Final-FS03 - Phys 208 Theoretical Physics Final Exam Dec....

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Phys 208 – Theoretical Physics – Final Exam – Dec. 19, 2003 1. (17 pts) a) Find and plot the roots of . 8 3 i b) Evaluate in Cartesian form, i.e., form. () i i xi y + c) Determine the points in the ( x , y ) plane satisfying the equation . || zi − + = 12 2. (17 pts) Given that , evaluate . ed x a ax = 2 1 2 0 π xe d x ax 2 0 2 3. (17 pts) The normalized ground and first excited stationary state wave functions of the one-dimensional harmonic oscillator are given by: and , where ψ 0 14 2 2 / / a ax = ( ) 1 2 2 2 / / xax e a ax = −∞ < < ∞ x and where a is a constant. These states have energies and . E 0 2 = h ω / E 1 32 = h / a) Show that in each of the stationary states, and . ⟨⟩= x 0 0 x 1 x b) Consider a time-dependent wave function defined as: . [] Ψ (,) // xt iE t iE t =+ −− 1 2 01 ψψ hh Determine . You do not need to substitute the explicit form for and . |( , ) | Ψ 2 0 x 1 x c) Determine the average position of the particle as a function of time, i.e. , . ⟨⟩ 4. (17 pts) a) Express as a linear combination of Legendre polynomials.
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Final-FS03 - Phys 208 Theoretical Physics Final Exam Dec....

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