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5. Consider a square potential barrier in figure below: V(x) 0 x <0 a) Assume that...
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...
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...
Consider a finite square barrier potential shown below. Figure A. For a<x<b, the space part of...
Consider a finite square barrier potential shown below. Figure A. For a<x<b, the space part of the electron wave function has the form: k? = 2mE/h? and gu2m(V,-E)/h2 (a) Aeikx (b) Aegn (c) Ae*** + Be** (d) Ae* (e) Aelkx + Be-ika For the finite square barrier potential shown below, Figure A. For x<a, the space part of the electron wave function has the form: k = 2mE/h? and g=2m(Vo-E) /h (a) Aeikx (b) Aetex (c) Ae*EN + Bet* (d)...
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...
(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 incident from the left on the potential step. Where E = 2 eV...
Consider a particle incident from the left on the potential step. Where E = 2 eV V(x) {5 eV lo x < 0 x > 0 1) Find the wave function of the particle in two regions 2) Find reflection and transmission coefficients R and T
Consider a traveling (electrons) wave moving in the +x direction approaching a step barrier of height 1 eV; that is V =...
Consider a traveling (electrons) wave moving in the +x direction
approaching a step barrier of height 1 eV; that is V = 0 for x <
0, V = 1.0 eV for x ≥ 0. For x < 0, there will be both the
traveling wave in the +x direction. For x ≥ 0, only a solution
corresponding to motion in the +x direction exists. By solving the
Schrödinger wave equation in both x < 0 and x ≥ 0...
Problem # 5 ( 6 pts) Consider the potential: V(x) αδ(x) Where α is a positive...
Problem # 5 ( 6 pts) Consider the potential: V(x) αδ(x) Where α is a positive constant, and with E>0 Calculate the reflection and transmission coefficients.
toward a Problem 4 (30 points): Consider a current of particles of energy E moving from...
toward a Problem 4 (30 points): Consider a current of particles of energy E moving from x = - potential step as shown in the figure. x > 0 V(x) = {v. x<0 TE Where E > V a) (8 points) Derive the general solution of Schrödinger equation for x < 0 and for x > 0 b) (14 points) Apply the boundary conditions and calculate the transmission and the reflection coefficients. c) (8 points) What is the value of...
18. Consider the potential barrier as shown in the figure. Calculate the transmission coefficcient. Discuss the...
18. Consider the potential barrier as shown in the figure. Calculate the transmission coefficcient. Discuss the depence of this coefficient. This is so called tunneling effect. Give a physical phenomenum that can be explained by this fact. E 00 0 V1 V(x) = x < 0 -V 0 < x <a V2 a< x <b 0 x <b
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...
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...
Consider a finite square barrier potential shown below. Figure A. For a<x<b, the space part of the electron wave function has the form: k? = 2mE/h? and gu2m(V,-E)/h2 (a) Aeikx (b) Aegn (c) Ae*** + Be** (d) Ae* (e) Aelkx + Be-ika For the finite square barrier potential shown below, Figure A. For x<a, the space part of the electron wave function has the form: k = 2mE/h? and g=2m(Vo-E) /h (a) Aeikx (b) Aetex (c) Ae*EN + Bet* (d)...
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...
(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 incident from the left on the potential step. Where E = 2 eV V(x) {5 eV lo x < 0 x > 0 1) Find the wave function of the particle in two regions 2) Find reflection and transmission coefficients R and T
Consider a traveling (electrons) wave moving in the +x direction
approaching a step barrier of height 1 eV; that is V = 0 for x <
0, V = 1.0 eV for x ≥ 0. For x < 0, there will be both the
traveling wave in the +x direction. For x ≥ 0, only a solution
corresponding to motion in the +x direction exists. By solving the
Schrödinger wave equation in both x < 0 and x ≥ 0...
Problem # 5 ( 6 pts) Consider the potential: V(x) αδ(x) Where α is a positive constant, and with E>0 Calculate the reflection and transmission coefficients.
toward a Problem 4 (30 points): Consider a current of particles of energy E moving from x = - potential step as shown in the figure. x > 0 V(x) = {v. x<0 TE Where E > V a) (8 points) Derive the general solution of Schrödinger equation for x < 0 and for x > 0 b) (14 points) Apply the boundary conditions and calculate the transmission and the reflection coefficients. c) (8 points) What is the value of...
18. Consider the potential barrier as shown in the figure. Calculate the transmission coefficcient. Discuss the depence of this coefficient. This is so called tunneling effect. Give a physical phenomenum that can be explained by this fact. E 00 0 V1 V(x) = x < 0 -V 0 < x <a V2 a< x <b 0 x <b
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