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Quantum mechanics provides us with an understanding of atomic structure and atomic properties. Lasers are one of the most important applications of the quantummechanical properties of atoms and light. Chapter Goal: To understand the structure and properties of atoms. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Topics today: • The Electron’s Spin • Multielectron Atoms • The Periodic Table of the Elements • Excited States and Spectra • Lifetimes of Excited States • Stimulated Emission and Lasers Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. The electron has an inherent magnetic moment and inherent angular momentum called spin, designated vector-S. The z-component of this spin angular momentum is The quantity ms is called the spin quantum number. The ms = + ½ state, with Sz = + ½ h-bar, is called the spinup state and the ms = –½ state is called the spin-down state. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Example: For ℓ=2, m ℓ = -2, -1, 0, +1, +2 correspond to three different directions of orbital motion. A magnetic moment is associated with the electron current. The z-component of the magnetic moment appears to be quantized. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. • Like a compass needle, a magnetic moment interacts with an external magnetic field depending on its direction. • Low energy when aligned with field, high energy when anti-aligned • The total energy of a hydrogenic electron in an external magnetic field is approximately: This means that spectral lines will split in a magnetic field Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. 13.6 eV − µ • B 2 n 13.6 = − 2 eV − µz B n 13.6 = − 2 eV − m µB B n E =− Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. A neutral atom moving in uniform magnetic field feels no net force, only a torque resulting in spin precession. In an inhomogeneous magnetic field, the net force is not zero. It proportional to magnetic moment along the direction of magnetic field variation. An atomic beam passing through an inhomogeneous field is observed to separate into discrete beams, one per quantum angular momentum state. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. The currents and magnetism of electrons in atoms generally cancel out. The magnetism of an unbalanced electron should produce an odd number (2L+1 values of m) of beams. Observation of an even number of beams and oddities in spectral splitting motivated the idea of intrinsic electron angular momentum and magnetism. The electron has two magnetic states and two values of angular momentum. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. • Spin up Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Spin down Pauli discovered that two complex waves are required to describe a non-relativistic electron In a magnetic field, these are linked and describe “spin orientation/ polarization.” Paul Dirac generalized this. A relativistic electron is described by four coupled complex waves, two more because an electron can go backwards in time appearing as an antielectron/ positron. That’s eight waves total counting real and imaginary parts. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. • When analyzing a multielectron atom, each electron is treated independently of the other electrons. • This approach is called the independent particle approximation, or IPA. • This approximation allows the Schrödinger equation for the atom to be broken into Z separate equations, one for each electron. • A major consequence of the IPA is that each electron can be described by a wave function having the same four quantum numbers n, l, m, and ms used to describe the single electron of hydrogen. • A major difference, however, is that the energy of an electron in a multielectron atom depends on both n and l. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Notice for example 2p states are higher that 2s. They are further out in radius so see a nuclear charge partially screened by 1s and 2s electrons. Note that the scale and details will depend on Z! Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. In 1925, Pauli hypothesized that no two electrons in a quantum system can be in the same quantum state. In other words, no two electrons can have exactly the same set of quantum numbers n, l, n and ms. If one electron is present in a state, it excludes all others. This statement, which is called the Pauli exclusion principle, turns out to be an extremely profound statement about the nature of matter. It applies to all spin ½ kinds of particles (fermions) , but not to integer spin particles (bosons) like photons. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Putting electrons on atom • Electrons are obey exclusion principle • Only one electron per quantum state unoccupied occupied Hydrogen: 1 electron n=1 states one quantum state occupied Helium: 2 electrons two quantum states occupied Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. n=1 states Other elements: Li has 3 electrons ⎛ n=2 ⎞ ⎜ ⎟ ⎜ =0 ⎟ ⎜ m = 0 ⎟ ⎜ 1⎟ ⎜ ms = + ⎟ ⎝ 2⎠ ⎛ n=2 ⎞ ⎜ ⎟ ⎜ =0 ⎟ ⎜ m = 0 ⎟ ⎜ 1⎟ ⎜ ms = − ⎟ ⎝ 2⎠ ⎛ n=2 ⎞ ⎜ ⎟ ⎜ =1 ⎟ ⎜ m = 0 ⎟ ⎜ 1⎟ ⎜ ms = + ⎟ ⎝ 2⎠ ⎛ n=2 ⎞ ⎜ ⎟ ⎜ =1 ⎟ ⎜ m = 0 ⎟ ⎜ 1⎟ ⎜ ms = − ⎟ ⎝ 2⎠ ⎛ n=2 ⎞ ⎜ ⎟ ⎜ =1 ⎟ ⎜ m = 1 ⎟ ⎜ 1⎟ ⎜ ms = + ⎟ ⎝ 2⎠ ⎛ n=2 ⎞ ⎜ ⎟ ⎜ =1 ⎟ ⎜ m = 1 ⎟ ⎜ 1⎟ ⎜ ms = − ⎟ ⎝ 2⎠ ⎛ n=2 ⎞ ⎜ ⎟ ⎜ =1 ⎟ ⎜ m = −1 ⎟ ⎜ 1⎟ ⎜ ms = + ⎟ ⎝ 2⎠ ⎛ n=2 ⎞ ⎜ ⎟ ⎜ =1 ⎟ ⎜ m = −1 ⎟ ⎜ 1⎟ ⎜ ms = − ⎟ ⎝ 2⎠ n=2 states, 8 total, 1 occupied n=1 states, 2 total, 2 occupied one spin up, one spin down ⎛ n =1 ⎞ ⎛ n =1 ⎞ ⎜ ⎟ ⎜ ⎟ = 0 = 0 ⎜ ⎟ ⎜ ⎟ ⎜ m = 0 ⎟ ⎜ m = 0 ⎟ ⎜ Copyright ⎟ ⎜ ⎟ Pearson Education, Inc., publishing as Pearson Addison-Wesley. ⎝ ms = +1/2⎠ ⎝©m2008 s = −1/2⎠ Atom Configuration H 1s1 He 1s2 Li 1s22s1 Be 1s22s2 B 1s22s22p1 Ne etc 1s shell filled (n=1 shell filled noble gas) 2s shell filled 1s22s22p6 2p shell filled Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. (n=2 shell filled noble gas) • Elements are arranged in the periodic table so that atoms in the same column have ‘similar’ chemical properties. • Quantum mechanics explains this by similar ‘outer’ electron configurations. • If not for Pauli exclusion principle, all electrons would be in the 1s state! H 1s1 Li 2s1 Na 3s1 Be 2s2 Mg 3s2 B 2p1 Al 3p1 C 2p2 Si 3p2 Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. N 2p3 P 3p3 H 1s1 O F 2p4 2p5 S Cl 3p4 3p5 He 1s2 Ne 2p6 Ar 3p6 • Orbital electron currents and magnetism generally cancel in multi electron atoms. • Spin and its magnetism generally cancels. • Unpaired orbital currents explain diamagnetism • Unpaired spin magnetism underlies ferromagnetism • When the magnetism of a bulk of atoms aligns, the material is magnetized. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. An atom can jump from one stationary state, of energy E1, to a higher-energy state E2 by absorbing a photon of frequency In terms of the wavelength: Note that a transition from a state in which the valence electron has orbital quantum number l1 to another with orbital quantum number l2 is allowed only if Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. QUESTIONS: Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Consider an experiment in which N0 excited atoms are created at time t = 0. The number of excited atoms remaining at time t is described by the exponential function where Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. QUESTIONS: Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. An extension of simple Schrodinger theory, Quantum Electrodynamics (QED) is the theory describing charged matter fields/particles interacting with time dependent quantum electromagnetic fields. It describes the creation and destruction (emission and absorption) of light quanta with the simultaneous evolution of matter wave states. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. The output beam contains identical photons with special coherence properties. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. LASER: Light Amplification by Stimulated Emission of Radiation Atoms ‘prepared’ in metastable excited states …waiting for stimulated emission Called ‘population inversion’ (atoms normally in ground state) Excited states stimulated to emit photon from a spontaneous emission. Two photons out, these stimulate other atoms to emit. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Ruby Laser • Ruby crystal has the atoms which will emit photons • Flashtube provides energy to put atoms in excited state. • Spontaneous emission creates photon of correct frequency, amplified by stimulated emission of excited atoms. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Ruby laser operation Relaxation to metastable state (no photon emission) 3 eV 2 eV 1 eV Metastable state PUMP Transition by stimulated emission of photon Ground state Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.