Chapter2_12 - . Solutions of this form correspond to states...

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PHY4604 R. D. Field Department of Physics Chapter2_12.doc University of Florida The Harmonic Oscillator (1) One Dimensional Simple Harmonic Oscillator: The simple harmonic oscillator has a linear restoring force ( i.e. Hooke’s Law spring) and Kx dx x dV F x = = ) ( and 2 2 1 ) ( Kx x V = where K is the spring constant. Classically the system oscillates with an angular frequency of m K / = ω and a linear frequency m K f π 2 1 2 = = . The energy of the system is the sum of the kinetic energy and the potential energy and classically can have any value including zero. Planck’s Postulate (Ad Hoc): Planck postulated that the energies of the harmonic oscillator were quantized and E n = nhf , where h is Planck’s constant and n = 0, 1, 2,. . . Time-Dependent Schrödinger Equation: The time dependent Schrodinger with V(x) = Kx 2 /2 is t t x i t x Kx x t x m Ψ = Ψ + Ψ ) , ( ) , ( 2 1 ) , ( 2 2 2 2 2 h h . Time-Independent Schrödinger Equation: Look for solutions of the form h / ) ( ) , ( iEt e x t x = Ψ ψ
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Unformatted text preview: . Solutions of this form correspond to states with definite energy since H op | > = E| > . Substituting (x,t) into the time dependent equation yields ) ( ) ( 2 1 ) ( 2 2 2 2 2 x E x Kx dx x d m = + h and hence ) ( 2 2 ) ( 2 2 2 2 2 = + x x mf mE dx x d h h where I set K = m(2 f) 2 . Setting h / 2 mf = and 2 / 2 h mE = yields ) ( ) ( ) ( 2 2 2 2 = + x x dx x d . We must find the allowed solutions ( i.e. (x) must be square-integrable) of this differential equation. The differential equation can be converted into the Hermite Differential Equation and the solutions are Hermite Polynomials . We will not solve for the energy levels in this way, instead we will solve the problem using operators!...
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