what is the 2-dimensional phase space density(=1-dimensional particle distribution function) of a simple harmonic oscillator? using...
Estimate the ground-state energy of a one-dimensional simple harmonic oscillator using (50) = e-a-l as a trial function with a to be varied. For a simple harmonic oscillator we have H + jmwºr? Recall that, for the variational method, the trial function (HO) gives an expectation value of H such that (016) > Eo, where Eo is the ground state energy. You may use: n! dH() ||= TH(c) – z[1 – H(r)], 8(2), dx S." arcade an+1 where (x) and...
7 Harmonic oscillator in "energy space" Consider the harmonic oscillator in "energy space", i.e., in terms of the basis of eigenvectors n) of the harmonic oscillator Hamiltonian, with Hn) -hwn1/2)]n). We computed these in terms of wavefunctions in position space, ie. pn(x)-(zln), but we can also work purely in terms of the abstract energy eigenvectors in Dirac notation. PS9.pdf 1. You computed the matrix elements 〈nleln) on an earlier problem set. Now find (nn) for general n,n' 2. Find the...
Question no 6.1, statistical physics by Reif Volume 5 Problems 6.1 Phase space of a classical harmonic oscillator The energy of a one-dimensional harmonic oscillator, whose position coordinate is x and whose momentum is p, is given by where the first term on the right is its kinetic and the second term its potential energy. Here m denotes the mass of the osellating particle and a the spring constant of the restoring force acting on the particle. Consider an ensemble...
Using the properties of the raising and lowering operators for the 1 dimensional simple harmonic oscillator to compute where is an integer and
The most general wave function of a particle in the simple harmonic oscillator potential is: V(x, t) = (x)e-1st/ where and E, are the harmonic oscillator's stationary states and their corresponding energies. (a) Show that the expectation value of position is (hint: use the results of Problem 4): (v) = A cos (wt - ) where the real constants A and o are given by: 1 2 Ae-id-1 " Entichtin Interpret this result, comparing it with the motion of a...
A particle with mass m is in a one-dimensional simple harmonic oscillator potential. At time t = 0 it is described by the state where lo and l) are normalised energy eigenfunctions corresponding to energies E and Ey and b and c are real constants. (a) Find b and c so that (x) is as large as possible. b) Write down the wavefunction of this particle at a time t later c)Caleulate (x) for the particle at time t (d)...
2. Consider a one-dimensional simple harmonic oscillator. Do the following algebraically. 2. Consider a one -dimensional simple harmonic oscillator. Do the following algebraically. a) Construct a linear superposition of I0) and |1) by choosing appropriate phases, such that (x) is as large as possible. i) Suppose the oscillator is in the state la) att 0. Find la) and x) ii) Evaluate the expectation value(x) at t 〉 0. ii Evaluate (A x)2) att > 0. b) Answer the same questions...
A particle with mass m is in a one dimensional simple harmonic oscillator potential. At timet0 it is described by the superposition state where Vo, 1 and Vz are normalised energy eigenfunctions of the harmonic oscillator potential corresponding to energies Eo, E1 and E2 (a) Show that the wavefunction is normalised (b) If an observation of energy is made, what is the most likely value of energy and with what probability would it be obtained? (c) If the experiment is...
Consider a simple single quantum particle with the energy levels of the harmonic oscillator En = (n + 1/2)ℏω. This particle is in thermal contact with a reservoir with temperature T. a) Calculate the partition function of this particle. b) Calculate the internal energy of the particle as a function of temperature. Deduce and interpret the state of this energy at low and high temperatures. c) Calculate the specific temperature of this particle at constant pressure.
Please do this problem about quantum mechanic harmonic oscillator and show all your steps thank you. Q1. Consider a particle of mass m moving in a one-dimensional harmonic oscillator potential. 1. Calculate the product of uncertainties in position and momentum for the particle in 2. Compare the result of (a) with the uncertainty product when the particle is in its the fifth excited state, ie. (OxơP)5. lowest energy state. Q1. Consider a particle of mass m moving in a one-dimensional...