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2. Consider a potential U = Uod-b) cos(2πkx), Uo is a positive constant. 2.1) Draw the phase diag...
Answers can be more than one: VII. (12pts) Consider the following potential energy: region 1: U(X)...
Answers can be more than one:
VII. (12pts) Consider the following potential energy: region 1: U(X) = U. x < 0 region 2: U(X) = 0 0<x</ Uo region 3: U(x) = U. x>L ТЕ where U. >0. We want to consider a particle with energy E such that 0 < E<Uo. There are two possible forms for the wave function that might be used to represent the particle: (x) = 4 sin kyx+ B, coskx v(x) = 4e** +...
Consider the potential energy described in problem 1.14. For low amplitudes, the motion of the object...
Consider the potential energy described in problem 1.14. For low
amplitudes, the motion of the object is well described by simple
harmonic motion, so that the period is independent of amplitude.
However, once the amplitude gets high enough this is no longer
true. As the amplitude increase, does the period increase or
decrease? Explain your reasoning thoroughly, and assume that the
amplitude is always less than pi/B.
(You do not need to answer question 1.14)
1.13 In research-level theel pnysles,...
(15 points) Encounter with a semi-infinite potential "well" In this problem we will investigate one situation involving a a semi-infinite one-dimensional po- tential well (Figure 1) U=0 regio...
(15 points) Encounter with a semi-infinite potential "well" In this problem we will investigate one situation involving a a semi-infinite one-dimensional po- tential well (Figure 1) U=0 region 1 region 2 region 3 Figure 1: Semi-infinite potential for Problem 3 This potential is piecewise defined as follows where Uo is some positive value of energy. The three intervals in x have been labeled region 1,2 and 3 in Figure 1 Consider a particle of mass m f 0 moving in...
0 Figure 2: The potential barrier setup for Problem 4 4. (10 points) "Burrowing a hole...
0 Figure 2: The potential barrier setup for Problem 4 4. (10 points) "Burrowing a hole in the wall" Some particles of mass m and energy E move from the left to the potential barrier shown in Figure 2 below 0 <0 Uo 20 U(x) where Uo is some positive value (a) (5 points) Write the Time-Independent Schrödinger equations and the physically acceptable general solutions for the wave function (x) in regions I and II as labeled in Figure 2...
A particle is introduced to a region with a potential described by U(x)--2x2 +x*+1 Joules. 3. a. ...
A particle is introduced to a region with a potential described by U(x)--2x2 +x*+1 Joules. 3. a. (2 pts) In software, plot the potential U) Set your axis ranges: -2 SxS2 and 0s b. (5 pts) Find the equilibrium positions and determine whether they are stable or c. (8 pts) Describe the motion of the particle for total energy values E-О.0.05. 1.0, 2.0 unstable. Explain how you arrived at your answers. (all in Joules). What I am looking for here...
(III) Quantum Tunneling Consider an electron in 1D in presence of a potential barrier of width...
(III) Quantum Tunneling Consider an electron in 1D in presence of a potential barrier of width L represented by a step function ſo I<0 or 1>L V U. r>0 and 2<L The total wavefunction is subject to the time-independent Schrödinger equation = EV (2) 2m ar2 +V where E is the energy of the quantum particle in question and m is the mass of the quantum particle. A The total wavefunction of a free particle that enters the barrier from...
Consider a particle of a potential energy V(x) = {î where 1 is a positive constant....
Consider a particle of a potential energy V(x) = {î where 1 is a positive constant. ( V arises, for example from a force field - e.g a uniform electric field). Write Ehrenfest theorem for (h) and (p). Integrate these equations and compare the results to the classical results.
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...
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...
2. 2.1 Draw the indifference curves for the utility function U(21, 22) = x1 + 3x2....
2. 2.1 Draw the indifference curves for the utility function U(21, 22) = x1 + 3x2. 2.2 What is the marginal rate of substitution evaluated at an arbitrary consumption bundle (21, 22)? 2.3 Suppose that p1 = 5, P2 = 2, and M = 10. Find the utility-maximizing consump- tion bundle (among those that satisfy the budge constraint) for this agent. You should be able to do this without using any calculus: it should be clear from your indifference curves....
Consider the symmetrical finite square well potential shown below. U(x) = 46 eV for xs-L/2 U(x)...
Consider the symmetrical finite square well potential shown below. U(x) = 46 eV for xs-L/2 U(x) 0 eV for-L/2 < x < L/2 U(x) 46 eV for x 2 L/2 L-0.27mm Note: 46 ev 1. the width L is unchanged from the infinite well you previously considered 2, the potential outside x-±L/2 is finite with U-46 eV. 3. you found the three lowest energy levels for that infinite -8.135 0.135 potential well were: 5.16 ev, 20.64 ev, and46.45 ev. 1)...
Answers can be more than one:
VII. (12pts) Consider the following potential energy: region 1: U(X) = U. x < 0 region 2: U(X) = 0 0<x</ Uo region 3: U(x) = U. x>L ТЕ where U. >0. We want to consider a particle with energy E such that 0 < E<Uo. There are two possible forms for the wave function that might be used to represent the particle: (x) = 4 sin kyx+ B, coskx v(x) = 4e** +...
Consider the potential energy described in problem 1.14. For low
amplitudes, the motion of the object is well described by simple
harmonic motion, so that the period is independent of amplitude.
However, once the amplitude gets high enough this is no longer
true. As the amplitude increase, does the period increase or
decrease? Explain your reasoning thoroughly, and assume that the
amplitude is always less than pi/B.
(You do not need to answer question 1.14)
1.13 In research-level theel pnysles,...
(15 points) Encounter with a semi-infinite potential "well" In this problem we will investigate one situation involving a a semi-infinite one-dimensional po- tential well (Figure 1) U=0 region 1 region 2 region 3 Figure 1: Semi-infinite potential for Problem 3 This potential is piecewise defined as follows where Uo is some positive value of energy. The three intervals in x have been labeled region 1,2 and 3 in Figure 1 Consider a particle of mass m f 0 moving in...
0 Figure 2: The potential barrier setup for Problem 4 4. (10 points) "Burrowing a hole in the wall" Some particles of mass m and energy E move from the left to the potential barrier shown in Figure 2 below 0 <0 Uo 20 U(x) where Uo is some positive value (a) (5 points) Write the Time-Independent Schrödinger equations and the physically acceptable general solutions for the wave function (x) in regions I and II as labeled in Figure 2...
A particle is introduced to a region with a potential described by U(x)--2x2 +x*+1 Joules. 3. a. (2 pts) In software, plot the potential U) Set your axis ranges: -2 SxS2 and 0s b. (5 pts) Find the equilibrium positions and determine whether they are stable or c. (8 pts) Describe the motion of the particle for total energy values E-О.0.05. 1.0, 2.0 unstable. Explain how you arrived at your answers. (all in Joules). What I am looking for here...
(III) Quantum Tunneling Consider an electron in 1D in presence of a potential barrier of width L represented by a step function ſo I<0 or 1>L V U. r>0 and 2<L The total wavefunction is subject to the time-independent Schrödinger equation = EV (2) 2m ar2 +V where E is the energy of the quantum particle in question and m is the mass of the quantum particle. A The total wavefunction of a free particle that enters the barrier from...
Consider a particle of a potential energy V(x) = {î where 1 is a positive constant. ( V arises, for example from a force field - e.g a uniform electric field). Write Ehrenfest theorem for (h) and (p). Integrate these equations and compare the results to the classical results.
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...
2. 2.1 Draw the indifference curves for the utility function U(21, 22) = x1 + 3x2. 2.2 What is the marginal rate of substitution evaluated at an arbitrary consumption bundle (21, 22)? 2.3 Suppose that p1 = 5, P2 = 2, and M = 10. Find the utility-maximizing consump- tion bundle (among those that satisfy the budge constraint) for this agent. You should be able to do this without using any calculus: it should be clear from your indifference curves....
Consider the symmetrical finite square well potential shown below. U(x) = 46 eV for xs-L/2 U(x) 0 eV for-L/2 < x < L/2 U(x) 46 eV for x 2 L/2 L-0.27mm Note: 46 ev 1. the width L is unchanged from the infinite well you previously considered 2, the potential outside x-±L/2 is finite with U-46 eV. 3. you found the three lowest energy levels for that infinite -8.135 0.135 potential well were: 5.16 ev, 20.64 ev, and46.45 ev. 1)...
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