that each element (as a gas) emits a unique set of spectral lines when heated or subjected to high voltage. Why these lines and only these lines? (this uniqueness extends into the infrared and ultraviolet ranges as well) Why don’t electrons quickly lose their energy? Energy is proportional to frequency Photoelectric effect demands that light come in discrete quantities. Energy is proportional to frequency 𝐸 = ℎ𝑓 Absorption lines demand that • matter absorbs and emits energy in ONLY discrete quantities • electrons do not behave the way macroscopic particles do Quantization of angular momentum Neils Bohr offered a math trick in 1912 Angular momentum of electrons is quantized So… Photon of light is emitted when electron drops from ‘excited’ energy state to less energetic state, stopping at ‘ground state’ And… Frequency determine by difference in energy levels ℎ𝑓 = 𝐸𝑢𝑝𝑝𝑒𝑟 − 𝐸𝑙𝑜𝑤𝑒𝑟 Implications Reasons we trust the model Does a nice job explaining spectral lines for hydrogen Reasons we question the model: Math trick without a very good explanation: • Angular momentum conserved?? Why? • Ground state? What? • Moving charged particles don’t emit light? Sorry, Maxwell. Wave nature of matter Louis de Broglie proposed particles (e.g., electrons) also have wave nature (1923) • Bohr’s quantum numbers correspond to number of full ‘electron’ wavelengths • ONLY resonant ‘standing waves’ can persist. 𝑚𝑎𝑡𝑡𝑒𝑟 Where ℎ ℎ = = 𝑝 𝑚𝑣 ℎ = Planck’s constant 𝑝 = momentum = mass x velocity (at non-relativistic speeds) Wave nature of matter Louis de Broglie proposed particles (e.g., electrons) also have wave nature (1923) • Bohr’s quantum numbers correspond to number of full ‘electron’ wavelengths • ONLY resonant ‘standing waves’ can persist. Falsifiable predictions Electrons diffract? • Yes: Demonstrated 1927 • Davison – Germer Electrons interfere with each other in double-slit experiment? • Yes: Demonstrated in 1961