Harmonic oscillator Definition and 743 Threads

Homework Statement A harmonic oscillator with a vertical mass on a string has a hanging mass of 2m and a spring constant of K. It oscillates with an amplitude of Z. When its position is at a distance Z/2 of the equilibrium point, its potential energy is Ui. What is the maximum kinetic energy...

Homework Statement Show that the underdamped oscillator solution can be expressed as x(t)=x_{0}e^{-γt}[cos(Ω't+((v_{o}+γx_{o})/(x_{o}Ω')sinΩ't] and demonstrate by direct calculation that x(0)=x_{o} and \dot{x}(0)=v_{o} Homework Equations The underdamped oscillator solution is...

In the context of normal modes, what is a free mode? When the whole system is in motion?

Hey, I'm doing a vacation scholarship at my university where I am helping a masters student with some of his research. We have a 3x3 lattice of coupled oscillators which we have determined the Hamiltonian of and applied the squeeze operator. We constructed a 18x18 conical Hamiltonian...

Homework Statement I was wondering if there was a general method for finding a function that fits a set of data for a damped harmonic oscillator I'm currently writing up a presentation on the experiment for the gravitational constant and the way i did the experiment was to use a torsion...

Homework Statement A simple harmonic oscillator with mass m = 1/2 and k = 2 is initially at the point x = √3 when it is projected towards the origin with speed 2. Find the equation of motion describing x(t). Homework Equations x=Asin(ωt+θ) The Attempt at a Solution At t=0...

I have an interesting problem I have come across in my research. It results in the differential equation as follows: x''+2γ(x')^\nu+\omega_{o}^2x=g(t) Primes indicate the derivative with respect to t. \gamma and \omega are constants. The non-linearity comes from the first derivative x'...

Homework Statement A particle is in a region with the potential V(x) = κ(x2-l2)2 What is the approximate ground state energy approximation for small oscillations about the location of the potential's stable equilibrium? Homework Equations ground state harmonic oscillator ~ AeC*x2...

Homework Statement Consider an harmonic oscillator with time-dependent frequency as: \omega (t)=\omega_0 * \exp^{- \lambda t} Find the time dependence of the ground state energy of this oscillator for \lambda << 1 situation. Homework Equations H=H_{0} + V(t) H_{0} = \frac{p^2}{2m} +...

Hello fellow computer physics nerds, I'm trying to write a program to plot the positions of the three particles connected by two springs (one dimensional) in Fortran 90. I have a main program block and a module that calls a PGPLOT. My problem is that the positions of the second and third...

I REALLY need help with this one guys! As of right now I believe I only need help with just the set up of the problem. The rest is just solving a differential equation and I assume the frequencies they want will just pop out. Homework Statement Two identical springs and two identical...

Homework Statement A particle with mass m moves in 3-dimensions in the potential V(x,y,z)=\frac{1}{2}m\omega^{2}x^{2}. What are the allowed energy eigenvalues?Homework Equations The Attempt at a Solution The Hamiltonian is given by H=\frac{P^{2}}{2m}+\frac{1}{2}m\omega^{2}X^{2} where P is the...

Hi everybody, This is my first post in this forum although I started following it some time ago. My question is related to rotational properties involving harmonic oscillator model. Homework Statement We are told to evaluate the expectation value of the rotational constant of a...

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...

Homework Statement "Vibrational spectroscopic studies of HCl show that the radiation absorbed in a transition has frequency 8.63*10^13 Hz. Calculate the vibrational frequency of the molecule in this transition." Homework Equations E_n=(n+1/2)hv v=(1/(2pi))(sqrt(k/μ)) The Attempt...

Homework Statement To test the resiliency of its bumper during low-speed collisions, a 1000 kg automobile is driven into a brick wall. The car's bumper behaves like a spring with a force constant 5.00 x 106 N/m and compresses 3.16cm as the car is brought to rest. What was the speed of the car...

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!

Homework Statement Consider a capacitor consisting of two metal plates with a charge +Q on one plate and −Q on the other. In the gap of the capacitor we have two perfectly harmonic springs attached to the top plate—one with a H atom and the other with a H ion attached to the end of the...

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...

Homework Statement A damped harmonic oscillator is being forced. I have to say whether it is direct forcing or forcing by displacement. I have the equation of motion which is expressed in terms of the particle's height above the equilibrium point and an expression for the force being...

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...