En 2a 3. A particle of mass m moves in a one-dimensional infinite potential well defined by V (2) = 0 when ||

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En 2a 3 A Particle Of Mass M Moves In A One Dimensional Infinite Potential Well Defined By V 2 0 When A V X 1 (59.74 KiB) Viewed 42 times

En 2a 3. A particle of mass m moves in a one-dimensional infinite potential well defined by V (2) = 0 when || <a, V(x) = 2 otherwise. The energy eigenvalues are: narh2 Cn?, with n=1,2,3,.... 8ma? The corresponding orthonormal eigenfunctions are the even and odd functions La cos(nbx) for n=1,3,5... ta sin(nbx) for n = 2,4,6... with b = The potential between -a < x <a is modified to: V (x) = AC sin(3bx) (a) Using Perturbation Theory: i. Determine the first-order order correction to the energy of the ground state. ii. Determine the second-order correction of the ground state energy. ii. For what values I would you expect perturbation theory to be useful? (b) Use the variational principle to obtain an energy for the ground state using the trial function 1 1 Or (20) cos(bx) + 9 + sin(25x) = 4, (x) + qu2 (x) va va where q is a variational parameter. For the case where I = } find the best value for q and hence the optimal energy. Suggest an improved trial function which might further lower the energy. [3] [6] [2] (5) [2] [2]