Question

Note: You may not use integral tables or programs like Maple or Mathematica to evaluate integrals in your solution to this problem set. 1. Griffiths, Problem 2.21. (24 points) Useful Information You may use the following result without proof

help with quantum. im really lost in the algebra

Answer 1

The wave function at time t = 0 is, psi (x, 0) = A e^- a x^v (a) Now the normalization condition is integral^infinity_- infinity psi^* (x) psi (x) dx = 1 or A^v integral^infinity_- infinity e^- integral a x^2 dx = 1 or, A^v/squareroot 2 a integral^infinity_- infinity e^- (squareroot 2 a x)^2 (squareroot 2 a dx = 1) or, A^v/squareroot 2 a times squareroot pi = 1 [Using integral^infinity_- infinity e^- x^2 dx = squareroot pi] or A^v = squareroot 2 a/pi Therefore A = (2 a/pi)^1/4 (b) now, the time evolution in quantum mechanizes is given by psi (x, t) = psi (x, 0) e^- i H t/h where H is the Hamiltonian of the system Now, the Hamiltonian is straight forward in terms of wave vector k H = p^v/2 m = h^v k^v/2 m