= is an un-normalized trial function (see the blue curve), perform

Consider a particle exposed t o t h e ramp potential V = b x for x > 0 and V = +∞ for x ≤ 0 (see the picture for the straight-line). The Hamiltonian can be expressed as: 𝐠= − ℏ% & ( )% )*% + 𝐵. If 2 2 y ( ) 1 a x x c x e – = is an un-normalized trial function (see the blue curve), perform the following tasks: 1) Normalize the wavefunction; 2 ) Using α as a variational parameter estimate the ground-state energy 3) By setting b = 0.1, m = 1 and = 1, plot vs α and show that your variational energy converges towards the true energy from above. Note, you will need to limit the range of your integration to (0, +∞) in computing integrals. 4) Compute the uncertainty product ∆x×∆p. 5) A linear combination of properly chosen trial functions can improve the quality of approximation. Use 2 2 ( ) 2( 2) a x x c x kx e – F = + as the trial wavefunction, and determine the optimal value of k and the corresponding normalization coefficient, c2 . [Note: use the a value that you obtained from your answer to question 3]. Hints: a). In Mathematica, you can impose the condition a > 0 by: Refine[expr, Assumptions-> α >0], or Assuming[a > 0, expr] b). In Mathematica, “SetAttributes[X, Constant]” can be used to set X as a constant. c). In quantum mechanics, Δx and Δp can be calculated as: Δx = (x − x )2 =% 312 𝐨𝐩(𝐠− 𝐠)&𝐨𝐩 𝐵 Δp = ( p − p )2 = % 312 𝐨𝐩 𝐠− 𝐠&𝐨𝐩 𝐵 where < > means the expectation value, and the momentum operator can be expressed as: dx d p = -i!  < E >

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