Announcements. Lecture 20 Chapter. 7 QM in 3-dims & Hydrogen Atom. The Radial Part of Schrodinger Equation for Hydrogen Atom

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1 Announcements! HW7 : Chap.7 18, 20, 23, 32, 37, 38, 45, 47, 53, 57, 60! Physics Colloquium: Development in Electron Nuclear Dynamics Theory on 3:40pm! Quiz 2 (average: 9), Quiz 3: 4/19 *** Course Web Page ** Lecture Notes, HW Assignments, Schedule for thephysics Colloquium, etc.. Lecture 20 Chapter. 7 QM in 3-dims & Hydrogen Atom Outline:!The Schrödinger Eq. in 3-Dimensions! The 3D Infinite Well! Energy Quantization & Spectral Lines in Hydrogen! The Schrödinger Eq. for a Central Force! Angular Behavior in a Central Force! The Hydrogen Atom! Radial Probability! Hydrogen-like Atoms The Radial Part of Schrodinger Equation for Hydrogen Atom Energy quantization The principal quantum number n Hydrogen P.E. and l must obey Hydrogen Radial Equation Quantum number l is limited to (n-1)

2 Quantum numbers Now, we can discuss where hydrogen s e might be found Traditional naming scheme Spectroscopic notation accidental degeneracy Because of 1/r S: sharp, p: principle, d: diffuse, f: fundamental 3d state: n=3 & l = 2 2p state: n=2 & l = 1 Electron Prob. Densities in the Hydrogen atom, through n = 3 State are labeled using spectroscopic notation; n, l 2d state is possible? -- No!! Because, here, n=2, d =2; Remember n > d Spectral Lines Hydrogen energies & specral lines; A photon is emitted when the electron jumps downward.

3 Summary: Hydrogen Atom Outline: Chapter. 8 Spin & Atomic Physics! Evidence of Angular Momentum Quantization! Identical Particles! The Exclusion Principle! Multi-electron Atoms & the Periodic Table! Characteristic X-Rays It s open said that in Q.M. there re only 3 bound-state problems solvable (w/o numerical approximation tech.) 1.! Infinite well, 2. Harmonic oscillation, 3. hydrogen atom all 1-particle problem. Most real application: multiple system. so, start an atom with multiple electrons Bohr Radius Radial Distribution of the Electron Probability Density in a Hydrogen Atom How Small is Small? Ground State

4 Electron Probability densities in the hydrogen atom, through n = 3 Lecture Today How Small is Small? How Fast is Fast? Quantum numbers Lecture Today s p L = l( l + 1)! = 0 L = l( l + 1)! = 2! accidental degeneracy Because of 1/r

5 L GroundState Lecture l( l + 1) = 0Today s =! L s = l( l + 1)! = 0 = l( l + 1)! = 2! L p p L GroundState = l( l + 1)! = 0 Ground State: The Electron is NOT Orbiting around the proton Classical Physics: The Electron is Orbiting around the Proton Conventional current is opposite electron motion fundamental charge Orbiting in Classical Physics 2 right-hand rules: µ = IA L = r x p A charge with angular m/m has a magnetic dipole moment Orbiting in Classical Physics Potential energy of a dipole µ in a magnetic field B period of revolution Magnetic Dipole Moment L Magnetic Dipole Moment

6 Magnetic force on a system with dipole moment µ The Stern-Gerlach Experiment U F = negative gradient of potential energy An atom with a magnetic dipole moment passing through a non-uniform B- field F can be measured The Stern-Gerlach Experiment Classical Expectation The Quantum Stern-Gerlach Theory Expectation Experiment important factor governing the effect of B- field; so, magnetic quantum # Ground State -> l = 0 -> L = 0 -> F = 0

7 Quantum Theory Expectation Ground State -> l = 0 -> L = 0 -> F = 0 Surprise: Real Experimental Result Ground State -> l = 0 -> L = 0 -> F = 0 (???) Oops! The Solution: INTRINSIC MAGNETIC MOMENT and ANGULAR MOMENTUM called SPIN is carried by every electron Like for L: SPIN A given particle s intrinsic magnetic dipole moment is related to intrinsic angular m/m, S gyromagnetic ratio s the quantum number of SPIN Intrinsic property of a particle

8 For and electron: s = 1/2 Intrinsic angular momentum: Spin Orientation 2 possible spin states of an electron e.g. for proton s = 1/2, for W, s = 1, see Table 8.1 (p295) Like before, for the z-component of angular momentum L Stern-Gerlach Experiment? spin quantum number: allowed values are from s to +s in integral steps Now let us return to the Stern-Gerlach experiment. For l =0, the orbital magnetic moment µ L = 0, so it shouldn t be subject to a force, but there should still be a force on the intrinsic magnetic dipole moment. Sz 1 = ±! 2 for electrons spin quantum #: +- 1/2

9 The Stern-Gerlach Experiment 2 lines corresponding to m s = -1/2 & +1/2 S z 1 = ± 2! F = ±F 0 SPIN = 1/2