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1 Particle in a Box with a Bump (based on B&J 4.11) Consider a particle of...
3. [Total: 24 pts] a) (8 pts) Calculate the probability of finding a particle in the classically forbidden regime for the ground state of the 1D harmonic oscillator. Simplify the integral expression for the probability as much as possible - the integral can only be solved numerically. b) (8 pts) For the 1D harmonic oscillator, the energy eigenstates are either even or odd. This is indeed a special case of a more general statement: If V(x) is an even function...
2. A particle is confined to the interval (-L/2. L/2) by infinite potentials for rs -L/2 and * 1/2 - Votol - ܘܐ V(x) - [+ o 0 (+ for Is-L/2 for -L/2<x<L/2 for r 1/2 ܝ 02 This is the same as the "particle-in-one-dimensional-box" model of Problem 1, except the origin of the coordinate is taken at the midpoint of the interval. With this choice of the ori gin, potential energy function V () of the particle-in-one-dimensional-box" model becomes...
A particle is trapped in a one-dimensional potential energy well given by: 100 x < 0 0 < x <L U(x) = L < x < 2L (20. x > 2L Consider the case when U, < E < 20., where E is the particle energy. a. Write down the solutions to the time-independent Schrödinger equation for the wavefunction in the four regions using appropriate coefficients. Define any parameters used in terms of the particles mass m, E, U., and...
Problem 4.1 - Odd Bound States for the Finite Square Well Consider the finite square well potential of depth Vo, V(x) = -{ S-V., –a sx sa 10, else In lecture we explored the even bound state solutions for this potential. In this problem you will explore the odd bound state solutions. Consider an energy E < 0 and define the (real, positive) quantities k and k as 2m E K= 2m(E + V) h2 h2 In lecture we wrote...
3 At a given time, the normalised wave function for a particle in a one-dimensional infinite square well -a < x < a is given by 2 sin2 V inside the well and zero outside. Find the probability that a measurement of energy yields the eigenvalue En. (Hint: use data on page 6.) [6] Useful Data and Formulas = 1.60 x 10-19 C Elementary charge e h/2T=1.05 x 10-34 Js Planck's constant 3.00 x 108 m s-1 Speed of light...
Consider a particle of mass m moving in a one-dimensional potential of the form V. for 0<x<b, V(a) = 0 for Islal<e, for 1212, with V., b and c positive constants and c>b. a Explain why the wave function of the particle can be assumed to be cither an even function or an odd function of a. b For the case that the energy E of the particle is in the range 0<ESV., find the (unnormalized) even cigenfunctions and give...
Please answer a,b and c. Now, consider a 1-d infinite square well of width a, between x = 0 and a, such that V(x) = 0 for 0<x<a and too elsewhere. A perturbation is then added to it so that V(x) = V. for 0 <x <a/2, and the same as before elsewhere. In other words, a flat bump of width a/2 and height V. in the left half of the well. (a) (5 pts) Carefully sketch the potential and...
P10.6 Consider a charged particle of mass m in a SHO potential but which is also subject to an external electric field E. The potential for this problem is now given by where q is the charge of the particle. terms of the standard SHO solutions. Hint: Complete the square. Find the new eigenfunctions and energy eigenvalues. Show that for a particular value of E the ground state Does this mean that there is no zero point energy in this...
please solve all 3 Differential Equation problems 3.8.7 Question Help Consider the following eigenvalue problem for which all of its eigenvalues are nonnegative y',thy-0; y(0)-0, y(1) + y'(1)-0 (a) Show that λ =0 is not an eigenvalue (b) Show that the eigenfunctions are the functions {sin α11,o, where αη įs the nth positive root of the equation tan z -z (c) Draw a sketch indicating the roots as the points of intersection of the curves y tan z and y...
In class, we considered a box with walls at \(x=0\) and \(x=L\). Now consider a box with width \(L\) but centered at \(x=0\), so that it extends from \(x=-L / 2\) to \(x=L / 2\) as shown in the figure. Note that this box is symmetric about \(x=0 .\) (a) Consider possible wave functions of the form \(\psi(x)=A \sin k x\). Apply the boundary conditions at the wall to obtain the allowed energy levels.(b) Another set of possible wave functions...