In a 2D harmonic oscillator the single particle energies are:
show that the degeneracy of states should be
In a 2D harmonic oscillator the single particle energies are: show that the degeneracy of states...
Quantum mechanics
Consider a two-dimensional harmonic oscillator
. If
find the energy of the base state until second order in theory of
disturbances and the energies of the first level excited to first
order in
.
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The variational method can be used to solve for the ground state wavefunction and energy of a harmonic oscillator. Using a trail wavefunction of , where the function is defined between . The Hamiltonian operator for a 1D harmonic oscillator is Solving for the wavefunction gives Find that gives the lowest energy and compare from the trial function to the exact value, where coS We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to...
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...
3 Problem Three [10 points] (The Quantum Oscillator) We have seen in class that the Hamiltonian of a particle of a simple Harmonic oscillator potential in one dimension can be expressed in term of the creation and annihilation operators àt and à, respectively, as: or with In >, n = 0,1,..) are the nth eigenstates of the above Hamiltonian. Part A A.1. Show that the energy levels of a simple harmonic oscillator are E,' Aw (nti), n=0, 12, A.2. Calculate...
Solve the harmonic oscillator motion for initial conditions x(0)
= 0, V(0) = V0 in the case of (a) underdamped
(b) overdamped
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8.4 The Two-Dimensional Central-Force Problem The 2D harmonic oscillator is a 2D central force problem (as discussed in TZD Many physical systems involve a particle that moves under the influence of a central force; that is, a force that always points exactly toward, or away from, a force center O. In classical mechanics a famous example of a central force is the force of the sun on a planet. In atomic physics the most obvious example is the hydrogen atom,...
For a spin-1/2 particle in a magnetic field B, with energies and , (a) calculate the partition function. (b) Show that the mean energy of this particle is given by ̅ For a system of noninteracting spins, (c) what is the total partition function and (d) mean energy? We were unable to transcribe this image2 2kT 2 2kT
Quantum Mechanics. Consider a one-dimensional harmonic oscillator of frequency found in the ground state. At a perturbation is activated. Obtain an expression for the expected value of as a function of time using time-dependent perturbation theory. A step by step process is deeply appreciated. The best handwriting possible, please. Thank you very much. We were unable to transcribe this imageWe were unable to transcribe this imageV (t) = Fox cos (at) We were unable to transcribe this image V (t)...
Quantum Mechanics.
Find the energies, degenerations and wave functions for the first
three energy levels (ground state
and first two excited states) of a system of two identical
particles with spin , which move in a
one-
dimensional infinite well of size .
Find corrections of energies to first order in if an
attracting potential of contact
is added.
Show that in the case of "spinless" fermions, the previous
perturbation has no effect.
Step by step process with good handwriting,...
Quantum Mechanics. Find the energies, degenerations and wave functions for the first three energy levels (ground state and first two excited states) of a system of two identical particles with spin , which move in a one- dimensional infinite well of size . Find corrections of energies to first order in if an attracting potential of contact is added. Show that in the case of "spinless" fermions, the previous perturbation has no effect. Step by step process with good handwriting,...