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zone 1 Consider the following piecewise continuous, finite potential energy: ro; x < -a V(x)={-U, ;...
Consider the symmetrical finite square well potential shown below. U(x) = 46 eV for xs-L/2 U(x)...
Consider the symmetrical finite square well potential shown below. U(x) = 46 eV for xs-L/2 U(x) 0 eV for-L/2 < x < L/2 U(x) 46 eV for x 2 L/2 L-0.27mm Note: 46 ev 1. the width L is unchanged from the infinite well you previously considered 2, the potential outside x-±L/2 is finite with U-46 eV. 3. you found the three lowest energy levels for that infinite -8.135 0.135 potential well were: 5.16 ev, 20.64 ev, and46.45 ev. 1)...
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** +...
[Finite potential well] Consider a symmetric square well potential of a finite depth, i.e., V(x) =...
[Finite potential well] Consider a symmetric square well potential of a finite depth, i.e., V(x) = 0 inside the well, V(x) = V outside the well. NOTE: for a general discontinuous potential the boundary conditions are the continuity of both the wave function and its first derivative at the point(s) of the discontinuity of the potential y (x_)=y(x),y'(x_)=y'(x4) (i) What are the functional forms of the solutions for y(x) inside and outside the well? (ii) What are the explicit continuity...
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...
Consider a potential well defined as U(x) = for x < 0, U(x) = 0 for 0 < x < L, and U(x) = U0 > 0 for x > L (see the following figure).
Consider a potential well defined as \(U(x)=\infty\) for \(x<0, U(x)=0\)for \(0<x<L,\)and \(U(x)=U_{0}>0\) for \(x>L\) (see the following figure). Consider a particle with mass \(m\) and kinetic energy \(E<U_{0}\)that is trapped in the well. (a) The boundary condition at the infinite wall ( \(x=\)0) is \(\psi(x)=0\). What must the form of the function \(\psi(x)\) for \(0<x<L\)be in order to satisfy both the Schrödinger equation and this boundary condition? (b) The wave function must remain finite as \(x \rightarrow \infty\). What must...
(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...
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...
Consider a particle in a 1-d well with potential V(x) =-U for-d < x < d,...
Consider a particle in a 1-d well with potential V(x) =-U for-d < x < d, and V(z) 0 elsewhere. We will use the variational wave function v(z) = A(b + r), t(x)-A(b-x), -b < r < 0, 0 < x < b, to show that a bound state exists for any U0. a) Normalize the wave function. Find the expectation values of the kinetic and potential energies b) Show that for sufficiently large b, with b> d, the expectation...
Instead of assuming that a one-dimensional particle has no energy (v(x)=0), consider the case of a...
Instead of assuming that a one-dimensional
particle has no energy (v(x)=0), consider the case of a
one-dimensional particle which has finite, but constant, energy
V(x)= V sub zero.. Show that the ID particle in a box wave
functions. n(x)= A sin ((pi n x)/a). Also solve the Schrödinger
equation for this potential, and determine the energies En
Problem 2: Particle in a Box with Non-Zero Energy (2 points) Instead of assuming that a one-dimensional particle has no energy (V(x) =...
Consider the finite difference matrix operator for the 1D model problem u(/d2- f(x) on domain [0,...
Consider the finite difference matrix operator for the 1D model problem u(/d2- f(x) on domain [0, 1] with boundary conditions u(0) = 0 and u(1) = 0, given by [-2 1 1-2 1 E RnXn h2 1 -2 1 This matrix can be considered a discrete version of the continuous operator d/da2 that acts upon a function(r). (a) Show that the n eigenvectors of A are given by the vectors ) (p-1,... , n) with components and with eigenvalues h2...
Consider the symmetrical finite square well potential shown below. U(x) = 46 eV for xs-L/2 U(x) 0 eV for-L/2 < x < L/2 U(x) 46 eV for x 2 L/2 L-0.27mm Note: 46 ev 1. the width L is unchanged from the infinite well you previously considered 2, the potential outside x-±L/2 is finite with U-46 eV. 3. you found the three lowest energy levels for that infinite -8.135 0.135 potential well were: 5.16 ev, 20.64 ev, and46.45 ev. 1)...
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** +...
[Finite potential well] Consider a symmetric square well potential of a finite depth, i.e., V(x) = 0 inside the well, V(x) = V outside the well. NOTE: for a general discontinuous potential the boundary conditions are the continuity of both the wave function and its first derivative at the point(s) of the discontinuity of the potential y (x_)=y(x),y'(x_)=y'(x4) (i) What are the functional forms of the solutions for y(x) inside and outside the well? (ii) What are the explicit continuity...
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...
(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...
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...
Consider a particle in a 1-d well with potential V(x) =-U for-d < x < d, and V(z) 0 elsewhere. We will use the variational wave function v(z) = A(b + r), t(x)-A(b-x), -b < r < 0, 0 < x < b, to show that a bound state exists for any U0. a) Normalize the wave function. Find the expectation values of the kinetic and potential energies b) Show that for sufficiently large b, with b> d, the expectation...
Instead of assuming that a one-dimensional
particle has no energy (v(x)=0), consider the case of a
one-dimensional particle which has finite, but constant, energy
V(x)= V sub zero.. Show that the ID particle in a box wave
functions. n(x)= A sin ((pi n x)/a). Also solve the Schrödinger
equation for this potential, and determine the energies En
Problem 2: Particle in a Box with Non-Zero Energy (2 points) Instead of assuming that a one-dimensional particle has no energy (V(x) =...
Consider the finite difference matrix operator for the 1D model problem u(/d2- f(x) on domain [0, 1] with boundary conditions u(0) = 0 and u(1) = 0, given by [-2 1 1-2 1 E RnXn h2 1 -2 1 This matrix can be considered a discrete version of the continuous operator d/da2 that acts upon a function(r). (a) Show that the n eigenvectors of A are given by the vectors ) (p-1,... , n) with components and with eigenvalues h2...
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