The Bohr Model of Hydrogen, a Summary, Review
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1 The Bohr Model of Hydrogen, a Summary, Review Allowed electron orbital radii and speeds: Allowed electron energy levels:
2 Problems with the Bohr Model Bohr s model for the atom was a huge success in that it could explain the spectrum of Hydrogen to very high precision. However, it left many important questions unanswered. For example: Multiple Electron Atoms: When applied to atoms with more than one electron, the model failed to reproduce spectra. Arnold Sommerfeld developed theories that included elliptical orbits, but these didn t work. (Bohr s model does work for any one electron ion, like singly ionized Helium.) The Intensities of Spectral Lines: Why are some emission lines brighter than others? This means that some electron transitions are more probable than others. Bohr says nothing about this. Also, some of these lines are actually closely spaced multiple lines fine structure. Postulates: Ultimately, the Bohr Model gets some things right and gives a useful mental picture of the atom, but as for a model of the atom, it is wrong. However, the postulate question leads us in the right path.
3 de Broglie and his Waves Recall what we have for a photon: (from special relativity*) In 1923, Louis de Broglie (in his PhD thesis!) proposed that the above equations hold for all objects even particles have wave properties! What motivated de Broglie s thinking? No experiment had been done indicating that particles like electrons behave like waves. He was more inspired by symmetry in nature: If photons have particle properties, shouldn t particles, like electrons, have wave properties? So, for every particle, there is an associated wavelength: *Don t use the equation p = E/c for particles. The E in this equation must be the total relativistic energy which is hf for a photon, but is not the kinetic energy for particles. For particles, use wavelength = h/p.
4 Whiteboard QM-7: de Broglie Wavelength Part 1: What kind of numbers are we talking about? Consider a bullet: Calculate the de Broglie wavelength of the bullet in meters. (LC) If we wanted to do an experiment to demonstrate the wave nature (e.g. diffraction) of this bullet, what would we have to do? We should pass the bullet through a hole about the same size as the wavelength! Part 2: The spacing between atoms in a crystal solid is on the order of 1 nm. What kinetic energy (in ev) should an electron have so that its wave properties can be observed when passing through or reflecting from this crystal solid? (LC) Solution: It is very easy to produce electrons in the laboratory with kinetic energies of a few ev. The wave nature of electrons was demonstrated experimentally (by accident) by Davisson and Germer in 1927.
5 How did de Broglie know that he was on to something really big? De Broglie realized this connection between his wavelength and the Bohr theory of hydrogen: The allowed electron orbits in the hydrogen atom are those orbits for which the circumference contain an integral number of de Broglie wavelengths, i.e. circular standing waves. For any Bohr orbit (any n), de Broglie says: (Bohr s angular momentum postulate!) PhET H-atom But, is an electron a particle or a wave? And, if it has wave properties, what is doing the waving? Cass_qm_debr
6 Whiteboard Problem: QM-8 The diameter of the atomic nucleus is about 10 fm. A simple model of the nucleus is that protons and neutrons are confined in a one dimensional box of length 10 fm. Consider a proton in such a box: a) What are the first three energy levels of the proton (in MeV)? Hint: Find the three longest wavelengths that can form a standing wave in the box of length, L = 10 fm. Then use these wavelengths to find the kinetic energies. b) What is the wavelength (in nm) of a photon that is emitted by a nucleus when the proton makes a transition for the n = 2 to the n =1 state? (LC) What type of electromagnetic radiation is this?
7 Quantum Mechanics Everything that we covered up to this point (especially the Bohr atom) falls in the category of Old Quantum Mechanics. We re now ready to start to use what the old quantum mechanics reveals to develop what is known today as Quantum Mechanics. We start with de Broglie s discovery that particles have wave properties, and ask: If a particle has wave properties, What is doing the waving? An Assertion (formally it s a postulate of quantum mechanics): Where did such an idea come from?
8 Quantum Mechanics What should a wave function for a free particle in 1D look like? Classically, a particle of mass m has a position, x, and momentum, p: In QM, what if we use a travelling sinusoidal wave to represent the particle in 1D? Now for the particle: But, where is the particle located? (a well-defined momentum) The particle is equally likely to be anywhere on the x-axis! For a sinusoidal wave: our knowledge of p is perfect: But, our knowledge of x is totally imperfect:
9 Quantum Mechanics - HUP To represent a free particle, we would like a wave that looks like this: small amplitude where the particle isn t. large amplitude where the particle is located. To get a wave packet like this, you have to add together many many waves of different wavelengths, and So, to get more knowledge of the particle s position, we must lose knowledge of it s momentum. (position of particle is somewhere in here) Heisenberg Uncertainty Principle (HUP): In 1927, Werner Heisenberg quantified this relation between how well you can simultaneously know the position and momentum of a particle: Important point: HUP is not a statement about instrument imperfection and how well we can measure things in the lab. It is an inherent property of the particle due to its wave characteristics. Cass_qm_heis
10 Whiteboard QM-9: HUP Can a particle confined in a 1D box of width L have zero energy? Use the HUP to find an expression for the minimum kinetic energy of the particle. (LC) (does this look familiar?) More: Think of the Hydrogen atom; the electron is confined in a region of space, can it have zero energy?
11 Extra QM Credit Quiz: What if h was Bigger? In a minute, we re going to have you take a basic physics evaluation test, and we want to give you some reward for it. So go to MP and begin the QM extra credit quiz. HUP shows that if a particle of mass m is confined (in 1D) to a region of space L that the particle s minimum kinetic energy is For a macroscopic object, this a very tiny energy. But, what if h was bigger, say h = 1000 J. s? Suppose a 50 kg person is confined to a region of space with width 1.0 m. Using this value of h, calculate the minimum speed of the person. So, if h was this big, we would want to avoid narrow areas!
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