Oscillator Definition and 1000 Threads

Hello all, I've breadboarded a simple astable multivibrator to generate 38kHz (identical to http://upload.wikimedia.org/wikipedia/commons/6/6a/Transistor_Multivibrator.svg) using 2N3904s. Upon power up, it's a solid 38Khz. After 60 seconds, it's drifted up to ~44Khz... and stabilizes to...

Homework Statement Consider a time-dependent harmonic oscillator with Hamiltonian \hat{H}(t)=\hat{H}_0+\hat{V}(t) \hat{H}_0=\hbar \omega \left( \hat{a}^{\dagger}\hat{a}+\frac{1}{2} \right) \hat{V}(t)=\lambda \left( e^{i\Omega t}\hat{a}^{\dagger}+e^{-i\Omega t}\hat{a} \right) (i)...

Homework Statement The equation \psi(x) = \frac{1}{sqrt(2)}\psi_0 (x) + \frac{i}{sqrt(5)}\psi_1 (x) + \gamma\psi_2 (x) is a combination of the first three eigenfunctions in the 1D harmonic oscillator. So, \psi_0 = Ae^{-mωx^2 /2\hbar} and so on for the first and second excited states. If...

I am doing page 4 of the lab attached, and I am trying to simulate the relaxation oscillator in Figure 2. However, I am not getting a square wave like the lab suggests I should get. I have been able to do the derivation at the bottom of the page. I have attached my multisim diagram as well as...

Homework Statement The pendulum of a grandfather clock activates an escapement mechanism every time it passes through the vertical. The escapement is under tension (provided by a hanging weight) and gives the pendulum a small impulse a distance l from the pivot. The energy transferred by...

Hello, my book explains detailed the proofs of these three formulas: y = Asin(ωt + φo) v = ωAcos(ωt + φo) a = -ω²Asin(ωt + φo) Where a is acceleration, v is velocity, ω is angular velocity, A is amplitude. The book uses the following figures: Figure a) -->...

Hi guys. Recentely I'm approaching Quantum Mechanics starting from the mathematical basics. In order to understand the benefit of representing a certain matrix in its eigenvectors basis my book makes the example I attached ( Principles of Quantum Mechanics by Shankar ). Using matrix form it...

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 know amplitude doesn't depend on mass but I came across a problem where the amplitude changed with a change in mass. The problem was a horizontal frictionless spring with a box of mass m1 attached to it with a stone of mass m2 inside of the box. At its equilibrium point at its maximum...

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 Take as a trial wavefunction for the hydrogen atom the 3D oscillator ground state wavefunction ψ(r) = N exp (-br^2 / 2). Calculate the value of b that gives the best energy and calculate this energy. Homework Equations Radial part of ∇^2 = 1/r2 (∂/∂r) (r^2 ∂/∂r)...

I came up with the following idea of a device: We have a short circuited type-2 (to allow more current) superconducting solenoid. A current is flowing through the solenoid, and it creates magnetic induction inside, B1. Inside the solenoid we place a piece of type-1 superconductor. B1<critical...

For a damped mechanical oscillator, the energy of the system is given by $$E = \frac{1}{2}m \dot{x}^2 + \frac{1}{2}k x^2$$ where ##k## is the spring constant. From there, I've seen it dictated that the average kinetic energy ##\langle T \rangle ## is half of the total energy of the system. This...

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" =...

Hey guys I'm new to the forum and having a little trouble with this conceptual problem. 1. A block of mass m is connected to a spring, the other end of which is fixed. There is also a viscous damping mechanism. The following observations have been made of this system: i) If the block is...

Hello, I have a triangle wave generator working nicely with video op amps but I am trying to substitute much slower op amps. I am required to get it to oscillate at 30 kHz. I performed calculations and got it to about 29.8kHz you can see it attached for the AD829's I tried substituting 741...

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 *...

Hello folks, I've been trying to understand how crystals work in crystal oscillator circuits. I understand the piezoelectric effect to the following extent: If we apply an electric field to the crystal it will deform and when the field is removed, the crystal will generate an electric field...

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...

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##...

I has read the link http://physics.stackexchange.com/questions/75411/frequency-of-small-oscillation-of-particle-under-gravity-constrained-to-move-in and, i don't unsderstand why the integral \int^1_0 \frac{dy}{\sqrt{1+y^4}} is proportional to \Gamma (5/4) / \Gamma (3/4) . I had read the...

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...

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...

Hi, I'm considering this set up: A vessel with a curved (parabolic?) inner surface is rotating at an angular speed ω. Two heavy balls are placed near the axis of rotation of the vessel. Due to centrifugal force, the balls move outwards towards the edge. This increases the moment of...

Hi everyone, long time lurker, first time poster. I've just begun a phd which involves nanoribbons (a small strip of a 2D material connected at either end to a larger 'bulk' section of the same 2D material). A question has occurred to me. These nanoribbons look a lot like a piece of string...

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...

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...

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...

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...

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)...

Hi, in this article: http://dx.doi.org/10.1016/S0021-9991(03)00308-5 damped molecular dynamics is used as a minimization scheme. In formula No. 9 the author gives an estimator for the optimal damping frequency: Can someone explain how to find this estimate? best, derivator

For a set of energy eigenstates |n\rangle then we have the energy eigenvalue equation \hat{H}|n\rangle = E_{n}|n\rangle. We also have a commutator equation [\hat{H}, \hat{a^\dagger}] = \hbar\omega\hat{a}^{\dagger} From this we have \hat{a}^{\dagger}\hat{H}|n\rangle =...

Homework Statement A particle is in the ground state of a simple harmonic oscillator, potential → V(x)=\frac{1}{2}mω^{2}x^{2} Imagine that you are in the ground state |0⟩ of the 1DSHO, and you operate on it with the momentum operator p, in terms of the a and a† operators. What is the...

Just one short question about something I didn't understand in my book: "At low temperaturs only vibration modes where hω<kT will be excited to any appreciable extent. The excitation of these modes will be approimately classical each with an energy close to kT." I don't understand the last...

Homework Statement Find the eigenvalues of the following Hamiltonian. Ĥ = ħwâ^{†}â + \alpha(â + â^{†}) , \alpha \in |RHomework Equations â|\phi_{n}>=\sqrt{n}|\phi_{n-1}> â^{†}|\phi_{n}>=\sqrt{n+1}|\phi_{n+1}> The Attempt at a Solution By applying the Hamiltonian to a random state n I...

Hello The energy of harmonics oscillator, started of U=-\frac{\partial}{\partial \beta} \ln Z is equal to \frac{\hbar \omega}{2} + \frac{\hbar \omega}{exp(\beta \hbar \omega)-1}. At high temperature, i could say that exp (\beta \hbar \omega ) \approx 1 + (\beta \hbar \omega ), and then...

Homework Statement Okay, I am trying to solve this Anharmonic Oscillator equation. Now I am given with the potential U=(1/2)x^2-(1/4)x^4 and Kinetic energy T=(1/2)x' ^2 So the Lagrangian becomes \mathcal L=T-U Now I have taken all the k's and m to be 1 Homework Equations...

Homework Statement The question is from Sakurai 2nd edition, problem 3.21. (See attachments) ******* EDIT: Oops! Forgot to attach file! It should be there now.. *******The Attempt at a Solution Part a, I feel like I can do without too much of a problem, just re-write L as L=xp and then...

Homework Statement A force Fext(t)=F0[1-e(-a*t)] acts, for time t>0, on an oscillator which is at rest at x=0 at time 0. The mass is m; the spring constant is k; and the damping force id -b dx/dt. The parameters satisfy these relations: b=mq and k=4mq2 where q is a constant with units...

Homework Statement (A) The damped oscillator is described by the equation mx''+bx'+kx=0. What is the condition for critical damping expressed in terms of m,b,k. Assume this is satisfied. (B) For t<0 the mass is at rest (x=0). This mass is set in motion at t=0 by a sharp impulsive force so...

Homework Statement The problem is long so I will post the whole thing but ask only for help on part C. The steady-state motion of a damped oscillator driven by an applied force F0 cos(ωt) is given by xss(t) = A cos(ωt + φ). Consider the oscillator which is released from rest at t = 0...

Homework Statement Given (\mathcal{L} + k^2)y = \phi(x) with homogeneous boundary conditions y(0) = y(\ell) = 0 where \begin{align} y(x) & = \frac{2}{\ell}\sum_{n = 1}^{\infty} \frac{\sin(k_nx)}{k^2 - k_n^2},\\ \phi(x) & = \frac{2}{\ell}\sum_{n = 1}^{\infty}\sin(k_nx),\\ u_n(x) &=...

Homework Statement The generalized Green function is $$ G_g(x, x') = \sum_{n\neq m}\frac{u_n(x)u_n(x')}{k_m^2 - k_n^2}. $$ Show G_g satisfies the equation $$ (\mathcal{L} + k_m^2)G_g(x, x') = \delta(x - x') - u_m(x)u_m(x') $$ where \delta(x - x') = \frac{2}{\ell}\sum_{n =...

Given \((\mathcal{L} + k^2)y = \phi(x)\) with homogeneous boundary conditions \(y(0) = y(\ell) = 0\) where \begin{align} y(x) & = \frac{2}{\ell}\sum_{n = 1}^{\infty} \frac{\sin(k_nx)}{k^2 - k_n^2},\\ \phi(x) & = \frac{2}{\ell}\sum_{n = 1}^{\infty}\sin(k_nx),\\ u_n(x) &=...

Homework Statement Which of the following statements about the harmonic oscillator (HO) is true? a) The depth of the potential energy surface is related to bond strength. b) The vibrational frequency increases with increasing quantum numbers. c) The HO model does not account for bond...