Consider the linear homogneous differential equation (DE) of second order D cap
ID: 1469258 • Letter: C
Question
Consider the linear homogneous differential equation (DE) of second order D cap psi i(x) sigma i with the operator D cap = - d^2/dx^2 The allowed values of x are from -1 to + 1. The boundary conditions are that psi i(x)must vanish x = -1 and x = +1. Further, sigma i is a real or complex number (eigenvalue). What kind of quantum system is related to this DE What is the general expression for the eigenvalues Provide the lowest 3 eigenvalues to three significant figures of precision. Provide expressions in terms of sine and cosine functions for the first three exact eigenfunctions, or dered from lowest to highest eigenvalue. Classify the functions by their symmetry behavior with respect to reflection at the vertical coordinate axis. Use the symbols g for even symmetry, i.e. psi i(x) = psi i(-x) and the symbol u for odd symmetry, i.e. psi i(x) = -psi(-x). For an integration interval [-a, +a] (a = 1 in our example), what is the result of integrating over a function with u symmetry What symmetry, g or u, do you get for the product function when you multiply two functions: (i) g times g,(ii) u times u, (iii) g times u or u times g Use the following functions as a basis set: x mu = ( x^2 - 1) x^mu , mu = 0, 1, 2. These basis functions satisfy the boundary conditions. Classify the basis functions by their symmetry, g or u. Calculate the 'overlap matrix'S for the basis functions. which has elements S_mu v = ( x mu/x v). Is this an orthonormal basis Are the basis functions mutually orthogonal (some/all) Identify the role of the symmetry indices in the orthogonality or lack thereof. Do not calculate.S01 and S12 by integration. Instead, use the results from part (d) to show that these elements must be zero. Further, save yourself some l time and show why S02 = S20 instead of calculating both values explicitly. (some correct answers are: S22 = 16/315, S02 = 16/105) Calculate the three functions D cap x mu and classify by symmetry. Calculate the matrix representation D of the operator D cap in the basis, i.e. calculate the matrix elements D mu v = (x mu/D cap x v). Do not calculate D01 and D12. Instead, use the results from part (d) to show that these elements must be zero. What about D02 versus D20 Remember that the operator D cap is self-adjoint (some correct answers are: D00 =8/3;d11 = 8/5; D22 = 88/105) Re-order the basis functions, and therefore the entries in the matrices S and D, such that the basis functions are grouped by symmetry. Calculate the secular determinant, and then the three eigenvalues from setting the determinant to zero, in order to solve the generalized eigenvalue problem Dci = Sci sigma j. Here, ci is a column vector with three coefficientsc mu i that approximate the exact solutions as psi i = sigma mu x mu c mu i for each eigenvalue. You don't need to calculate the vectors ci here, but feel free to do so as an ungrded optional homework. If you do so, I suggest that you also plot the and the approximate eigenfunctions and see how they compare. Compare the eigenvalues (3 .significant figures) that you obtained with the basis set approximation with the exact values from (b). Discuss the relative and absolute errors in the approximate solutions.Explanation / Answer
(a) A system with zero potential energy. Correspond to an infinite square well.
(b)We know the solution for schrodinger quation for an infinite square well {(h2/2*4*pi2*m)DX= EnX
energy(En)=n2h2/8ma2 . But the equation given DX= ei*X
So, ei*(h2/2*4*pi2*m)= En;
So, ei=n2*pi2/a2 where a=1-(-1)=2 {width of the well)
lowest 3 eigen values correspond to n=1,2 and 3
So, lowest eigen values=2.47, 9.87 , 22.21