Question

First four harmonic oscillator normalized wavefunctions 1/4 Y.-(4)"-** 4, = 1/4 v2y ev2 1/4 Y, =| -1)ev¾2 1/4 - 3y)e¬v³½ y =ax 1. Consider a harmonic oscillator with a = 1. a) Prove that these eigenstates are all orthonormal b) Plot the first four eigenstates. How would doubling the mass change the eigenfunctions? c) Pick one eigenstate, and show that it is a solution to the Schrodinger Equation, that is, show that V? on (x) + w²ma? ¢n (x) = En&n (x) with E, = (n +5) ħw. (x)with the corresponding eigenstate (e.g. cos ( ) ) for a particle of the same mass in an infinite well. d) Overlay each пла ) or sin ( Notice that the value for a must get bigger for increasing n to get a good "fit". (2n+1) e) Explain why serves a good estimate for the effective size of the well for each eigenstate of the harmonic oscillator. Hint: think about energy and classical analogy. (2n+1) Explain where an = V f) Plot P1o (x) = $i, (x) $10 (x)and overlay it with the classical probability function P10 (*) why Pio (x) isn't flat, and why the effective wavelength of ø10 (x)gets larger as x goes from 0 to a10. g) Is [H,p) Øn (x) = 0? Are the eigenstates of the p = -iħVoperator the same as the eigenstates of the Hamiltonian?

Answer 1

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- Pick to = Lava & 8% - Me oleme ho (6) +ym wz? 4. (8) * - y 4o(2) + x2 mw? x2 40(2) = ) Ya Hale*) +xmw.x24() = ) %*|*(-42. 2027)] +x_mw?s? () = ** (*)*** [******j vme? 46) = Set 1 - * **) exsmotane yutuce) + ymwne 2,2 1 Imus mw x x² + + hW + Y₂ mw? x2 l to (4) mw x 2 /hW 40 (*) = Now Yath W 40 12) = Eo 40(?) ( 9 = I mu? m ko mua En - P4, \\\) Ev=Y2hw

- ® For first case Ho (a) e cos (en mm) and case Ti (x) = sin(at han ) 30d Case 49(*) ^ - Cos 0 a ) 4th case 4g (2) = -sinfant ) If a must bigger then exact fit