Homework Answers
In
the second part one can derive the exact form of transmission
coefficient (Cr/Ar) in terms of k, k' and a.
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4. Answer the following short answer questions. a. For the particle in a square well, when...
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....
2.5 ty which will be discussed in chapter 4 2.3 Consider a particle of mass m...
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...
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...
(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...
(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...
1. A particle of mass m moves in the one-dimensional potential: x<-a/2 x>a/2 Sketch the potential....
1. A particle of mass m moves in the one-dimensional potential: x<-a/2 x>a/2 Sketch the potential. Sketch what the wave functions would look like for α = 0 for the ground state and the 1st excited state. Write down a formula for all of the bound state energies for α = 0 (no derivation necessary). a) b) Break up the x axis into regions where the Schrödinger equation is easy to solve. Guess solutions in these regions and plug them...
A NON stationary state A particle of mass m is in an infinite square well potential of width L, a...
A NON stationary state A particle of mass m is in an infinite square well potential of width L, as in McIntyre's section 5.4. Suppose we have an initial state vector lv(t -0) results from Mclntrye without re-deriving them, and you may use a computer for your math as long as you include your code in your solution A(3E1) 4iE2)). You may use E. (4 pts) Use a computer to plot this probability density at 4 times: t 0, t2...
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...
Lcarning Goal: Submit My Answers Glve Up To understand the qualities of the finite square-well potential...
Lcarning Goal: Submit My Answers Glve Up To understand the qualities of the finite square-well potential and how to connect solutions to the Schrödinger equation from different regions. Correct The case of a particle in an infinite potential well, also known as the particle in a box, is one of the simplest in quantum mechanics. The closely related finite potential well is substantially more complicated to solve, but it also shows more of the qualities that are characteristic of quantum...
From the previous problem, Problem 2.17, answer the questions below; (20=3+3+4+5+5 points) Does the particle stop...
From the previous problem, Problem 2.17,
answer the questions below; (20=3+3+4+5+5 points)
Does the particle stop for time t between -4 s and +4
s? If it stops, when? Explain your answer.
Does this particle change direction? Explain your answer.
Describe the motion of the particle for time t between
-4 s and +4 s.
Graph v(t) versus t for -4 s≤ t ≤ 4
s.
[Please use a graph sheet to answer f). You can download and print
it...
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....
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...
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...
(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...
(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...
1. A particle of mass m moves in the one-dimensional potential: x<-a/2 x>a/2 Sketch the potential. Sketch what the wave functions would look like for α = 0 for the ground state and the 1st excited state. Write down a formula for all of the bound state energies for α = 0 (no derivation necessary). a) b) Break up the x axis into regions where the Schrödinger equation is easy to solve. Guess solutions in these regions and plug them...
A NON stationary state A particle of mass m is in an infinite square well potential of width L, as in McIntyre's section 5.4. Suppose we have an initial state vector lv(t -0) results from Mclntrye without re-deriving them, and you may use a computer for your math as long as you include your code in your solution A(3E1) 4iE2)). You may use E. (4 pts) Use a computer to plot this probability density at 4 times: t 0, t2...
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...
Lcarning Goal: Submit My Answers Glve Up To understand the qualities of the finite square-well potential and how to connect solutions to the Schrödinger equation from different regions. Correct The case of a particle in an infinite potential well, also known as the particle in a box, is one of the simplest in quantum mechanics. The closely related finite potential well is substantially more complicated to solve, but it also shows more of the qualities that are characteristic of quantum...
From the previous problem, Problem 2.17,
answer the questions below; (20=3+3+4+5+5 points)
Does the particle stop for time t between -4 s and +4
s? If it stops, when? Explain your answer.
Does this particle change direction? Explain your answer.
Describe the motion of the particle for time t between
-4 s and +4 s.
Graph v(t) versus t for -4 s≤ t ≤ 4
s.
[Please use a graph sheet to answer f). You can download and print
it...
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