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4.) A spherical square well is a square well in three spatial dimensions which satisfies -Vo,...
Question 2: finite square well in three dimensions 12 marks *Please note: in PHYS2111 we have...
Question 2: finite square well in three dimensions 12 marks *Please note: in PHYS2111 we have not discussed multi-dimensional systems, but please keep in mind that in order to answer this question all you need is the knowledge about a particle moving in one dimension in a finite square well. Consider a particle of mass m moving in a three-dimensional spherically symmetric square-well potential of radius a and depth V. (see also figure on pag. 3): V(r) = { S-Vo...
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 one-dimensional well with one impenetrable wall. The potential energy is given by 0 x...
Consider a one-dimensional well with one impenetrable wall. The potential energy is given by 0 x < 0 V(x) = { -V. 0 < x < a 10 x > a We showed in the homework that the allowed energies for the eigenstates of a bound particle (E < 0) in this potential well satisfy the transcendental function -cotĚ = 16 - 52 $2 where 5 = koa, and ko = V2m(Vo + E)/ħ, and 5o = av2mV /ħ (a)...
tthe-independent Help: The operator expression dimensions is given by H 2m r ar2 [2] A particle of mass m is in a three-dimensional, spherically symmetric harmonic oscillator potential given by V...
tthe-independent Help: The operator expression dimensions is given by H 2m r ar2 [2] A particle of mass m is in a three-dimensional, spherically symmetric harmonic oscillator potential given by V(r)2r2. The particle is in the I-0 state. Noting that all eigenfunetions must be finite everywhere, find the ground-state radial wave-function R() and the ground-state energy. You do not have to nor oscillator is g (x) = C x exp(-8x2), where C and B are constants) harmonic malize the solution....
4 Spherical harmonics in action Consider a particle in three dimensions in a state (2, y,...
4 Spherical harmonics in action Consider a particle in three dimensions in a state (2, y, z) = A2 + y + 2z)e-ar where a > 0 is a constant and A is a normalization factor. (a) Write the spherical harmonics Y0+ in terms of x,y,z, and r. (b) Using your result from part (a), show that can be written as a product of a purely r-dependent wavefunction and a linear combination of spherical harmonics.
The one-dimensional infinite potential well can be generalized to three dimensions.
(25 marks) The one-dimensional infinite potential well can be generalized to three dimensions. The allowed energies for a particle of mass \(m\) in a cubic box of side \(L\) are given by$$ E_{n_{p} n_{r, n_{i}}}=\frac{\pi^{2} \hbar^{2}}{2 m L^{2}}\left(n_{x}^{2}+n_{y}^{2}+n_{z}^{2}\right) \quad\left(n_{x}=1,2, \ldots ; n_{y}=1,2, \ldots ; n_{z}=1,2, \ldots\right) $$(a) If we put four electrons inside the box, what is the ground-state energy of the system? Here the ground-state energy is defined to be the minimum energy of the system of electrons. You...
2. Calculate th first order energy shift for the first three states of the infinite square...
2. Calculate th first order energy shift for the first three states of the infinite square for 0-x-L. well in one dimension due to a ramp-shaped perturbation: V(r)- Use the following unperturbed eigenstates for the square well: Solution: The first order corrections for each state is given by E -(vn) 1, n°)), and there- fore: Sinn what happens to the sin4? 2 Vo L2 LL14
1. Consider a 1D finite square well potential defined as follows. Vo-a<x<a V(x) = 0otherwise a)...
1. Consider a 1D finite square well potential defined as follows. Vo-a<x<a V(x) = 0otherwise a) What are the energy eigenfunctions n of the Hamiltonian for a single particle bound in this potential? You may write your answer in piece-wise form, with an arbitrary normalization. b) Derive the characteristic equation that the energy eigenvalues E, must satisfy in order to satisfy the eigenvalue equation Hy,-EnUn for eigen function Un c) Write a computer program1 to find the eigenvalues E, for...
4. (20 points) Infinite Wells in Three Dimensions a) Consider a three dimensional in- finite rect...
4. (20 points) Infinite Wells in Three Dimensions a) Consider a three dimensional in- finite rectangular well for which L -L, Ly-2L, ald L2-3L. In terms of quantum numbers (e.g. nz, ny, and n.), M. L, and ћ. write down an expression for the energies of all quantum states. (b) Find the energies of the ground state and the first three lowest lying energies. As in part (b), for each energy level, give the quantum numbers n, ny, n and...
1) (60 points) The ground state of the hydrogen atom: In three dimensions, the radial part...
1) (60 points) The ground state of the hydrogen atom: In three dimensions, the radial part of the Schrodinger equation appropriate for the ground state of the hydrogen atom is given by: ke2 -ħ2 d2 (rR) = E(rR) 2me dr2 where R(r) is a function of r. Here, since we have no angular momentum in the ground state the angular-momentum quantum number /=0. (a) Show that the function R(r) = Ae-Br satisfies the radial Schrodinger equation, and determine the values...
Question 2: finite square well in three dimensions 12 marks *Please note: in PHYS2111 we have not discussed multi-dimensional systems, but please keep in mind that in order to answer this question all you need is the knowledge about a particle moving in one dimension in a finite square well. Consider a particle of mass m moving in a three-dimensional spherically symmetric square-well potential of radius a and depth V. (see also figure on pag. 3): V(r) = { S-Vo...
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 one-dimensional well with one impenetrable wall. The potential energy is given by 0 x < 0 V(x) = { -V. 0 < x < a 10 x > a We showed in the homework that the allowed energies for the eigenstates of a bound particle (E < 0) in this potential well satisfy the transcendental function -cotĚ = 16 - 52 $2 where 5 = koa, and ko = V2m(Vo + E)/ħ, and 5o = av2mV /ħ (a)...
tthe-independent Help: The operator expression dimensions is given by H 2m r ar2 [2] A particle of mass m is in a three-dimensional, spherically symmetric harmonic oscillator potential given by V(r)2r2. The particle is in the I-0 state. Noting that all eigenfunetions must be finite everywhere, find the ground-state radial wave-function R() and the ground-state energy. You do not have to nor oscillator is g (x) = C x exp(-8x2), where C and B are constants) harmonic malize the solution....
4 Spherical harmonics in action Consider a particle in three dimensions in a state (2, y, z) = A2 + y + 2z)e-ar where a > 0 is a constant and A is a normalization factor. (a) Write the spherical harmonics Y0+ in terms of x,y,z, and r. (b) Using your result from part (a), show that can be written as a product of a purely r-dependent wavefunction and a linear combination of spherical harmonics.
2. Calculate th first order energy shift for the first three states of the infinite square for 0-x-L. well in one dimension due to a ramp-shaped perturbation: V(r)- Use the following unperturbed eigenstates for the square well: Solution: The first order corrections for each state is given by E -(vn) 1, n°)), and there- fore: Sinn what happens to the sin4? 2 Vo L2 LL14
1. Consider a 1D finite square well potential defined as follows. Vo-a<x<a V(x) = 0otherwise a) What are the energy eigenfunctions n of the Hamiltonian for a single particle bound in this potential? You may write your answer in piece-wise form, with an arbitrary normalization. b) Derive the characteristic equation that the energy eigenvalues E, must satisfy in order to satisfy the eigenvalue equation Hy,-EnUn for eigen function Un c) Write a computer program1 to find the eigenvalues E, for...
4. (20 points) Infinite Wells in Three Dimensions a) Consider a three dimensional in- finite rectangular well for which L -L, Ly-2L, ald L2-3L. In terms of quantum numbers (e.g. nz, ny, and n.), M. L, and ћ. write down an expression for the energies of all quantum states. (b) Find the energies of the ground state and the first three lowest lying energies. As in part (b), for each energy level, give the quantum numbers n, ny, n and...
1) (60 points) The ground state of the hydrogen atom: In three dimensions, the radial part of the Schrodinger equation appropriate for the ground state of the hydrogen atom is given by: ke2 -ħ2 d2 (rR) = E(rR) 2me dr2 where R(r) is a function of r. Here, since we have no angular momentum in the ground state the angular-momentum quantum number /=0. (a) Show that the function R(r) = Ae-Br satisfies the radial Schrodinger equation, and determine the values...
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