(a) Write down wave functions that describe the behavior of the particle in region 1, region...
(a) Write down wave functions that describe the behavior of the particle in region 1, region 2, and region
those coefficients and explain why they are equal to zero. Write down the expression of ?? as well. ??
(b) Sketch the probability distributions you would expect for the ground state and the first excited state.
(c) Use the continuity conditions at x = 0 to show how the coefficients of the wave function in region 2 are related to the coefficients of the wave function in region 1.
Homework Answers
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(a) Write down wave functions that describe the behavior of the
particle in region 1, region...
Answers can be more than one: VII. (12pts) Consider the following potential energy: region 1: U(X)...
Answers can be more than one:
VII. (12pts) Consider the following potential energy: region 1: U(X) = U. x < 0 region 2: U(X) = 0 0<x</ Uo region 3: U(x) = U. x>L ТЕ where U. >0. We want to consider a particle with energy E such that 0 < E<Uo. There are two possible forms for the wave function that might be used to represent the particle: (x) = 4 sin kyx+ B, coskx v(x) = 4e** +...
1. Suppose we didn't actually know the wave functions for a particle in a box. Reasonable...
1. Suppose we didn't actually know the wave functions for a particle in a box. Reasonable guesses for the ground- and first-excited-state wave functions might be functions of the form 1 = a y (1 - y) 02 = by (y-c)(y-1), where y = (x/L), L is the length of the box, and a, b, and care constants. (a) These functions have quite a number of features that make them sensible guesses. Sketch both functions and list these special features....
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...
1. A particle, initially (t -> 0) in the ground state of an infinite, 1D potential...
1. A particle, initially (t -> 0) in the ground state of an infinite, 1D potential box with walls at r 0 and = a, is subjected at time t = 0 to a time-dependent perturbation V (r, t) et/7, with eo a small real number a) Calculate to first order the probability of finding the particle in an excited state for t 0. Consider all final states. Are all possible transitions allowed? b) Examine the time dependence of the...
3. A particle of mass m in a one-dimensional box has the following wave function in...
3. A particle of mass m in a one-dimensional box has the following wave function in the region x-0 tox-L: ? (x.r)=?,(x)e-iEy /A +?,(X)--iE//h Here Y,(x) and Y,(x) are the normalized stationary-state wave functions for the n = 1 and n = 3 levels, and E1 and E3 are the energies of these levels. The wave function is zero for x< 0 and forx> L. (a) Find the value of the probability distribution function atx- L/2 as a function of...
Extra Credit (3 points to Mideterm-2) Q1. A particle is described by the wave function (x)...
Extra Credit (3 points to Mideterm-2) Q1. A particle is described by the wave function (x) b(a2-x2) for -a sx s a and (x) 0 for x -a and x +a, where a and b are positive real constants. (a) Using the normalization condition, find b in terms a. (b) What is the probability to find the particle at x = +a/2 in a small interval ofwidth 0.01 a ? (c) What is the probability for the particle to be...
1. For the one-dimensional particle in a box of length L=1A a. Write an integral expression for the probability of findi...
1. For the one-dimensional particle in a box of length L=1A a. Write an integral expression for the probability of finding the particle between L/4 and L/3, for the fourth excited state. b. Write the wavefunction for the fourth excited state c. Calculate the numerical probability of finding the particle between 0 and L/3, for the ground state. I am having trouble understanding these questions for my practice assignment, I have an exam tonight and I want to be able...
1. The wave-functions of the states [4) and (0) are given by y(x) and Q(x), respectively....
1. The wave-functions of the states [4) and (0) are given by y(x) and Q(x), respectively. Derive the expression for the inner product (14) in terms of the wave- functions Q(x) and (x). What is the physical meaning of y(x) and (x)/2? 2. Fig. 1 shows a sketch of y(x). Sketch y(x) such that the states (4) and (o) are orthogonal: (014) = 0. (x) M Figure 1 3. Assume a particle has a wave-function y(x) sketched in Fig. 2....
1. The wave-functions of the states [4) and (0) are given by y(x) and Q(x), respectively....
1. The wave-functions of the states [4) and (0) are given by y(x) and Q(x), respectively. Derive the expression for the inner product (14) in terms of the wave- functions Q(x) and (x). What is the physical meaning of y(x) and (x)/2? 2. Fig. 1 shows a sketch of y(x). Sketch y(x) such that the states (4) and (o) are orthogonal: (014) = 0. (x) M Figure 1 3. Assume a particle has a wave-function y(x) sketched in Fig. 2....
2. Derivation of the Kronig-Penney result: Write the solution for the wave function in the region...
2. Derivation of the Kronig-Penney result: Write the solution for the wave function in the region 0 < r < a as a linear combi nation of sin and cos functions and in the region-b< 0 as a linear combination of the hvperbolic sin and cos functions: ψ(z) Asin Kr +Bcos Kx (0<x <a) = By invoking the continuity and differentiability conditions at a- 0, show that B D and AK - CQ, so essentially two constants, say A and...
Answers can be more than one:
VII. (12pts) Consider the following potential energy: region 1: U(X) = U. x < 0 region 2: U(X) = 0 0<x</ Uo region 3: U(x) = U. x>L ТЕ where U. >0. We want to consider a particle with energy E such that 0 < E<Uo. There are two possible forms for the wave function that might be used to represent the particle: (x) = 4 sin kyx+ B, coskx v(x) = 4e** +...
1. Suppose we didn't actually know the wave functions for a particle in a box. Reasonable guesses for the ground- and first-excited-state wave functions might be functions of the form 1 = a y (1 - y) 02 = by (y-c)(y-1), where y = (x/L), L is the length of the box, and a, b, and care constants. (a) These functions have quite a number of features that make them sensible guesses. Sketch both functions and list these special features....
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...
1. A particle, initially (t -> 0) in the ground state of an infinite, 1D potential box with walls at r 0 and = a, is subjected at time t = 0 to a time-dependent perturbation V (r, t) et/7, with eo a small real number a) Calculate to first order the probability of finding the particle in an excited state for t 0. Consider all final states. Are all possible transitions allowed? b) Examine the time dependence of the...
3. A particle of mass m in a one-dimensional box has the following wave function in the region x-0 tox-L: ? (x.r)=?,(x)e-iEy /A +?,(X)--iE//h Here Y,(x) and Y,(x) are the normalized stationary-state wave functions for the n = 1 and n = 3 levels, and E1 and E3 are the energies of these levels. The wave function is zero for x< 0 and forx> L. (a) Find the value of the probability distribution function atx- L/2 as a function of...
Extra Credit (3 points to Mideterm-2) Q1. A particle is described by the wave function (x) b(a2-x2) for -a sx s a and (x) 0 for x -a and x +a, where a and b are positive real constants. (a) Using the normalization condition, find b in terms a. (b) What is the probability to find the particle at x = +a/2 in a small interval ofwidth 0.01 a ? (c) What is the probability for the particle to be...
1. The wave-functions of the states [4) and (0) are given by y(x) and Q(x), respectively. Derive the expression for the inner product (14) in terms of the wave- functions Q(x) and (x). What is the physical meaning of y(x) and (x)/2? 2. Fig. 1 shows a sketch of y(x). Sketch y(x) such that the states (4) and (o) are orthogonal: (014) = 0. (x) M Figure 1 3. Assume a particle has a wave-function y(x) sketched in Fig. 2....
1. The wave-functions of the states [4) and (0) are given by y(x) and Q(x), respectively. Derive the expression for the inner product (14) in terms of the wave- functions Q(x) and (x). What is the physical meaning of y(x) and (x)/2? 2. Fig. 1 shows a sketch of y(x). Sketch y(x) such that the states (4) and (o) are orthogonal: (014) = 0. (x) M Figure 1 3. Assume a particle has a wave-function y(x) sketched in Fig. 2....
2. Derivation of the Kronig-Penney result: Write the solution for the wave function in the region 0 < r < a as a linear combi nation of sin and cos functions and in the region-b< 0 as a linear combination of the hvperbolic sin and cos functions: ψ(z) Asin Kr +Bcos Kx (0<x <a) = By invoking the continuity and differentiability conditions at a- 0, show that B D and AK - CQ, so essentially two constants, say A and...
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