1. A particle is described by wave function: = A exp(-alphax^2). Find the potential energy V(x)...
Consider a particle of mass m that is described by the wave function (x, t) = Ce~iwte-(x/l)2 where C and l are real and positive constants, with / being the characteristic length-scale in the problem Calculate the expectation values of position values of 2 and p2. and momentum p, as well as the expectation Find the standard deviations O and op. Are they consistent with the uncertainty principle? to be independent What should be the form of the potential energy...
A particle of mass m is in a potential energy field described by, V(x, y) = 18kx² +8ky? where k is a positive constant. Initially the particle is resting at the origin (0,0). At time t = 0 the particle receives a kick that imparts to it an initial velocity (vo, 2vo). (a) Find the position of the particle as a function of time, x(t) and y(t). (b) Plot the trajectory for this motion (Lissajous figure) using Vo = 1,...
Extra Credit (3 points to Mideterm-2) Q1. A particle is described by the wave function (x) b(a2-x2) for -a sx s a and (x) 0 for x -a and x +a, where a and b are positive real constants. (a) Using the normalization condition, find b in terms a. (b) What is the probability to find the particle at x = +a/2 in a small interval ofwidth 0.01 a ? (c) What is the probability for the particle to be...
Potential energy function, V(x) = (1/2)mw2x2 Assuming the time-independent Schrödinger equation, show that the following wave functions are solutions describing the one-dimensional harmonic behaviour of a particle of mass m, where ?2-h/v/mK, and where co and ci are constants. Calculate the energies of the particle when it is in wave-functions ?0(x) and V1 (z) What is the general expression for the allowed energies En, corresponding to wave- functions Un(x), of this one-dimensional quantum oscillator? 6 the states corresponding to the...
The most general wave function of a particle in the simple harmonic oscillator potential is: V(x, t) = (x)e-1st/ where and E, are the harmonic oscillator's stationary states and their corresponding energies. (a) Show that the expectation value of position is (hint: use the results of Problem 4): (v) = A cos (wt - ) where the real constants A and o are given by: 1 2 Ae-id-1 " Entichtin Interpret this result, comparing it with the motion of a...
a) Show that the wave function y(x) = N exp( – x²/(2a?)) with a? = () is a solution of the Schrödinger equation for harmonic oscillator with potential V(x) = k x2/2. (10 pt) b) What is the energy of harmonic oscillator with the wave function y(x) in terms of k and m? (5 pt) c) Sketch the potential energy of harmonic oscillator, the energy level corresponding to y(x), the wave function (x), and the probability density associated with y(x)...
Problem 2: Time development of a free particle The wave function of a free particle at time t- 0 is given by exp(2K1T Now answer the questions below. l. what is the time evolved wave function ψ(z,t) ? points 2. What is the average momentum at any future time? 4 points 3. What is the average energy at any future time ? 3 points Problem 2: Time development of a free particle The wave function of a free particle at...
h2 4. In a region of the x-axis, a particle has a wave function given by y(x) = Ae-*4722° and energy where L is some length. (a) Find the potential energy as a function of x, and sketch V (x) versus x. (b) What is the classical potential (or corresponding force function) that has this dependence? (c) Find the kinetic energy as a function of x. (d) Show that x = L is the classical turning point (i.e. the place...
A fellow student proposes that a possible wave function for a free particle with mass \(m\) (one for which the potential-energy function \(U(x)\) is zero ) is$$ \psi(x)=\left\{\begin{array}{ll} e^{-k x}, & x \geq 0 \\ e^{+\kappa x}, & x<0 \end{array}\right. $$where \(\kappa\) is a positive constant. (a) Graph this proposed wave function.(b) Determine the energy of the particle if the proposed wave function satisfies the Schrödinger equation for \(x<\)0.(c) Show that the proposed wave function also satisfies the Schrödinger equation...
A particle is described by the wave function where A0. Find the normalization constant A. A particle is described by the wave function where A0. Find the normalization constant A.