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2. Consider two different systems: a square well 2 1 and a linear wel -1 (don't...
The Finite Square Wel A more realistic version of the infinite square well potential has a...
The Finite Square Wel A more realistic version of the infinite square well potential has a finite well depth: -a V(x)--V for -a<x <a for x <-a,'r > a =0 This assignment will consider the bound states of a particle (of mass m) in this potential (i.e. total energy E <0). (1) Determine the general solutions to the time-independent Schrödinger equation for the three regions x <-a, -a<x <a, and > a. Write these solutions in terms of k and...
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...
Questions 1 - 5 deal with a particle in a one-dimensional infinite square well of width...
Questions 1 - 5 deal with a particle in a one-dimensional infinite square well of width a where 0, 0 SX Sa V(x) = 100, Otherwise. The stationary states are Pn(x) = sin(**) with energies En = "forn = 1,2,3.. Question 1 (14 pts) Which of the following is correct? A. The Hilbert space for this system is one dimensional. B. The energy eigenstates of the system form a ID Hilbert space. C. Both A and B are correct. D....
Q.7) Consider a systems of N>>1 identical, distinguishable and independent particles that can be placed in...
Q.7) Consider a systems of N>>1 identical, distinguishable and independent particles that can be placed in three energy levels of energies 0, E and 2€, respectively. Only the level of energy sis degenerate, of degeneracy g=2. This system is in equilibrium with a heat reservoir at temperature T. a) Obtain the partition function of the system. b) What is the probability of finding each particle in each energy level? c) Calculate the average energy <B>, the specific heat at constant...
2. Goal of this problem is to study how tunnelling in a two-well system emerges. In particular, w...
2. Goal of this problem is to study how tunnelling in a two-well system emerges. In particular, we are interested in determining how the tunnelling rate T' of a particle with mass m scales as a function of the (effective) height Vo - E and width b of an energy barrier separating the two wells. The following graphics illustrates the set-up. Initially the particle may be trapped on the left side corresponding to the state |L〉, we are now interested...
1-) Consider two systems A and B. Each system has 5 particles and allowed energy levels...
1-) Consider two systems A and B. Each system has 5 particles and allowed energy levels are ε=0, ε=1 , ε=2. Suppose total energy of the system is Utotal=UA+UB=4. For all possible distributions calculate the total multiplicity of the system. Which energy distribution maximizes multiplicity? 2) Suppose system A has 3, where as B has 7 particles. How does your answer change for 1)?
1. An electron in a finite well An electron is in a finite square well that is 20 eV deep and 0.2...
1. An electron in a finite well An electron is in a finite square well that is 20 eV deep and 0.25 nm wide. You may use all results from class/textbook without re-deriving therm A. (2 pts) By graphing both sides of the quantization condition like we did in class, determine how many bound energy eigenstates exist for this well. Don't forget that there are two quantization conditions, one for the even solutions, and one for the odd solutions! B....
1 Particle in a Box with a Bump (based on B&J 4.11) Consider a particle of...
1 Particle in a Box with a Bump (based on B&J 4.11) Consider a particle of mass m in a 1-D double well with potential given by Vo, 05\x\<b V(x) = { 0, b<\x<c 100, [x]>c . We will study the lowest energy states, for which 0 <E<V, corresponding to tunnelling between the two wells. (a) Write down the time-independent Schödinger equation in the three regions -c<x<-b, –b< <b, and b< I< c. Write down the most general wavefunction solution...
(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. Construct a table of the possible linear systems as follows: (a) The first column contains the...
1 and 2
1. Construct a table of the possible linear systems as follows: (a) The first column contains the type of the system (sink, spiral sink, source, if it has a name. (b) The second column contains the condition on the eigenvalues that corresponds (c) The third column contains a small picture of two or more possible phase por- (d) The fourth column contains x(1)-and y(a)-graphs of typical solutions indicated Hint: The most complete table contains 14 cases. Don't...
The Finite Square Wel A more realistic version of the infinite square well potential has a finite well depth: -a V(x)--V for -a<x <a for x <-a,'r > a =0 This assignment will consider the bound states of a particle (of mass m) in this potential (i.e. total energy E <0). (1) Determine the general solutions to the time-independent Schrödinger equation for the three regions x <-a, -a<x <a, and > a. Write these solutions in terms of k and...
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...
Questions 1 - 5 deal with a particle in a one-dimensional infinite square well of width a where 0, 0 SX Sa V(x) = 100, Otherwise. The stationary states are Pn(x) = sin(**) with energies En = "forn = 1,2,3.. Question 1 (14 pts) Which of the following is correct? A. The Hilbert space for this system is one dimensional. B. The energy eigenstates of the system form a ID Hilbert space. C. Both A and B are correct. D....
Q.7) Consider a systems of N>>1 identical, distinguishable and independent particles that can be placed in three energy levels of energies 0, E and 2€, respectively. Only the level of energy sis degenerate, of degeneracy g=2. This system is in equilibrium with a heat reservoir at temperature T. a) Obtain the partition function of the system. b) What is the probability of finding each particle in each energy level? c) Calculate the average energy <B>, the specific heat at constant...
2. Goal of this problem is to study how tunnelling in a two-well system emerges. In particular, we are interested in determining how the tunnelling rate T' of a particle with mass m scales as a function of the (effective) height Vo - E and width b of an energy barrier separating the two wells. The following graphics illustrates the set-up. Initially the particle may be trapped on the left side corresponding to the state |L〉, we are now interested...
1. An electron in a finite well An electron is in a finite square well that is 20 eV deep and 0.25 nm wide. You may use all results from class/textbook without re-deriving therm A. (2 pts) By graphing both sides of the quantization condition like we did in class, determine how many bound energy eigenstates exist for this well. Don't forget that there are two quantization conditions, one for the even solutions, and one for the odd solutions! B....
1 Particle in a Box with a Bump (based on B&J 4.11) Consider a particle of mass m in a 1-D double well with potential given by Vo, 05\x\<b V(x) = { 0, b<\x<c 100, [x]>c . We will study the lowest energy states, for which 0 <E<V, corresponding to tunnelling between the two wells. (a) Write down the time-independent Schödinger equation in the three regions -c<x<-b, –b< <b, and b< I< c. Write down the most general wavefunction solution...
(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 and 2
1. Construct a table of the possible linear systems as follows: (a) The first column contains the type of the system (sink, spiral sink, source, if it has a name. (b) The second column contains the condition on the eigenvalues that corresponds (c) The third column contains a small picture of two or more possible phase por- (d) The fourth column contains x(1)-and y(a)-graphs of typical solutions indicated Hint: The most complete table contains 14 cases. Don't...
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