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Guest Lecture Stephen Hill University of Florida – Department of Physics Cyclotron motion and the Quantum Harmonic Oscillator •Reminder about HO and cyclotron motion •Schrodinger equation •Wave functions and quantized energies •Landau quantization •Some consequences of Landau quantization in metals Reading: My web page: http://www.phys.ufl.edu/~hill/ The harmonic oscillator E K U 1 m 2 x 2 2 1 2 1 E mv m 2 x 2 2 2 V(x) when v 0, E E x 1 A 2 2 m k /m A 0 +A x Classical turning points at x = ±A, when kinetic energy = 0, i.e. v = 0 Cyclotron motion: classical results y Lorentz force: R dp F q vB dt F x e, m 2 B out of page mv mv evB R R eB v eB c R m What does this have to do with today’s lecture? Cyclotron motion.... Lorentz force: y dp F q vB dt x e, m Restrict the problem to 2D: B Bkˆ v v xiˆ v y ˆj B out of page p p xiˆ p y ˆj It looks just like the Harmonic Oscillator px2 e2 B 2 2 1 2 1 E x 2 mv x 2 mc2 x 2 2m 2m d 2 2 1 2m dx 2 2 mc x E 2 2 eB c m The harmonic oscillator '' d 2 1 1 2 22 2 '' 2 2 m2 mcxc xE E 22 mm dx 22 '' '' V(x) E Constraints: 0as x ; dx 1 2 A 0 +A x Classical turning points at x = ±A, when kinetic energy = 0, i.e. v = 0 The quantum harmonic oscillator solutions Due to symmetry, one expects: ( x) ( x) 2 2 Thus, the solutions must be either symmetric, (x) = x), or antisymmetric, (x) = x). Orthogonality m n ,m * n http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hosc.html The quantum harmonic oscillator solutions Due to symmetry, one expects: ( x) ( x) 2 2 Thus, the solutions must be either symmetric, (x) = x), or antisymmetric, (x) = x). Further discussion regarding the symmetry of can be found in the Exploring section on page 268 of Tippler and Llewellyn. http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hosc.html The correspondence principle The harmonic oscillator wave functions ( x) 1 2 2 m x ( x ) E ( x ) 2 2m x 2 2 2 Solutions will have a form such that '' (Ax2 + B). A function that works is the Gaussian: ( x ) Ae x 2 1 x 2 1 2 2 ; ''( x ) 4 A ( x )e 4 ( x ) 2 2 2 2 For higher order solutions, things get a little more complicated. n ( x) Cne m x 2 / 2 H n ( x) where Hn(x) is a polynomial of order n called a Hermite polynomial. The first three wave functions 0 ( x ) A0e 1 ( x ) A1 m x 2 / 2 m xe En n 12 c m x 2 / 2 n 1, 2,3.... 2m x 2 m x2 / 2 2 ( x ) A2 1 e A property of these wavefunctions is that: mdx m,n ; also * n n* x mdx 0 unless n m 1 This leads to a selection rule for electric dipole radiation emitted or absorbed by a harmonic oscillator. The selection rule is Dn = ±1. Thus, a harmonic oscillator only ever emits or absorbs radiation at the classical oscillator frequency c = eB/m. The quantum harmonic oscillator Landau levels (after Lev Landau) e 28 GHz/T 2 me eB 2 f c m c 7 2 c 5 2 c c 9 2 c 3 2 1 2 c What happens if we have lots of electrons? Landau levels Empty EF Filled # states per LL = 2eB/h What happens if we have lots of electrons? Landau levels c Cyclotron resonance Empty EF Filled # states per LL = 2eB/h Electrons in an ‘effectively’ 2D metal Width of resonance a measure of scattering time (lifetime/uncertainty) cyclotron resonance BCR 1 | cos | f = 62 GHz T = 1.5 K But these are crystals – electrons experience lattice potential 2 d2 1 2 2 2m dx 2 2 mc x + Vcrystal ( x ) E 2 c 3 c c 5 2 3 2 1 2 c • n no longer strictly a good quantum number 9 2 c • no longer form 7 an orthogonal 2 c basis set c c Anharmonic oscillator •See harmonic resonances •wc depends on E Electrons in an ‘effectively’ 2D metal Look more carefully – harmonic resonances: measure of the lattice potential Electrons in an ‘effectively’ 2D metal Even stronger anharmonic effects f = 54 GHz T = 1.5 K Harmonic cyclotron frequencies Heavy masses: m = 9me eB 2 f m What if we vary the magnetic field? Landau levels Empty EF Filled # states per LL = 2eB/h # LLs below EF = mEF/eB What if we vary the magnetic field? Landau levels Empty EF Filled # states per LL = 2eB/h # LLs below EF = mEF/eB What if we vary the magnetic field? Landau levels kBT ~ meV eV Empty EF Filled Properties oscillate as LLs pop through EF What if we vary the magnetic field? Period 1/B Properties oscillate as LLs pop through EF Microwave surface impedance an for organic conductor Conductivity (arb. units - offset) Shubnikov-de Haas effect 52 GHz 4.2 K 2K 1.34 K 10 15 20 25 Magnetic field (tesla) 30 Magnetoresistance for an for organic superconductor Shubnikov-de Haas effect Guest Lecture Stephen Hill University of Florida – Department of Physics Cyclotron motion and the Quantum Harmonic Oscillator •Reminder about HO and cyclotron motion •Schrodinger equation •Wave functions and quantized energies •Landau quantization •Some consequences of Landau quantization in metals Reading: My web page: http://www.phys.ufl.edu/~hill/