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8. Consider a particle encountering a barrier...
Consider a particle encountering a barrier with potential U = U.>0 between x = -a and...
Consider a particle encountering a barrier with potential U = U.>0 between x = -a and x = a with incoming energy E > U. a) Write the symbolic wave functions before and after passing through the barrier (i.e., for x<-a and x>a; regions I and III). U1 b) Write down the Schrodinger equation for the wave function in the middle (region II) where the potential is non-zero i.e., where -a<x<a; region II). c) What solution would you try for...
A free electron moving in the positive x-direction encountering a potential energy barrier in the region...
A free electron moving in the positive x-direction encountering a potential energy barrier in the region x 0 is described by W(x) Aexp(-i2ax/A1) Bexp(-12x/A1) x< 0 (zone I) WI(X) Cexp(i27ox/A) x 20 (zone II) with A 0.80 m-1/2, B 0.20 m-1/2 and C 1.00 m-12. a) Show that the wave function is continaous at x 0. b) Is the electron showing barrier-penetration behavior? Or barrier-transmission behavior? Justify your answer. c) Calculate the probability the electron is reflected at x 0.
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...
(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...
Problem 16.1 P16.1 In this problem, you will calculate the transmission probability through the barrier illustrated...
Problem 16.1
P16.1 In this problem, you will calculate the transmission probability through the barrier illustrated in Figure 16.10. We first go through the mathematics leading to the solution. You will then carry out further calculations. The domain in which the calculation is carried out is divided into three regions for which the potentials are Aetikx + Be-ikx Region I ψ(x)-cexpFPWh-x] - 1 V(x) =0 for x 0 V(x) = Vo for 0 < x < a V(x) =0 for...
In class we considered quantum tunneling of a particle of energy Eo through a barrier of...
In class we considered quantum tunneling of a particle of energy Eo through a barrier of potential Vofor Vo > Eo. Here we focus on two aspects of the problem we ignored in class. In order to simplify we will only consider the initial first half of the barrier as shown below RegionI xS0 Regionx 20 Il There are two cases to consider: Eo< Vo Considered in class E>Vo Not considered in class Here we will focus on the second...
A potential modified from and infinitely deep well is shown below. It is located at x...
A potential modified from and infinitely deep well is shown below. It is located at x = 0 and is narrow enough that the wave function does not change across the well. please not E2= -4U. 0 E1 -300 E2 1) Sketch the wave function for E1 and Ez ii) Write down the time independent Schrödinger equation for the regions x > 0 and x <0. Then find a general solution for each region. iii) Simplify the solutions using the...
Problem 4.1 - Odd Bound States for the Finite Square Well Consider the finite square well...
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...
6. The Particle in a Box problem refers to a potential energy function called the infinite square...
6. The Particle in a Box problem refers to a potential energy function called the infinite square well, aka the box: ; x < 0 (Region I) V(x) = 0 : 0 L (Region II) x x >L (Region III) Let's investigate a quantum particle with mass m and energy E in this potential well of length L We were unable to transcribe this image6d (continued) write down an equation relating ψ, (x = 0) to ψ"(x I and II....
(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...
Consider a particle encountering a barrier with potential U = U.>0 between x = -a and x = a with incoming energy E > U. a) Write the symbolic wave functions before and after passing through the barrier (i.e., for x<-a and x>a; regions I and III). U1 b) Write down the Schrodinger equation for the wave function in the middle (region II) where the potential is non-zero i.e., where -a<x<a; region II). c) What solution would you try for...
A free electron moving in the positive x-direction encountering a potential energy barrier in the region x 0 is described by W(x) Aexp(-i2ax/A1) Bexp(-12x/A1) x< 0 (zone I) WI(X) Cexp(i27ox/A) x 20 (zone II) with A 0.80 m-1/2, B 0.20 m-1/2 and C 1.00 m-12. a) Show that the wave function is continaous at x 0. b) Is the electron showing barrier-penetration behavior? Or barrier-transmission behavior? Justify your answer. c) Calculate the probability the electron is reflected at x 0.
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...
(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...
Problem 16.1
P16.1 In this problem, you will calculate the transmission probability through the barrier illustrated in Figure 16.10. We first go through the mathematics leading to the solution. You will then carry out further calculations. The domain in which the calculation is carried out is divided into three regions for which the potentials are Aetikx + Be-ikx Region I ψ(x)-cexpFPWh-x] - 1 V(x) =0 for x 0 V(x) = Vo for 0 < x < a V(x) =0 for...
In class we considered quantum tunneling of a particle of energy Eo through a barrier of potential Vofor Vo > Eo. Here we focus on two aspects of the problem we ignored in class. In order to simplify we will only consider the initial first half of the barrier as shown below RegionI xS0 Regionx 20 Il There are two cases to consider: Eo< Vo Considered in class E>Vo Not considered in class Here we will focus on the second...
A potential modified from and infinitely deep well is shown below. It is located at x = 0 and is narrow enough that the wave function does not change across the well. please not E2= -4U. 0 E1 -300 E2 1) Sketch the wave function for E1 and Ez ii) Write down the time independent Schrödinger equation for the regions x > 0 and x <0. Then find a general solution for each region. iii) Simplify the solutions using the...
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...
6. The Particle in a Box problem refers to a potential energy function called the infinite square well, aka the box: ; x < 0 (Region I) V(x) = 0 : 0 L (Region II) x x >L (Region III) Let's investigate a quantum particle with mass m and energy E in this potential well of length L We were unable to transcribe this image6d (continued) write down an equation relating ψ, (x = 0) to ψ"(x I and II....
(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...
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