We have a free particle in one dimension at a time t = 0, the initial wave function is ¥ (x,0) = Ae¬r|æ| where A andrare positive real constants. Calculate the expectation value (x).

Consider a particle moving in a 2D infinite rectangular well defined by V = 0 for 0 < x < L₁ and 0 ≤ y ≤ L2, and V = ∞ elsewhere. Outside the well, the wavefunction (x, y) is zero. Inside the well, the wavefunction (x, y) obeys the standing wave condition in the x and y direction, so it is given as: where A is a constant. (x, y) = Asin(k₁x) sin(k₂y), (a) The wavenumber k₁ in the x direction is quantized in terms of an integer n₁. Using the standing wave condition, find the possible values of k₁. (1) (b) The wavenumber k2 in the y direction is quantized in terms of a different integer n₂. Using the standing wave condition, find the possible values of k₂. (1) (c) Each state of the 2D infinite rectangular well is defined by the pair of quantum numbers (n₁, n₂). What is the energy of the state Eni,n₂? JXZ1