You will answer this question by dragging and dropping elements from the list below intoboxes. The elements below are colour-coded: you can only drop an element into a boxwith a matching colour. Please note that you may need to leave some of the boxes blank.Do not include any items that are equal to zero.The following wavefunction is a solution for the time-independent Schrödinger equation fora particle inside a one-dimensional finite potential energy barrier:y=A exp (-ax).(a) The particle has total energy E

Step 1: (a) Completing the Schrödinger EquationThe Schrödinger equation is:−2mℏ2dx2d2ψ+V(x)ψ=EψFor…

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Modern Physics

3rd Edition

ISBN:9781111794378

Author:Raymond A. Serway, Clement J. Moses, Curt A. Moyer

Publisher:Raymond A. Serway, Clement J. Moses, Curt A. Moyer

Chapter6: Quantum Mechanics In One Dimension

Section: Chapter Questions

Problem 29P

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Photos - B4.jpg A See all photos + Add to * Edit & Create v IA Share A particle of mass, m, moves freely inside an infinite potential well spanning the range, 0
Below is a figure that depicts the potential energy of an electron (a finite square well), as well as the energies associated with the first two wave-functions. a) Sketch the first two stationary wavefunctions (solutions to the Schrődinger equation) for an electron trapped in this fashion. Pay attention to detail! Use the two dashed lines as x-axes. U(x) E2 E1 b) If the potential energy were an infinite square well (not finite well as shown above), what would the energy of the first two allowed energy levels be (i.e., E1 and E2). Write the expressions in terms of constants and a (the width of the well) and then evaluate numerically for a = 6.0*10-10 m. [If you don't remember the formula, you can derive it by using E = ħ²k² /2m, together with the condition on 1 = 2a/n.] c) Let's say I adjust the width of the well, a, such that E1 = 3.5 eV. In that case, calculate the wavelength (in nanometers) of a photon that would be emitted in the electron's transition from E2 to E1. (Remember: hc =…
A proton is confined in box whose width is d = 750 nm. It is in the n = 3 energy state. What is the probability that the proton will be found within a distance of d/n from one of the walls? Include a sketch of U(x) and ?(x). Sketch the situation, defining all your variables
We will consider the Schrödinger equation in this problem as well as the analogies between the wavefunction and how boundary conditions are an essential part of developing this equation for various problems (situations). a) Write the form of the time-independent Schrödinger equation if the potential is that of a spring with spring constant k. Write the form of the time-dependent Schrödinger equation with the same potential. Briefly describe all the terms and variables in these equations. b) One solution to the time independent Schrödinger equation has the form Asin(kx). Why might it be called the wavefunction? If this form represents a wave of light, what is the energy for one photon? (Notek here stands for the wavevector and not the spring constant.) c) Why must all wavefunctions go to zero at infinite distance from the center of the coordinate system in all systems where the potential energy is always finite?
A particle in the ground state of the quantum harmonic oscillator has a wavefunction that can be written as found in image where a = mω/hbar , and where N, m and ω are all constants. For this particle: a) Find an expression for the normalization constant N, in terms of a. b) Derive an expression for the uncertainty in the momentum, ∆p. N.B. You may wish to refer to the Appendix when answering this question.
The uncertainties of a position and a momentum of a particle (Ax) and (Ap) are defined as Ax = /(x²) – (x)² Ap = /(p*) – {p}² 1. For the particle in the box at the ground eigenstate (n = 1) and first excited state ( 2), what is the uncertainty in the value x? How would you interpret the results of these calculations? n = 2. For the particle in the box at the ground eigenstate (n = 1) and first excited state ( n = 2), what is the uncertainty in the value p? How would you interpret the results of these calculations? 3. What is the product for the ground and first excited state: AxAp. 4. Does the Heisenberg Uncertainty Principle hold for a particle in each of these states?
Calculate: a. The mean of the displacement of the oscillator from equilibrium when a harmonic oscillator is in the v=0 and v=1 quantum states? Explain the origin of similarity and differences. b. The mean of the square of the displacement when a harmonic oscillator is in the v=0 and v=1 quantum states? Explain the origin of similarity and differences. 6.
Y:0. 4 %YO N Derive an equation which is represent the time-dependent Schrödinger equation. * 1 Add file This is a required question If Å is the position operator, and P, is the linear momentum operator, solve the following commutative, [X² ,P.] =?. 1 Add file If the operator Â = 3 in , and the wave function Y = 8 e-4ix , find the eigenvalue of (Â¥),Is ¥ an eigen function of the operator Â? dx 1 Add file Page 4 of 5 Вack Next Never submit passwords through Google Forms. العربية الإنجليزية
Background: In quantum mechanics, one can understand a lot about the wave nature of particles by solving simple one- dimensional scattering problems. We covered how to solve scattering from a step potential. In this problem, consider a sequence of steps for a free particle moving to the right and encounters a potential energy disturbance in space as shown to the right. (a) For the case where E > V5 write down the time independent Schrodinger equation that must be solved within each distinct region. Label the regions. (b) Write down the form of the solution for each region in such a way that all parameters in the equations are real. (c) Specify the boundary conditions that are necessary to impose on the solutions. (d) Repeat parts (a) and (b) for the case 0 0 V = 0 V, <0 x= 0 x = b
Suppose a particle of mass m is in a region where its potential energy varies as ax4, where a is a constant. Write down the corresponding Schrödinger equation. Do not evaluate or solve... just write the equation.
1. a. Identify the three characteristics of Separable Solutions. b. Explain each of the characteristic of Separable Solution. (You can refer to Introduction of Quantum Mechanics by Griffiths. You can find this on Chapter 2 of the book but I'm quite confused with the calculus details and the identified characteristics. Hope you can answer it though! The separable solutions I'm talking about here is the one used in obtaining Time Dependent Schrodinger Equation.)
a) For a particle in a one-dimensional box of length L, do the energy levels move up ordown if the box gets longer? Explain your answer clearly.b) Consider a particle in a two-dimensional box of side lengths a and b, where b = 2a.In which direction is it more expensive (in terms of energy) to add nodes? Explain youranswer clearly.c) Consider a particle in a two-dimensional box of side lengths a and b, where b = 2a.Write down the quantum numbers corresponding to the rst (lowest) 5 energy levels of thissystem. Note any degeneracies.

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