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

### = 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 = +âˆž for x â‰¤ 0 (see the picture for the straight-line). The Hamiltonian can be expressed as: ð = âˆ’ â„% & ( )% )*% + ðµ. 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 Î± as a variational parameter estimate the ground-state energy 3) By setting b = 0.1, m = 1 and = 1, plot vs Î± 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, +âˆž) in computing integrals. 4) Compute the uncertainty product âˆ†xÃ—âˆ†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-> Î± >0], or Assuming[a > 0, expr] b). In Mathematica, â€œSetAttributes[X, Constant]â€ can be used to set X as a constant. c). In quantum mechanics, Î”x and Î”p can be calculated as: Î”x = (x âˆ’ x )2 =% 312 ð¨ð©(ð âˆ’ ð )&ð¨ð© ðµ Î”p = ( p âˆ’ p )2 = % 312 ð¨ð© ð âˆ’ ð &ð¨ð© ðµ where < > means the expectation value, and the momentum operator can be expressed as: dx d p = -i! ï¨ < E >

### Disclaimer

### Quick Links

### Quick Links

### Stay Connected

© 2019 AssignmentFreelancers. All Rights Reserved.