1. Basis Function (40 pts)
Basis function is a useful mathematical tool often used in quantum
mechanics. We learned in class that using the basis function,
Schrödinger equation in the form of
Homework Answers
Add Answer to:
1. Basis Function (40 pts) Basis function is a useful mathematical tool often used in quantum mec...
1 Schrödinger Equation At the stationary regime (without temporal dependence), the wave function of a stationary...
1 Schrödinger Equation At the stationary regime (without temporal dependence), the wave function of a stationary quantum particle confined in a rectangular box of sides a, b and c satisfies the Schrödinger equation (1) h2 Vay = EU (1) 2m where E is the energy at the stationary state of the particle and one of vertices is located at the coordinated origin. It is required that the wave function is annulled at the borders of the box. (a) Using this...
1 Schrödinger Equation At the stationary regime (without temporal dependence), the wave function of a stationary...
1 Schrödinger Equation At the stationary regime (without temporal dependence), the wave function of a stationary quantum particle confined in a rectangular box of sides a, b and c satisfies the Schrödinger equation (1) h2 Vay = EU (1) 2m where E is the energy at the stationary state of the particle and one of vertices is located at the coordinated origin. It is required that the wave function is annulled at the borders of the box. (a) Using this...
5. The z transform is a very useful tool for studying difference equations. Often difference and ...
5. The z transform is a very useful tool for studying difference equations. Often difference and differential equations are used to describe causal systems and only the causal solution is of interest. This is the "initial condition" problem of a differential equations course. But both difference and differential equations describe more than just the causal system. For instance, "backwards" solutions and "two point boundary value" solutions. One way in which to think about the problem is the ROC of the...
1. (10 pts) There are problems a) and b) a)(4 pts) We often refer to a...
1. (10 pts) There are problems a) and b) a)(4 pts) We often refer to a turbulence model as "complete" or "incomplete". Explain what makes a model "complete". Is the one equation model Spalart-Allmaras model complete? b)(6 pts) Turbulence dissipation, e, is found to be related to length scale, L. and turbulent kinetic energy, k. Using dimensional analysis to obtain the relation in € Cof(k.) where Cp is a constant for dissipation.
MATLAB MATLAB MATLAB Model description The logistic map is a function that is often used to...
MATLAB MATLAB MATLAB Model description The logistic map is a function that is often used to model population growth. It is defined by P(t+1) = rP(t) (1 -P(t)/K) Here, P(t) represents the density of a population at year 1, the parameter r is a growth rate and the parameter K is the maximum possible population density (known as the carrying capacity). This equation says that if we know the density at one year, we can substitute it into the right-hand...
In this optional assignment you will find the eigenfunctions and eigenenergies of the hydrogen atom using an operator method which involves using Supersymmetric Quantum Mechanics (SUSY QM). In t...
In this optional assignment you will find the eigenfunctions and eigenenergies of the hydrogen atom using an operator method which involves using Supersymmetric Quantum Mechanics (SUSY QM). In the SUSY QM formalism, any smooth potential Vx) (or equivalently Vr)) can be rewritten in terms of a superpotential Wix)l (Based upon lecture notes for 8.05 Quantum Krishna Rajagopal at MIT Physics II as taught by Prof Recall that the Schroedinger radial equation for the radial wavefunction u(r)-r Rfr) can be rewritten...
1. (40 points) Write down the phasor representation for each of the following signals a. (5 pts) ...
1. (40 points) Write down the phasor representation for each of the following signals a. (5 pts) x, () 12cos(21) b. pts) x(1)-20cos(3t+45) c. (5 pts)x,(t)-10sin(21 + 300) d. (5 pts) x,(t)-35cos(20+22 sinQ2 +45") e. (10 pts) )- d 135cos(21)+22 sin(2i+4')leas ao express the result in the time domain. f (10 pts) ,)-[B5cos(21)+22sin(2r +45°ydr. Please also express the result in the time domain. Notes: Phasors are ideal for steady-state analysis of sinusoidal signals at a fixed frequency, so we cannot...
Quantum Mechanics Problem 1. (25) Consider an infinite potential well with the following shape: 0 a/4...
Quantum Mechanics Problem
1. (25) Consider an infinite potential well with the following shape: 0 a/4 3al4 a h2 where 4 Using the ground state wavefunction of the original infinite potential well as a trial function, 2πχ trial = 1-sin- find the approximation of the ground state energy for this system with the variational method. (Note, this question is simplified by considering the two components of the Hamiltonian, and V, on their own) b) If we had used the 1st...
Special Problem (20 pts) Consider an undoped AljGa7As/GaAs/ Al3Ga7As quantum well (QW) of width W...
Special Problem (20 pts) Consider an undoped AljGa7As/GaAs/ Al3Ga7As quantum well (QW) of width W-15 nm. (a) Due the quantum mechanical confinement in the quantum well, the lowest energy states of in the conduction band is no longer the conduction band edge, but the CB edge plus the confined state energy (particle in the box problem), where the confinement energy relative to the CB edge is given by the solutions for infinite barriers where n-1,2,.is the quantum number, n-1 is...
(40 pts) 2a. Show that u(z) is the solution to the problem where k(x)-1 for x < 1/2 and k = 2 for...
(40 pts) 2a. Show that u(z) is the solution to the problem where k(x)-1 for x < 1/2 and k = 2 for x > 1 /2. 2b. Set up the weak form for the differential equation above and the resulting element stiffness and element load vector and calculate the element stiffness matrix and load vector for 4 quadratic elements by using the Gaussian quadrature that is going to exactly calculate the integrals Then set up the global K and...
1 Schrödinger Equation At the stationary regime (without temporal dependence), the wave function of a stationary quantum particle confined in a rectangular box of sides a, b and c satisfies the Schrödinger equation (1) h2 Vay = EU (1) 2m where E is the energy at the stationary state of the particle and one of vertices is located at the coordinated origin. It is required that the wave function is annulled at the borders of the box. (a) Using this...
1 Schrödinger Equation At the stationary regime (without temporal dependence), the wave function of a stationary quantum particle confined in a rectangular box of sides a, b and c satisfies the Schrödinger equation (1) h2 Vay = EU (1) 2m where E is the energy at the stationary state of the particle and one of vertices is located at the coordinated origin. It is required that the wave function is annulled at the borders of the box. (a) Using this...
5. The z transform is a very useful tool for studying difference equations. Often difference and differential equations are used to describe causal systems and only the causal solution is of interest. This is the "initial condition" problem of a differential equations course. But both difference and differential equations describe more than just the causal system. For instance, "backwards" solutions and "two point boundary value" solutions. One way in which to think about the problem is the ROC of the...
1. (10 pts) There are problems a) and b) a)(4 pts) We often refer to a turbulence model as "complete" or "incomplete". Explain what makes a model "complete". Is the one equation model Spalart-Allmaras model complete? b)(6 pts) Turbulence dissipation, e, is found to be related to length scale, L. and turbulent kinetic energy, k. Using dimensional analysis to obtain the relation in € Cof(k.) where Cp is a constant for dissipation.
In this optional assignment you will find the eigenfunctions and eigenenergies of the hydrogen atom using an operator method which involves using Supersymmetric Quantum Mechanics (SUSY QM). In the SUSY QM formalism, any smooth potential Vx) (or equivalently Vr)) can be rewritten in terms of a superpotential Wix)l (Based upon lecture notes for 8.05 Quantum Krishna Rajagopal at MIT Physics II as taught by Prof Recall that the Schroedinger radial equation for the radial wavefunction u(r)-r Rfr) can be rewritten...
1. (40 points) Write down the phasor representation for each of the following signals a. (5 pts) x, () 12cos(21) b. pts) x(1)-20cos(3t+45) c. (5 pts)x,(t)-10sin(21 + 300) d. (5 pts) x,(t)-35cos(20+22 sinQ2 +45") e. (10 pts) )- d 135cos(21)+22 sin(2i+4')leas ao express the result in the time domain. f (10 pts) ,)-[B5cos(21)+22sin(2r +45°ydr. Please also express the result in the time domain. Notes: Phasors are ideal for steady-state analysis of sinusoidal signals at a fixed frequency, so we cannot...
Quantum Mechanics Problem
1. (25) Consider an infinite potential well with the following shape: 0 a/4 3al4 a h2 where 4 Using the ground state wavefunction of the original infinite potential well as a trial function, 2πχ trial = 1-sin- find the approximation of the ground state energy for this system with the variational method. (Note, this question is simplified by considering the two components of the Hamiltonian, and V, on their own) b) If we had used the 1st...
Special Problem (20 pts) Consider an undoped AljGa7As/GaAs/ Al3Ga7As quantum well (QW) of width W-15 nm. (a) Due the quantum mechanical confinement in the quantum well, the lowest energy states of in the conduction band is no longer the conduction band edge, but the CB edge plus the confined state energy (particle in the box problem), where the confinement energy relative to the CB edge is given by the solutions for infinite barriers where n-1,2,.is the quantum number, n-1 is...
(40 pts) 2a. Show that u(z) is the solution to the problem where k(x)-1 for x < 1/2 and k = 2 for x > 1 /2. 2b. Set up the weak form for the differential equation above and the resulting element stiffness and element load vector and calculate the element stiffness matrix and load vector for 4 quadratic elements by using the Gaussian quadrature that is going to exactly calculate the integrals Then set up the global K and...
Most questions answered within 3 hours.
-
The definition of a partition
function involved a sum over all states, and knowledge of the...
asked 8 minutes ago -
Which of the following events would be considered Systematic?
(Choose all that apply.)
A. The Federal...
asked 18 minutes ago -
Questions of this part analyze the students’ understanding in
analyzing the given scenario and practical skills...
asked 24 minutes ago -
#C++
Write a function, named sortByCourseNumber, that takes a
reference to a vector of string (like...
asked 24 minutes ago -
A
cylunder of titanium (length = 5.00cm, diameter = 1.20 cm) has a
mass of 25.5...
asked 1 hour ago -
1) Answer Each question with 1-2 Sentences:
a) How can we use the continuity and change...
asked 2 hours ago -
5. What is the decimal equivalent of the largest binary
integer that can be obtained with...
asked 1 hour ago -
Why is the magnitude of the Type Ia supernova always the same
(-19.6)?
asked 2 hours ago -
5. You are a quality control engineer and you are
asked to analyze the lifetime (in hours)...
asked 2 hours ago -
How many ways can 2 people be chosen and arranged in a straight
line if there...
asked 3 hours ago -
(Problem 1d). Leandro has 16 hours per day that he can allocate
to work or leisure....
asked 4 hours ago -
Morganton Company
makes one product and it provided the following information to help
prepare the master...
asked 4 hours ago