1. A particle of mass m moves in the one-dimensional potential: x<-a/2 x>a/2 Sketch the potential....
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...
Consider the Kronig-Penney model discussed in the lectures, where the periodic potential corresponds to an array of delta functions: However, unlike the usual model, we will take α < 0, so that we have potential wells rather than barriers. In the following, we aim to solve the time-independent Schrödinger equation 2m r (a) First consider the case where the energy E 0. Write down the general solution for ψ(z) within the interval 0 < r < a, and use the...
2.5 ty which will be discussed in chapter 4 2.3 Consider a particle of mass m subject to a one-dimensional potential V(x) that is given by V = 0, x <0; V = 0, 0<x<a; V = Vo, x> Show that bound (E < Vo) states of this system exist only if k cotka = -K where k2 = 2mE/12 and k' = 2m(Vo - E)/h4. 2.4 Show that if Vo = 974/2ma, only one bound state of the system...
A For a particle with mass m moving under a one dimensional potential V(x), one solution to the Schrödinger equation for the region 0<x< oo is x) =2 (a>0), where A is the normalization constant. The energy of the particle in the given state is 0, Show that this function is a solution, and find the corresponding potential V(x)?
Figure 3. Double delta-function potential. X +a V(x) 2. Consider the symmetric, attractive double delta function potential illustrated in Fig. 3 where α is a positive constant. There are two lengths in this problem, the separation between the delta functions, 2a, and the decay lengthK-1-쁩)" of the wave function for an attractive delta function potential. [Note: In this problem, you may not need much math, but explain clearly the reasoning for your answers.] (a) How many bound states do you...
For a particle of mass m, consider a Morse potential of V. V(x) cosh (Bx)' where V> 0 and 8 >0. (a) Illustrate this potential graphically as a function of x. (b) Write the WKB quantization condition: of pladě = (n+ + ) 7, n=0,1,2,3,,.... in terms of the bound state energies En and V(x). What are I'min and Imax in this case, and what is the physical meaning/interpretation of Imin and Imax ? (C) Use WKB methods to determine...
Consider a one-dimensional well with one impenetrable wall. The potential energy is given by 0 x < 0 V(x) = { -V. 0 < x < a 10 x > a We showed in the homework that the allowed energies for the eigenstates of a bound particle (E < 0) in this potential well satisfy the transcendental function -cotĚ = 16 - 52 $2 where 5 = koa, and ko = V2m(Vo + E)/ħ, and 5o = av2mV /ħ (a)...
Consider the 1D square potential energy well shown below. A particle of mass m is about to be trapped in it. a) (15 points) Start with an expression for this potential energy and solve the Schrödinger 2. wave equation to get expressions for(x) for this particle in each region. (10 points) Apply the necessary boundary conditions to your expressions to determine an equation that, when solved for E, gives you the allowed energy levels for bound states of this particle....
*Problem 2.27 Consider the double delta-function potential V (x) -α[6(x + a) + δ( )], x- a where α and a are positive constants. (a) Sketch this potential. (b) How many bound states does it possess? Find the allowed energies, for a- h2/ma and for α h2/4ma, and sketch the wave functions.
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...