Harmonic Definition and 1000 Threads

I know I've seen this point about roots discussed somewhere but I can't for the life of me remember where. I'm hoping someone can point me the right direction. Here's the situation:- The standard derivation of the quantum HO starts with the classic Hamiltonian in the form H = p2 + q2...

I was hoping that someone could explain why these different equations can be found from different sources please. The time dependent position, x(t), of an underdamped harmonic oscillator is given by: x(t)=e^{-\gamma t}acos(\omega_{1}t-\alpha) where \gamma is the damping coefficient, and...

Homework Statement Hey guys, So I have this equation for the entropy of a classical harmonic oscillator: \frac{S}{k}=N[\frac{Tf'(T)}{f(T)}-\log z]-\log (1-zf(T)) where z=e^{\frac{\mu}{kT}} is the fugacity, and f(T)=\frac{kT}{\hbar \omega}. I have to show that, "in the limit of...

Homework Statement Determine the motion of this mechanical system satisfying the initial conditions :- y1(0) = 1 y2(0) = 2 y1'(0) = -2*sqrt(6) y2'(0) = sqrt(6) Hint : there are 4 different methods you can use to solve this problem. They all give the same exact result. I need to...

Hello, I have been studying Introduction to Quantum Mechanics by Griffith and in a section he solves the Schrodinger equation for a harmonic oscillator potential using the power series method. First he rewrites the shroedinger equation in the form d^2ψ/dε^2 = (ε^2 - K)ψ , where ε=...

I am reading an article on the "energy surface" of a Hamiltonian. For a simple harmonic oscillator, I am assuming this "energy surface" has one (1) degree of freedom. For this case, the article states that the "dimensionality of phase space" = 2N = 2 and "dimensionality of the energy surface" =...

Hello, When I have the differential equation \frac{dY(x)}{dx} = -k^2 Y(x) The solution is of course harmonic oscillation, however, looking at various places I see the solution given as: Y(x) = A cos(kx) + B sin(kx) instead of Y(x) = A cos(kx + \phi_1) + B sin(kx + \phi_2) Isnt...

Homework Statement Tension = 400 N Mass = 4g Length = .96m What is the speed of the wave on a string? What is the frequency of the 3rd harmonic? Homework Equations v=√T/(m/L) v=fλ The Attempt at a Solution v=√400N/(.004kg/.96m) = 310m/s...am I correct? f=v/λ...

If we apply the small angle approximation so that a simple pendulum can be considered to be under going SHM, what kind of potential energy would the pendulum bob be having? If the answer is gravitational potential energy, then we have a contradiction because this would mean that the bob would...

Homework Statement A block with a mass M is located on a frictionless, horizontal surface and is attached to a horizontal spring with spring stiffness k. The block is being pulled out to the right a distance x=x_0 of equilibrium and released at t = 0. At time t_1, corresponding to \omega...

For a mass on a spring (vertical set up) why is potential energy U defined as 1/2 kx^2? This is just the elastic potential energy. Shouldn't it be U = 1/2 kx^2 + mgh? Both the elastic AND potential energy? Also, for a simple pendulum at a very low amplitude, the potential energy is all...

For a mass on a spring (vertical set up) undergoing SHM, we equate the restoring force, -kx, to -ω^2 x, coming to a conclusion that ω = \sqrt{\frac{k}{m}}. My question is, is the restoring force |mg - T| Where T is the tension in the spring? Because this seems to be the net force. I am used to...

Simple harmonic motion (Again) :( This is not a question about a problem it is more about the position of a simple harmonic oscillator as a function of time:) I went through it in a lecture yesterday and found using the energy in simple harmonic motion to bex(t)= A cos(ωt +φ) Which is fine...

I have a question about the derivation that I have attached! I understand that both KE and U are 1/2 kA^2 So how is it that the two combine is also equal to 1/2kA^2Not sure if I'm missing something but I'm a little confused :(

A harmonic oscillator with frequency ω is in its ground state when the stiffness of the spring is instantaneously reduced by a factor f2<1, so its natural frequency becomes f2ω. What is the probability that the oscillator is subsequently found to have energy 1.5(hbar)f2ω? Thanks

Homework Statement Consider a one-dimensional linear harmonic oscillator perturbed by a Gaussian perturbation H' = λe-ax2. Calculate the first-order correction to the groundstate energy and to the energy of the first excited state Homework Equations ψn(x) = \frac{α}{√π*2n*n!}1/2 *...

Homework Statement This is a required analysis for my Physics II lab. We recorded the motion of an object oscillating on spring, and are asked to use the slopes of different graphs that we plotted using the data collected in lab in order to find the spring constant (k). Both graphs were...

Homework Statement Let's consider a Tokamak with major radius R=1m and minor radius a=0.3m, magnetic field B=5T with a deuterium plasma with central density 10^{20}m^{-3}, central temperature 1keV and parabolic temperature and density profiles \propto (1-r^2/a^2) a) Find the electronic...

Homework Statement Find the period of low-amplitude vertical vibrations of the system shown. The mass of the block is m. The pulley hangs from the ceiling on a spring with a force constant k. The block hangs from an ideal string...

Shouldn't the integrating factor be ##exp(\frac{m\omega x}{\hbar})##? \frac{\partial <x|0>}{\partial x} + \frac{m\omega x}{\hbar} <x|0> = 0 This is in the form: \frac{\partial y}{\partial x} + P_{(x)} y = Q_{(x)} Where I.F. is ##exp (\int (P_{(x)} dx)##

Homework Statement We want to prepare a particle in state ##\psi ## under following conditions: 1. Let energy be ##E=\frac{5}{4}\hbar \omega ## 2. Probability, that we will measure energy greater than ##2\hbar \omega## is ##0## 3. ##<x>=0## Homework Equations The Attempt at a...

What is a good way to memorize that ## \omega = \sqrt{\dfrac{k}{m}} ## ? I always confuse it with: ## T = 2\pi \sqrt{\dfrac{m}{k}}## , and can never tell them apart. (i guess part of it is that I'm not too familiar with it yet)

Homework Statement Potential energy of electron in harmonic potential can be described as ##V(x)=\frac{m\omega _0^2x^2}{2}-eEx##, where E is electric field that has no gradient. What are the energies of eigenstates of an electron in potential ##V(x)##? Also calculate ##<ex>##. HINT: Use...

Homework Statement One dimensional harmonic oscillator is at the beginning in state with wavefunction ##\psi (x,0)=Aexp(-\frac{(x-x_0)^2}{2a^2})exp(\frac{ip_0x}{\hbar })##. What is the expected value of full energy? Homework Equations ##<E>=<\psi ^{*}|H|\psi >=\sum \left | C_n \right |^2E_n##...

Here is the question: I have posted a link there to this thread so the OP can view my work.

Homework Statement A 1.15-kg mass oscillates according to the equation x = .650cos(8.40t) where x is in meters and t in seconds. Determine a)the amplitude, b)the frequency, c) the total energy of the system, and d) the kinetic and potential energy when x = 0.360m. Homework Equations x...

Homework Statement A solid wooden cylinder of radius r and mass M. It's weighted at one end so that it floats upright in calm seawater, having density ρ the buoy is pulled down a distance x from it's equilibrium position and released. a- Show that the block will undergo s.h.m b- Determine...

I have doubts of how can I put my frame of reference in a simple harmonic motion vertical spring. Normally the books choose the origin in the equilibrium position and the positive distance (x>0) downward, and in this conditions Newton´s second law is: ma=-kx; but instead of putting the positive...

I need help for the second part please. How do the oscillations being damped prevent them from being S.H.M. as well the other 2 points? I need an explanation so I can understand. The oscillations would still have the same time period eve if they are damped. Low frequency due to...

Homework Statement The masses in figure slide on a frictionless table.m1 ,but not m2 ,is fastened to the spring.If now m1 and m2 are pushed to the left,so that the spring is compressed a distance d,what will be the amplitude of the oscillation of m1 after the spring system is released...

Homework Statement A system consists of a spring with force constant k = 1250 N/m, length L = 1.50m, and an object of mass m = 5.00kg attached to the end. The object is placed at the level of the point of attachment with the spring unstretched, at position yi= L, an then is released so that it...

Homework Statement I need to show that for an eigen state of 1D harmonic oscillator the expectation values of the position X is Zero. Homework Equations Using a+=\frac{1}{\sqrt{2mhw}}(\hat{Px}+iwm\hat{x}) a-=\frac{1}{\sqrt{2mhw}}(\hat{Px}-iwm\hat{x}) The Attempt at a Solution...

A system exhibits simple harmonic motion with a frequency of 0.85 cycles per second. Calculate the acceleration experienced by the mass 3.0 m from the equilibrium This question seems simple, I haven't really tried anything cause I can't figure out how to start. I need help to start this...

Homework Statement Show that the energy of a simple harmonic oscillator in the n = 2 state is 5Planck constantω/2 by substituting the wave function ψ2 = A(2αx2- 1)e-αx2/2 directly into the Schroedinger equation, as broken down in the following steps. First, calculate dψ2/dx, using A for A, x...

Homework Statement The terminal speed of a freely falling object is v_t (assume a linear form of air resistance). When the object is suspended by a spring, the spring stretches by an amount a. Find the formula of the frequency of oscillation in terms of g, v_t, and a. Homework Equations the...

Homework Statement A harmonic oscillator oscillates with an amplitude A. In one period of oscillation, what is the distance traveled by the oscillator? Homework Equations I'm not sure which equation applies if any? The Attempt at a Solution My guess was 2A but the answer was 4A...

So we know that SHM can be described as: x(t) = Acos(ωt + ϕ) v(t) = -Aω sin(ωt + ϕ) a(t) = -Aω^2 cos(ωt + ϕ) it can be easily said that the max acceleration (in terms of magnitude) a SHM system can achieve is Aω^2 In Damped Harmonic Motion we know that: x(t) = (A)(e^(-bt/2m))cos(ωt + ϕ) given...

Homework Statement Given a quantum harmonic oscillator, calculate the following values: \left \langle n \right | a \left | n \right \rangle, \left \langle n \right | a^\dagger \left | n \right \rangle, \left \langle n \right | X \left | n \right \rangle, \left \langle n \right | P \left | n...

Homework Statement ) Two uniform, solid cylinders of radius R and total mass M are connected along their common axis by a short, massless rod. They are attached to a spring with force constant k using a frictionless ring around the axle. If the spring is pulled out and released, the cylinders...

An "attempt frequency" for a harmonic oscillator? Homework Statement What is the "attempt frequency" for a harmonic oscillator with bound potential as the particle goes from x = -c to x = +c? What is the rate of its movement from -c to +c? Homework Equations v...

Homework Statement Two points, each of charge Q, are fixed at either end of a frictionless rod of length 2R. Another point charge, of charge q (not Q) is free to move along the rod. Show that if charge q is displaced a small distance x (0<x<<R) from the centre of the rod, it will undergo...

I am a little confused with this subject. If you have a mass hanging from a spring, there is a specific equilibrium point, but what if you apply a force downwards on the mass, will this have an effect on the equilibrium position or will it remain the same? thanks!

Hello, I have two question regrading sound waves. The first one: The pressure P(x;t) at a point x at time t in a medium through which a harmonic wave is travelling can be described by: P(x,t) = Asin(wt -kx) If the equation describes a pressure wave traveling in air, with amplitude 2 Pa and...

Homework Statement So with pendulums in SHM, in my A level physics textbook (AQA Physics A), it shows a pendulum that has been displaced from equilibrium. It says that the restoring force is provided by the object's weight. Why isn't the restoring force provided by the tension in the string...

Homework Statement It's not a direct question, but it's an implied part of a larger question: can classical waves experience simple harmonic oscillator potentials, like a mass on a spring does? Homework Equations The Attempt at a Solution I'm thinking no, since I can't come up...

Well I was going through class lecture notes and my professor wrote this When x = A(the maximum value), v=0: E=1/2kA^2 When v = wA, x=0: E=1/2mw^2A^2 where w = omega, A = amplitude, k = spring constant, m = mass, v = velocity and apparently both equations are equal, i would like to...

Let a+,a- be the ladder operators of the harmonic oscillators. In my book I encountered the hamiltonian: H = hbarω(a+a-+½) + hbarω0(a++a-) Now the first term is just the regular harmonic oscillator and the second term can be rewritten with the transformation equations for x and p to the...

I have a couple of questions about what total harmonic distortion is, and what the measurement means. The definition I've read most places is: \frac{D}{S} × 100% , where S is the amplitude of the fundamental frequency, and D is the amplitude of the sum of all of the harmonics. A common...

Homework Statement A system is made of N 1D simple harmonic oscillators. Show that the number of states with total energy E is given by \Omega(E) = \frac{(M+N-1)!}{(M!)(N-1)!} Homework Equations Each particle has energy ε = \overline{h}\omega(n + \frac{1}{2}), n = 0, 1 Total energy is...

Homework Statement After four cycles the amplitude of a damped harmonic oscillator has dropped to 1/e of it's initial value. Find the ratio of the frequency of this oscillator to that of it's natural frequency (undamped value) Homework Equations x'' +(√k/m) = 0 x'' = d/dt(dx/dt)...