Consider the Hilbert space of a particle in one-dinension. Consider the operator where is a real, dimensionless parameter.
Use the eigenvalue equation show that
Consider the Hilbert space of a particle in one-dinension. Consider the operator where is a real,...
Consider the hamiltonian: where and Find the probability at to find .
a) Two instantaneous sources, each of strength M are symmetrically placed about the origin (x=0) at locations respectively and released at time t=0. Obtain the solution to the 1d diffusion equation describing the concentration c(x,t) at a time > 0 for . Plot the concentration field, c(x,t)/M, as a function of x for -4 What does the plot looks like for Dt > 5. b) Find the peak concentration at x=0 and the time to at which it develops. Plot...
(2) Consider the 'square' of the derivative operator in one dimension, ie., D"--2 ie. Find all solutions of without any restrictions on , ie.,-oo< (2b) Are there limitations on a? (e.g., any complex number, any real number, only positive real number, integers only, etc.). In mathematis? In physics, when the operator is meant to represent an observable? (2c) Can you find a solution to (4) which is not a solution to DIf(x) Vaf(x)? Hint: try trig functions! Note the 'philosophical'...
Problem Three (1) Write the expression that defines the expectation value of the operator <x> for any function Y. (II) Find the expectation value of <x> for the particle in a box defined by the wave function: Y=N(L x - X°) within limits ( < x < L. which you previously normalized in Problem Two on the previous page.
Consider a particle incident from the left on the potential step. Where E = 2 eV V(x) {5 eV lo x < 0 x > 0 1) Find the wave function of the particle in two regions 2) Find reflection and transmission coefficients R and T
Please answer all parts: Consider a particle in a one-dimensional box, where the potential the potential V(x) = 0 for 0 < x <a and V(x) = 20 outside the box. On the system acts a perturbation Ĥ' of the form: 2a ad αδα 3 Approximation: Although the Hilbert space for this problem has infinite dimensions, you are allowed (and advised) to limit your calculations to a subspace of the lowest six states (n = 6), for the questions of...
Particle in a box. (a) Let H=L?([0,L]) (square integrable wave functions on the interval 0 < x <L). Show that the wave functions Yn(x) = eilanx/L, n=0,1,-1,2, -2,... (6) form an orthonormal system in H. Is this system a basis? (b) Show that the wave functions Yn are eigenstates of the momentum operator p on H= L?([0,L]). Hence, show that the variance Ap in the state Yin vanishes. What is the variance Ať in the state Yn? Why is the...
Consider a particle confined to one dimension and positive r with the wave function 0, z<0 where N is a real normalization constant and o is a real positive constant with units of (length)-1. For the following, express your answers in terms of a: a) Calculate the momentum space wave function. b) Verify that the momentum space wave function is normalized such that (2.4) c) Use the momentum space wave function to calculate the expectation value (p) via (2.5)
6. (20pts) Consider a particle of mass m and energy E approaching the step potential V(x) = { 0x< V.>0 x > 0 from negative values of x. Consider the case E> Vo. a) Classically, what is the probability of reflection? b) Quantum mechanically, what is the probability of reflection? Express your result in terms of the ratio VIE. What is the probability of reflection if E= 2Vo?
Language: C Write a function that produces the next generation of cellular automata Background: Function signature: note that an unsigned long for the next gen is returned as a result. Function description: We compactly represent a generation as a 64-bit unsigned long, one bit for each cell. A 1 bit indicates the cell is live, 0 if not. Using this bit-packed representation, the code to read or update a cell is implemented as a bitwise operation Let's trace how one...