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Quantum Mechanics Problem 1. (25) Consider an infinite potential well with the following shape: 0 a/4...
Consider an infinite well for which the boltom is not lat, as sketched here. I the slope is small, the potential V = er/a may be considered as a per- turbation on the square-well potential over-a/2 < x < a/2. vox) a/2 -a/2 (a) Calculate the ground-state energy, correct to first order in perturbation theory. (b) Calculate the energy of the first excited state, correct to first order in perturbation theory. (c) Calculate the wave function in the ground state,...
Quantum Mechanics question about an infinite square well. A particle in an infinite square well potential has an initial state vector 14() = E1) - %|E2) where E) is the n'th eigenfunctions of the Hamiltonian operator. (a) Find the time evolution of the state vector. (b) Find the expectation value of the position as a function of time.
Q1) Consider 2.dimensional infinite "well" with the potential otherwise The stationary states are ny = (a) sin ( x) sin (y,) The corresponding energies are n) , 123 Note that the ground state, ?11 is nondegenerate with the energy E00)-E1)-' r' Now introduce the perturbation, given by the shaded region in the figure ma AH,-{Vo, if 0<x otherwise y<a/2 (a) What is the energy of the 1.st excited state of the unperturbed system? What is its degree of degeneracy,v? (b)...
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...
4.2 The potential energy in a MOFSET device near the metal oxide interface is approximately V(x) - qEx forx > 0 where q is the electron charge, and E is the electric field strength. Use the variational technique to estimate the ground state energy of an electron in this configuration. (Hints: a) use the un-normalized trial function ф(x)-x exp(-ax2)). b) Find the normalized trial wave-function c) Compute the energy functional (i.e. the expectation value of the Hamiltonian for the state...
4. 3D Infinite Square Well Perturbation (20 pts) Consider the three-dimensional infinite cubic well: otherwise The stationary states are where n, ny, and n, are integers. The corresponding allowed energies are Now let us introduce the perturbation otherwise a) Find the first-order correction to the ground state energy b) Find the first-order correction to the first excited state 4. 3D Infinite Square Well Perturbation (20 pts) Consider the three-dimensional infinite cubic well: otherwise The stationary states are where n, ny,...
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...
4. This problems deals with sudden approximation, please do not use perturbation theory. (a) (1 point) The system is described by the Hamiltonian t < 0, Ho+V, t>0 - and neither Ho nor V depends on time. Assuming that the system was in the eigenstate n) of Ho at t < 0, find the probability to find the system in the eigenstate V) of Ho V at t>0. (b) (5 points) Consider a particle in the ground state an infinite...
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...
Consider a quantum mechanical system with 4 states and an unperturbed Hamiltonian given by 1 0 0 0 Ho E0 0 2 0 a small perturbation is added to this Hamiltonian 0 0 1 0 where e is much smaller than E a) [10pts] What are the energy eigenvalues of the unperturbed system of the following states? 1 o 2o 0 and which energy levels are degenerate? b) [10pts Find a good basis for degenerate perturbation theory instead of c)...