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1. This problem is all about the variational method, as applied to the particle in a...
09 Estimate the ground state energy and wavefunction for a particle in a box using the...
09 Estimate the ground state energy and wavefunction for a particle in a box using the variational method with the following trial wavefunction, where N is the normalization constant and ß is a variational parameter that should be minimized. 14) = N exp(-Bx2) (7.6) 1. Is this a good trial wavefunction for this approximation (justify your answer)? 2. Why is this not a good wavefunction? 3. Can you solve this problem both analytically and numerically? Pay careful attention to limits...
3. The predecessor to Hartree-Fock was the Hartree method, where the main difference is that the ...
Questions 3-5
3. The predecessor to Hartree-Fock was the Hartree method, where the main difference is that the Hartree-Fock method includes an trial wavefunction by writing it as a Slater Determinant, while the Hartree method uses a simple product wavefunction that does not capture anti- symmetry. In particular, for a minimal-basis model of, the Hartree method's trial wavefunction is given in the while the Hartree-Fock trial wav is given by where and are molecular orbitals, and and coordinates of electron...
1. Variational method In this problem, you will approximate the ground state wave function of a...
1. Variational method In this problem, you will approximate the ground state wave function of a quantum system using the variational theory. Use the trial wave function below 2 cos/T) , 1x1 trial a/2 to approximate the ground state of a harmonic oscillator given by 2.2 2 using a as an adjustable parameter. (a) Calculate the expectation value for the kinetic energy, (?) trial 4 points (b) Calculate the expectation value for the potential energy, Virial. Sketch ??tria, (V)trial, and...
Particle in a box Figure 1 is an illustration of the concept of a particle in...
Particle in a box Figure 1 is an illustration of the concept of a particle in a box. V=00 V=00 V=0 Figure 1. A representation of a particle in a box, where the potential energy, V, is zero between x = 0 and x = L and rises abruptly to infinity at the walls. The Schrödinger equation for a particle in a box reads t² d²u Y +V(x)y = Ey 2m dx2 + (1) where ħ=h/21 , y represents the...
(b) What is the general form of the solution to the Schroedinger Equation where a) E>U...
(b) What is the general form of the solution to the Schroedinger Equation where a) E>U and b) E<U 2 marks] A particle is confined in a 1-D box of length L such that U=0 for 0〈x〈 and U is infinite elsewhere. Find an expression for the wavefunction inside the box which meets these boundary conditions, and show that it satisfies the Schroedinger equation. 2 marks] If the potential outside the box is Uo> O instead of infinite, describe with...
6. The Particle in a Box problem refers to a potential energy function called the infinite square...
6. The Particle in a Box problem refers to a potential energy function called the infinite square well, aka the box: ; x < 0 (Region I) V(x) = 0 : 0 L (Region II) x x >L (Region III) Let's investigate a quantum particle with mass m and energy E in this potential well of length L We were unable to transcribe this image6d (continued) write down an equation relating ψ, (x = 0) to ψ"(x I and II....
3. A particle is in a 1D box (infinite potential well) of dimension, a, situated symmetrically ab...
3. A particle is in a 1D box (infinite potential well) of dimension, a, situated symmetrically about the origin of the x-axis. A measurement of energy is made and the particle is found to have the ground state energy: 2ma The walls of the box are expanded instantaneously, doubling the well width symmetrically about the origin, leaving the particle in the same state. a) Sketch the initial potential well making it symmetric about x - 0 (note this is different...
2. The unational method is an incredibly simple but surprisingly powerful method for understanding the low- energy beha...
2. The unational method is an incredibly simple but surprisingly powerful method for understanding the low- energy behavior of quantu systems. It is used constantly in marny-body physics and in quantum chemistry. The main idea is thst for any physical Hamiltonian, there is a lowest energy state, i.e. the ground state Ipo). All other states (ignoring degeneracy) have higher energy that this one. Therefore we have Therefore, to get an upper bound on the energy of Eo, it suffices to...
Problem 3: Time-Independent Perturbation Theory Consider the particle in a 1D box of size L, as...
Problem 3: Time-Independent Perturbation Theory Consider the particle in a 1D box of size L, as in Fig. 3. A perturbation of the form. V,δ ((x-2)2-a2) with a < L is applied to the unperturbed Hamiltonian of the 1D particle in a box (solutions on the equation sheet). Here V is a constant with units of energy. Remember the following propertics of the Dirac delta function m,f(x)6(x-a)dx f(a) 6(az) が(z) = = ds( dz E, or Ψ(x)-En 10 0.0 0.2...
1. (50 points) Consider the particle in a one-dimensional box (0 s x S L). Assume a term is added to the Hamiltonian of the form: πχ V(x)g sin Sketch the potential and the expected eigenfunction (...
1. (50 points) Consider the particle in a one-dimensional box (0 s x S L). Assume a term is added to the Hamiltonian of the form: πχ V(x)g sin Sketch the potential and the expected eigenfunction (small g). In the limit of small g, find the second order correction to the ground state energy 2. (50 points) For a diatomic molecule rotating in free space, the Hamiltonian may be written: 12 21 Where L is the total angular momentum operator,...
09 Estimate the ground state energy and wavefunction for a particle in a box using the variational method with the following trial wavefunction, where N is the normalization constant and ß is a variational parameter that should be minimized. 14) = N exp(-Bx2) (7.6) 1. Is this a good trial wavefunction for this approximation (justify your answer)? 2. Why is this not a good wavefunction? 3. Can you solve this problem both analytically and numerically? Pay careful attention to limits...
Questions 3-5
3. The predecessor to Hartree-Fock was the Hartree method, where the main difference is that the Hartree-Fock method includes an trial wavefunction by writing it as a Slater Determinant, while the Hartree method uses a simple product wavefunction that does not capture anti- symmetry. In particular, for a minimal-basis model of, the Hartree method's trial wavefunction is given in the while the Hartree-Fock trial wav is given by where and are molecular orbitals, and and coordinates of electron...
1. Variational method In this problem, you will approximate the ground state wave function of a quantum system using the variational theory. Use the trial wave function below 2 cos/T) , 1x1 trial a/2 to approximate the ground state of a harmonic oscillator given by 2.2 2 using a as an adjustable parameter. (a) Calculate the expectation value for the kinetic energy, (?) trial 4 points (b) Calculate the expectation value for the potential energy, Virial. Sketch ??tria, (V)trial, and...
Particle in a box Figure 1 is an illustration of the concept of a particle in a box. V=00 V=00 V=0 Figure 1. A representation of a particle in a box, where the potential energy, V, is zero between x = 0 and x = L and rises abruptly to infinity at the walls. The Schrödinger equation for a particle in a box reads t² d²u Y +V(x)y = Ey 2m dx2 + (1) where ħ=h/21 , y represents the...
(b) What is the general form of the solution to the Schroedinger Equation where a) E>U and b) E<U 2 marks] A particle is confined in a 1-D box of length L such that U=0 for 0〈x〈 and U is infinite elsewhere. Find an expression for the wavefunction inside the box which meets these boundary conditions, and show that it satisfies the Schroedinger equation. 2 marks] If the potential outside the box is Uo> O instead of infinite, describe with...
6. The Particle in a Box problem refers to a potential energy function called the infinite square well, aka the box: ; x < 0 (Region I) V(x) = 0 : 0 L (Region II) x x >L (Region III) Let's investigate a quantum particle with mass m and energy E in this potential well of length L We were unable to transcribe this image6d (continued) write down an equation relating ψ, (x = 0) to ψ"(x I and II....
3. A particle is in a 1D box (infinite potential well) of dimension, a, situated symmetrically about the origin of the x-axis. A measurement of energy is made and the particle is found to have the ground state energy: 2ma The walls of the box are expanded instantaneously, doubling the well width symmetrically about the origin, leaving the particle in the same state. a) Sketch the initial potential well making it symmetric about x - 0 (note this is different...
2. The unational method is an incredibly simple but surprisingly powerful method for understanding the low- energy behavior of quantu systems. It is used constantly in marny-body physics and in quantum chemistry. The main idea is thst for any physical Hamiltonian, there is a lowest energy state, i.e. the ground state Ipo). All other states (ignoring degeneracy) have higher energy that this one. Therefore we have Therefore, to get an upper bound on the energy of Eo, it suffices to...
Problem 3: Time-Independent Perturbation Theory Consider the particle in a 1D box of size L, as in Fig. 3. A perturbation of the form. V,δ ((x-2)2-a2) with a < L is applied to the unperturbed Hamiltonian of the 1D particle in a box (solutions on the equation sheet). Here V is a constant with units of energy. Remember the following propertics of the Dirac delta function m,f(x)6(x-a)dx f(a) 6(az) が(z) = = ds( dz E, or Ψ(x)-En 10 0.0 0.2...
1. (50 points) Consider the particle in a one-dimensional box (0 s x S L). Assume a term is added to the Hamiltonian of the form: πχ V(x)g sin Sketch the potential and the expected eigenfunction (small g). In the limit of small g, find the second order correction to the ground state energy 2. (50 points) For a diatomic molecule rotating in free space, the Hamiltonian may be written: 12 21 Where L is the total angular momentum operator,...
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