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toward a Problem 4 (30 points): Consider a current of particles of energy E moving from...
Consider the 1D square potential energy well shown below. A particle of mass m is about to be tra...
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....
5. Consider a square potential barrier in figure below: V(x) 0 x <0 a) Assume that...
5. Consider a square potential barrier in figure below: V(x) 0 x <0 a) Assume that incident particles of energy E> v are coming from-X. Find the stationary states (the equations for region . 2 and 3 and the main equation for the all regions). Apply the matching limit conditions in the figure. Explain and find all the constants used in the equations in terms of the parameters provided and Planck's constant -(6) Find the transmission and reflection coefficients. -(4)
Scattering #1 Consider the "downstep" potential shown. A particle of mass m and energy E, inciden...
Scattering #1 Consider the "downstep" potential shown. A particle of mass m and energy E, incident from the left, strikes a potential energy drop-off of depth Vo 0 (2 pts) Using classical physics, consider a particle incident with speed vo. Use conservation of energy to find the speed on the right vf. ALSO, what is the probability that a given particle will "transmit" from the left side to the right side (again, classically)? A. B. (4 pts) This problem is...
A beam of particles, each of the same mass and the same energy, travels in the positive r-direction. The beam is incide...
A beam of particles, each of the same mass and the same energy, travels in the positive r-direction. The beam is incident on an abrupt potential energy step at 0 and some of the beam is transmitted into the region r > 0 and the rest reflected. The energy eigenfunction describing the beam is Aeik,Be-i for r< 0 )Cekfor T > 0, where the coefficients A, B and C are constants and ki and k are real constants (a) Write...
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...
In class we looked at the example of the potential energy step seen below (where E...
In class we looked at the example of the potential energy step seen below (where E > U_0). We wrote down the wave functions in complex exponential form as seen below: psi _0 (x) = A' e^i K_0 x + B' e^-i K_0 x x < 0 psi _1 (x) = C' e^i K_1 x + D' e^-i K_1 x x > 0 a) Assume the particles are incident on the barrier from the left, which coefficient can be set...
6. (20pts) Consider a particle of mass m and energy E approaching the step potential V(x)...
6. (20pts) Consider a particle of mass m and energy E approaching the step potential V(x) = { 0x< V.>0 x > 0 from negative values of x. Consider the case E> Vo. a) Classically, what is the probability of reflection? b) Quantum mechanically, what is the probability of reflection? Express your result in terms of the ratio VIE. What is the probability of reflection if E= 2Vo?
5. Consider a long rectangular "gutter" of length a in the x direction and infi- nite...
5. Consider a long rectangular "gutter" of length a in the x direction and infi- nite height in the y direction. The gutter is infinitely long in the z direction, so the potential V inside the gutter only depends on x andy. The left (x-0,y), and right (r- a,y) sides of the gutter are grounded so that the potential V(x,y) is zero on those surfaces. The bottom surface of the gutter is kept fixed at a potential given by V(r,y-0)-...
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...
(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 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....
5. Consider a square potential barrier in figure below: V(x) 0 x <0 a) Assume that incident particles of energy E> v are coming from-X. Find the stationary states (the equations for region . 2 and 3 and the main equation for the all regions). Apply the matching limit conditions in the figure. Explain and find all the constants used in the equations in terms of the parameters provided and Planck's constant -(6) Find the transmission and reflection coefficients. -(4)
Scattering #1 Consider the "downstep" potential shown. A particle of mass m and energy E, incident from the left, strikes a potential energy drop-off of depth Vo 0 (2 pts) Using classical physics, consider a particle incident with speed vo. Use conservation of energy to find the speed on the right vf. ALSO, what is the probability that a given particle will "transmit" from the left side to the right side (again, classically)? A. B. (4 pts) This problem is...
A beam of particles, each of the same mass and the same energy, travels in the positive r-direction. The beam is incident on an abrupt potential energy step at 0 and some of the beam is transmitted into the region r > 0 and the rest reflected. The energy eigenfunction describing the beam is Aeik,Be-i for r< 0 )Cekfor T > 0, where the coefficients A, B and C are constants and ki and k are real constants (a) Write...
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...
In class we looked at the example of the potential energy step seen below (where E > U_0). We wrote down the wave functions in complex exponential form as seen below: psi _0 (x) = A' e^i K_0 x + B' e^-i K_0 x x < 0 psi _1 (x) = C' e^i K_1 x + D' e^-i K_1 x x > 0 a) Assume the particles are incident on the barrier from the left, which coefficient can be set...
6. (20pts) Consider a particle of mass m and energy E approaching the step potential V(x) = { 0x< V.>0 x > 0 from negative values of x. Consider the case E> Vo. a) Classically, what is the probability of reflection? b) Quantum mechanically, what is the probability of reflection? Express your result in terms of the ratio VIE. What is the probability of reflection if E= 2Vo?
5. Consider a long rectangular "gutter" of length a in the x direction and infi- nite height in the y direction. The gutter is infinitely long in the z direction, so the potential V inside the gutter only depends on x andy. The left (x-0,y), and right (r- a,y) sides of the gutter are grounded so that the potential V(x,y) is zero on those surfaces. The bottom surface of the gutter is kept fixed at a potential given by V(r,y-0)-...
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...
(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...
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