Harmonic oscillator Definition and 699 Threads

Hi, so i am looking at the quantization of the harmonic oscillator and i have the following equation... ψ''+ (2ε-y^{2})ψ=0 I am letting y\rightarrow \infty to get... ψ''- y^{2}ψ=0 It says the solution to this equation in the same limit is... ψ= Ay^{m}e^{\pm y^{2}/2} The positive...

Hello everyone, I've been trying to figure out how to determine bifurcation values in a harmonic oscillator when either the spring constant α or damping coefficient β act as undefined parameters. I understand bifurcations in first-order DEs, but I can't figure them out in a second-order...

I'm not sure I'm in the right forum but I'll try and ask anyways. So I simulated a damped, driven pendulum in Java with the goal of showing period doubling/chaotic behavior. But then, as I was increasing the driving force, i saw the double period born. Then the 4-period...but then suddenly...

Homework Statement A harmonic oscillator is initially in the state \psi (x,0)=Ae^{-\frac{\alpha ^2 x^2}{2}} \alpha x (2\alpha x +i). Where \alpha ^2 =\frac{m \omega}{\hbar}. 1)Find the wavefunction for all t>0. 2)Calculate the probability to measure the values \frac{5\hbar \omega }{2} and...

Starting with the D-dim. harmonic oscillator and generators of SU(D) T^a;\quad [T^a,T^b] = if^{abc}T^c one can construct conserved charges Q^a = a^\dagger_i\,(T^a)_{ik}\,a_k;\quad [Q^a,Q^b] = if^{abc}Q^c satisfying the same algebra and commuting with the Hamiltonian H =...

Hey guys I was just looking over a past homework problem and found something I'm not too sure on - A particle is in the ground state of a Harmonic potential V (x) = 0.5mω2x2 If you measured the energy, what are the possible results, and with what probabilities? Now I know the answer...

Homework Statement Consider the Hamiltonian H=\frac{p^2}{2M}+\frac{1}{2}\omega^2r^2-\omega_z L_z Determine its eigenstates and energies. 2. The attempt at a solution I want to check my comprehension; by eigenstate they mean \psi(r) from the good old H\psi(r)=E\psi(r) and...

Find the expectation value of (px)2, keeping in mind that ψ0(x) = A0e−ax2 where A0 = (2mω0/h)^1/4, and <x2> = ∫x2|ψ|2dx = h_bar / 2mω0 <ψ(x)|px2|ψ(x)> = ∫ψ(x)(pop2)ψ(x) dx pop = [hbar / i] (\delta/\deltax) I'm not going to attempt to type out me solving the integral because it...

Hi everyone! Given that a harmonic oscillator has eigenkstates |n> where n = 1,2,3,..., how can we calculate <X>, <P>, <X^2>, etc. Is there a need to define a wavefunction in the |n> basis? Thanks!

Homework Statement Consider a classical particle in an unidimensional harmonic potential. Let A be the amplitude of the oscillation of the particle at a given energy. Show that the probability to find the particule between x and x+dx is given by P(x)dx=\frac{dx}{\pi \sqrt {A^2-x^2}}. 1)Graph...

Homework Statement A particle is in the ground state of a half harmonic oscillator (V=m/2 w^2 x^2 x>0, and infinity x<0). At t=0, the barrier at x=0 is suddenly removed. Find the possible energy measurements as a function of time and the wavefunction for all times. Homework Equations <H>...

Homework Statement A damped harmonic oscillator is displaced a distance xo from equilibrium and released with zero initial velocity. Find the motion in the underdamped, critically damped, and overdamped case. Homework Equations d2x/dt2 + 2K dx/dt + ω2x = 0 Underdamped: x =...

Homework Statement For the Harmonic Oscillator, the state |ψ> = (|0> + |1>) / √(2) Find \overline{x} = <ψ|x|ψ> \overline{p} = <ψ|p|ψ> \overline{x^2} = <ψ|x^{2}|ψ> and \overline{p^2} = <ψ|p^{2}|ψ> and <ψ| (x - \overline{x})^2 |ψ><ψ| (p - \overline{p})^2 |ψ> [b]2. Homework Equations...

Hello, I am confused about how to show that any two solutions of the damped harmonic oscillator equation equal another solution. Thanks!

hay guys, A three-dimensional harmonic oscillator is in thermal equilibrium with a temperature reservoir at temperature T. Finde The average total energy of the oscillator I have no idea, how can I solve this problem, can you hint me please:rolleyes:

Consider the usual 1D quantum harmonic oscillator with the typical hamiltonian in P and X and with the usual ladder operators defined. i) I have to prove that given a generic wave function \psi , \partial_t < \psi (t) |a| \psi (t)> is proportional to < \psi (t) | a | \psi (t) > and...

Homework Statement A particle of mass m is placed in the ground state of a one-dimensional harmonic oscillator potential of the form V(x)=1/2 kx2 where the stiffness constant k can be varied externally. The ground state wavefunction has the form ψ(x)\propto exp(−ax2\sqrt{k}) where a...

Well what is the partition function of harmonic oscillator with this energy E=hw(n+1/2) , n=1,3,5,... Z=e^(-BE) right? B=1/KT^2 How to expand this? Thank you.

Hi. I want to write the entropy of a 1d harmonic oscillator as a function of energy, but for each energy there is only one possible configuration. So is the entropy zero? I mean, the energy is E=hw(n+1/2), so there is only one microstate for each energy.

Homework Statement What are the stationary states of an isotropic 3D quantum harmonic oscillator in a potential U(x,y,z) = {1\over2}m\omega^2 (x^2+y^2+z^2) in the form \psi(x,y,z)=f(x)g(y)h(z) and how many linearly independent states have energy E=({3\over 2}+n)\hbar\omega? Homework...

Homework Statement A spring is elastically stretched 10 cm if a force of 3 Newtons is imposed. A 2 kg mass is hung from the spring and is also attached to a viscous damper that exerts a restraining force of 3 Newtons when the velocity of the mass is 5 m/sec. An external force time function...

Hi, I'm trying to resolve a problem (17-2) of Pauling's book (Introduction to Quantum Mechanics ), but I'm not achieving this integration. So, I ask for your help. The problem says: Calculate \overline{p_{z}²} (where p_{z} = momentum in z direction and \overline{x} = average value of x...

Homework Statement Let's consider a harmonic oscillator with a harmonic perturbation: H = \frac{p^2}{2} + \frac{x^2}{2} + a \frac{x^2}{2}. Exact solution is known, but we want to derive it using perturbation theory. More specifically, suppose we want to obtain a series for the ground state...

How are Hermite Polynomials related to the solutions to the Schrodinger equation for a harmonic oscillator? Are they the solutions themselves, or are the solutions to the equation the product of a Hermite polynomial and an exponential function? Thanks!

Hello: I am trying to understand how to build a hamiltonian for a general system and figure it is best to start with a simple system (e.g. a harmonic oscillator) first before moving on to a more abstract understanding. My end goal is to understand them enough so that I can move to symplectic...

The book derives the wavefunction for the ground state of a harmonic oscillator. It's found to be a Gaussian with dispersion l = \sqrt{\frac{\hbar}{2m\omega}}. The probability distribution for momentum is found to be Gaussian as well with dispersion \sigma_{p} = \frac{\hbar}{2l}. The following...

the general solution is given by x(t) = Acos(ωt) + Bsin(ωt). Express the total energy in terms of A and B and notice how it is independent of time. my book derives a formula earlier which says \frac{\partial{S_{cl}}}{\partial{t_f}} = -E where S_{cl} is the classical path defined by S_{cl} =...

Homework Statement Compute the partition function Z = Tr(Exp(-βH)) and then the average number of particles in a quantum state <nα > for an assembly of identical simple harmonic oscillators. The Hamiltonian is: H = \sum _{k}[(nk+1/2)\hbar - \mu nk] with nk=ak+ak. Do the calculations once...

Homework Statement The equation for a damped oscillator is d2x/dt2+2βdx/dt +ω02 x = 0. Let ω0=1.0 s−1 and β = 0.54 s−1. The initial values are x(0) = x0 and v(0)=0. Determine x(t)/x0 at t = 2π/ω0. Homework Equations the solution to equation is given by...

Homework Statement The logarithmic decrement δ of a lightly damped oscillator is defined to be the natural logarithm of the ratio of successive maximum displacements (in the same direction) of a free damped oscillator. That is, δ = ln(An/An+1) where An is the maximum displacement of the n-th...

Hi ! There's a lot of information about Harmonic Oscillator.But I'm just a beginner of physics.And my English is not excellent to understand all informations in the Internet.Is there anybody,who can explain me Harmonic Oscillator?

Homework Statement Please take a look at the attachment for the problem statement. Homework Equations For 1 dim Harmonic oscillator, E = (n+1/2)h.omega/2pi I don't know which equation to use for 2 dim The Attempt at a Solution I am unable to solve because I don't know which...

Homework Statement Kindly look at the attachment for the statement. Homework Equations L^2 (psi) = E (psi) The Attempt at a Solution For Part B, I wrote Lx, Ly, Lz in operator form. Thus I get L^2. L^2 (psi) = E (psi) psi = E^-alpha.r^2/2 So I get energy eigenvalue 2 h cross...

Hi I'm having problems with solving this question: a 90.0 kg skydiver hanging from a parachute bounces up and down with a period of 1.50 seconds. What is the new period of oscillation when a second skydiver, whose mass is 60.0 kg, hangs from the legs first? the answer is 1.94 seconds...

I'm sorry if the form of my post does not meet the general requirements(this is the first time i work with any kind of LaTeX) and I promise that my next posts will be more adequate. Right now I am in serious need of someone explaining me this problem, since on the 6th of June I'm supposed to...

I'm solving the 2D harmonic oscillator, numerically. -\frac{1}{2}\left( u_{xx} + u_{yy}\right) + \frac{1}{2}(x^2+y^2)u = Eu The solutions my solver spits out for say, the |01> state, are linear combinations of the form |u\rangle = \alpha_1 |01\rangle + \alpha_2 |10\rangle which is...

Homework Statement Folks, I am looking at a past exam question regarding the Harmonic Oscillator. The question ask 'Justify that the ground state of a harmonic oscillator a_\psi_0=0 equation 2.58 on page 45 of griffiths. THis was not covered in my notes. Any ideas how to justify this...

Why the reciprocal of the damping rate in this model equal to the phonon lifetime? Can somebody give me a detailed exaplanation. Thanks.

Hi there, I just started an intermediate classical mechanics course at university and was smacked upside the head with this question that I don't know how to even start. Homework Statement We are to find the response function of a damped harmonic oscillator given a Forcing function. The...

Homework Statement The period of a macroscopic pendulum made with a mass of 10 g suspended from a massless cord 50 cm long is 1.42 s. (a) Compute the ground state (zero-point) energy. (b) If the pendulum is set into motion so that the mass raises 0.1 mm above its equilibrium position, what will...

Homework Statement http://img191.imageshack.us/i/questionyw.png/ Homework Equations Given in problem The Attempt at a Solution a) I've been able to find expressions of operators x, p_x, y and p_y in terms of the creation/annihilation operators and hence been able to express the...

i have a quick question A particle in ground state of a S.H.O whose potential is given by V_1(X)=\frac{1}{2}mw^2_1x^2 suddenly changes to V_2(X)=\frac{1}{2}mw^2_2(x-x_o)^2 what is the wavefunction going to be like for the new potential? i'd think everything else stays the same in the...

Hello, I want to find <xftf|x(t)|xiti> in harmonic oscillator. I tried to insert the complete set of energy eigenstates to the right and the left side of x(t), but it yields somewhat more complicated stuff. Thank you

Homework Statement a block of mass m=.5kg is sliding on a horizontal table with coefficients of static and kinetic friction of .8 and .5 respectively. It is attached to a wall with a spring of unstretched length l=.13m and force constant 200 n/m. The block is released from rest at t=0 when...

Homework Statement There is a block attached to the wall via a spring. The only damping force is friction, where there is kinetic and static. Homework Equations m(d^2x/dt^2)=-kx-? The Attempt at a Solution I can solve this, except usually the damping force is given as...

Homework Statement I'm talking about the probability density plot of the harmonic oscillator. Is there some physical meaning to be extracted from this? Here's a link that contains the drawing of what I'm talking about http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hosc5.html...

Homework Statement The question states for a harmonic oscillator the wavefunction is: \mu = C*x*exp(-\alphax2/2) it then wants you to find \alpha. using the standard hamiltonian: H = -\hbar/2m d2/dx2 + 1/2 mw2x2 I have differentiated \mu twice and put it into the TISE. for the left hand...

Homework Statement A particle is in the ground state of a harmonic oscillator with classical frequency w. Suddenly the classical frequency doubles, w -> w' = 2w without initially changing the wavefunction. Instantaneously afterwards, what is the probability that a measurement of energy...

Homework Statement We know that a particle in SHM is in a state such that measurements of the energy will yield either E_0 or E_1 (and nothing else), each with equal probability. Show that the state must be of the form \psi = \frac{1}{\sqrt2} \psi_0 + \frac{e^{i \phi}}{\sqrt2} \psi_1 where...

Homework Statement Hi guys. I've been working on this problem for a while, it's starting to frustrate me. "Show that the function of Ѱ=e^(-bx^2) with b=mw/2ħ is a solution and that the corresponding energy is ħw/2." Homework Equations Schrodinger Eqn...