Oscillator Definition and 1000 Threads

The equations I'm getting when I solve the differential equations seem to imply that the amplitude of oscillation does not vary in time. For example, if I have x'' + ω02x = cos(ωt) If we suppose that ω≠ω0, then the general solution should look something like: x(t) = c1cos(ω0t) + c2sin(ω0t)...

Homework Statement consider any damped harmonic oscillator equation m(d2t/dt2 +bdy/dt +ky=0 a. show that a constant multiple of any solution is another solution b. illustrate this fact using the equation (d2t/dt2 +3dy/dt +2y=0 c. how many solutions to the equation do you get uf you use this...

Hi, why there is only odd eigenfunctions for a 1/2 harmonic oscillator where V(x) does not equal infinity in the +ve x direction but for x<0 V(x) = infinity. I understand that the "ground state" wave function would be 0 as when x is 0 V(x) is infinity and therefore the wavefunction is 0, and...

Well I numerically solved for the potential V(x)=x^4, the period: \begin{equation} T = \sqrt{8m} \int_0^a \frac{dx}{\sqrt{V(a) - V(x)}} \end{equation} where a was the amplitude of the oscillation and m the mass of the particle. Nevertheless, what I was asked to plot was the above period T(a)...

Homework Statement Due to the radial symmetry of the Hamiltonian, H=-(ħ2/2m)∇2+k(x^2+y^2+z^2)/2 it should be possible to express stationary solutions to schrodinger's wave equation as eigenfunctions of the angular momentum operators L2 and Lz, where...

Homework Statement Part d) of the question below. Homework Equations We are told NOT to use the ladder technique to find the position operator as that's not covered until our Advanced Quantum Mechanics module next year (I don't even know this technique anyway). I emailed my tutor and he...

Homework Statement The wave function for the three dimensional oscillator can be written ##\Psi(\mathbf r) = Ce^{-\frac{1}{2}(r/r_0)^2}## where ##C## and ##r_0## are constants and ##r## the distance from the origen. Calculate a) The most probably value for ##r## b) The expected value of ##r##...

So this is something that troubled me a bit- in Shankar's PQM, there's an exercise that asks you to find the position expectation value for the harmonic oscillator in a state \psi such that \psi=\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle) Where |n\rangle is the n^{th} energy eigenstate of...

Homework Statement verify that Ψ(x) = ( 1/a√π)½ exp(-(x2/2a2)) is a solution to the TISE for linear harmonic oscillator. Where a = √(hbar/mw). and V(x) = ½ mw2x2. Homework Equations HΨ=EΨ E_n = (n+½)hbar*wThe Attempt at a Solution I've started by differentiating the wave function twice to...

Homework Statement For a 1-dimensional simple harmonic oscillator, the Hamiltonian operator is of the form: H = -ħ2/2m ∂xx + 1/2 mω2x2 and Hψn = Enψn = (n+1/2)ħωψn where ψn is the wave function of the nth state. defining a new function fn to be: fn = xψn + ħ/mω ∂xψn show that fn is a...

For a 1D QHO we are given have function for ##t=0## and we are asked for expectation and variance of P at some time t. ##|\psi>=(1/\sqrt 2)(|n>+|n+1>)## Where n is an integer So my idea was to use Dirac operators ##\hat a## and ##\hat a^\dagger## and so I get the following solution ##<\hat...

Homework Statement The problem is attached Homework Equations f=2π/ω=2π√(m/k) The Attempt at a Solution My idea is that the mass doubles resulting in a √2 increase in the equation above. However, apparently the answer is (c). I have a strong feeling the book answer is wrong, but I wanted to...

Homework Statement Gain & phase response of TL072 op-amp Homework Equations 'Frequency of oscillation was observed to be about 15 % lower'[/B] Why ? The Attempt at a Solution so at this frequency the op-amp has 45 phase shift, but why will this impact the oscillator and lower the...

Homework Statement Homework EquationsThe Attempt at a Solution I'm not really sure how to go about this. They tell me the hint, and to use the simplification. I assume when they say (1+y)^n they take y to be in general, anything. It was confusing at first to see a y in the format when no y...

Homework Statement Solve for the steady-statem otion of a forced oscillator if the forcing is F_0*sin(wt) instead of F_0*cos(wt). Use complex representations. Homework Equations e^(i*x) = cos(x)+isin(x) The Attempt at a Solution I assume, First, steady-state means without damping. Next, we...

Homework Statement A particle with a mass(m) of 0.500kg is attached to a horizontal spring with a force constant(k) of 50.0N/m. At the moment t=0, the particle has its maximum speed of 20m/s and its moving to the left. Find the minimum time interval required for the particle to move from...

When I work out $$b^+b$$, I get $$\widehat{b^+} \widehat{b} = \frac{1}{2} (ξ - \frac{d}{dξ})(ξ + \frac{d}{dξ}) = \frac{1}{2} (ξ^2 - \frac{d^2}{dξ^2}) = \frac{mωπx^2}{h} - \frac{h}{4mωπ} \frac{d^2}{dx^2}$$ So base on what I have about, (9) should be $$(9) = \frac{hω}{2π} (\frac{1}{2}...

Homework Statement A simple pendulum consists of a mass m suspended by a ball to a yarn (massless) of length l. We neglect friction forces. Give the list of every forces applied to this system and then the motion of equation. Why is the following equations necessary to find the motion...

Homework Statement I am trying to obtain the hermite polynomial from the schrødinger equation for a har monic oscillator. My attempt is shown below. Thank you! The derivation is based on this site: http://www.physicspages.com/2011/02/08/harmonic-oscillator-series-solution/ The Attempt at a...

Homework Statement In the exercise, we solved the 2D Harmonic Oscillator in kartesian (x,y) and polar (r,φ) coordinates. We found out that both have the same energy levels, but they look very different, when I plot them. What am I missing? The polar solution seems more like it. Homework...

Homework Statement Consider a linear harmonic oscillator with the solution defined by the ladder operators a and a†. Use the number basis |n⟩ to do the following. a) Construct a linear combination of |0⟩ and |1⟩ to form a state |ψ⟩ such that ⟨ψ|X|ψ⟩ is as large as possible. b) Suppose that...

Hi i want to find propagator of inverse harmonic oscillator to find time dependent wave function, but I can't have any ideas about this. Is it possible to help me to find it Thanks

hy physics forum, I am doing an advanced qm course, but i still have some doubts about the correct formalism and so on.. so i have this problem of a one-dimensional harm. oscillator including 2 bosons. the hamiltonian would thus be ## \hat{H}=1/2 (\hat{p_1}^2+\hat{p_2}^2+\omega...

Homework Statement A linear harmonic oscillator with frequency ω = hbar / M is at time t = 0 in the state described by the wave-function: Ψ(x,0) = C( 1 + √2x) e-x2/2 Determine the values of energy which can be measured in this state. I'm not really sure where to start this question and was...

Homework Statement Consider the one-dimensional harmonic oscillator of frequency ω0: H0 = 1/2m p2 + m/2 ω02 x2 Let the oscillator be in its ground state at t = 0, and be subject to the perturbation Vˆ = 1/2 mω2xˆ2 cos( ωt )at t > 0. (a) Identify the single excited eigenstate of H0 for...

I am wondering why LC tank only oscillates at resonant frequency not other frequencies? Is there a physics explanation for that?

Homework Statement Show that for the one-dimensional linear harmonic oscillator the Hamiltonian is: [; H = \frac{1}{2}[P^2+\omega ^2 X^2]-\frac{1}{2}\omega \hbar ;] [; =\frac{1}{2}[P+i\omega X][P-i\omega X]+\frac{1}{2} \omega \hbar ;] where P, X are the momentum and position operators...

It is known R4=R5=1.2k. My task is to calculate amplitude of oscillations at v0: Opamps and diodes are ideal. I did analysis of circuit without amplitude stabilization and I got oscillation frequency, but I can't figure out how diodes stabilize amplitude in this circuit. Any sugestion...

Not a textbook/homework problem so I'm not using the format (hopefully that's ok). Can someone offer an explanation of normal modes and how to calculate the degrees of freedom in a system of coupled oscillators? From what I've seen the degrees of freedom seems to be equal to the number of...

Homework Statement I'm given a driven, dampened harmonic oscillator (can it be thought of as a spring-mass system with linear friction?) Is it possible to solve the equation of motion using Lagrangian mechanics? I could solve it with the usual differential equation x''+βx'+ωₒ²x=fₒcos(ωt) but as...

Hello. Here is my task: In oscillator circuit ideal opamps are used, it is known R=10K, C=10nF. a) Calculate frequency of oscillations, b) If Vcc=Vee=15V, calculate amplitude of voltages in points A, B and C. a) Here is my solution. I cut output loop, applied test generator and found loop gain...

Homework Statement An object of mass 0.2kg is hung from a spring whose spring constant is 80N/m. The object is subject to a resistive force given by -bv, where v is it's velocity in meters per second. If the damped frequency is √(3)/2 of the undamped frequency, what is the value of b...

Homework Statement Homework Equations http://www.maths.manchester.ac.uk/~gajjar/MATH44011/notes/44011_note4.pdf The Attempt at a Solution I obtain i) and ii), but I can't understand the way to study iii) and iv), particularly to choose the rigth paths.[/B]

Homework Statement At t=0 the wave function of a two-dimensional isotropic harmonic oscilator is ψ(x,y,0)=A(4α^2 x^2+2αy+4α^2 xy-2) e^((-α^2 x^2)/2) e^((-α^2 y^2)/2) where A its the normalization constant In which instant. Wich values of total energy can we find and which probability...

Homework Statement See attached image. Match the type of system or situation to the appropriate energy level diagram. 1) hadronic (such as +) 2) idealized quantized spring-mass oscillator 3) nuclear (such as the nucleus of a carbon atom) 4) vibrational states of a diatomic molecule such as O2...

Homework Statement For a quantum harmonic oscillator in an electric field, using ##\hat{V}=q\epsilon\hat{x}##, with the following trial state: $$|\psi\rangle=|0\rangle+b|1\rangle$$ Show that the energy can be written as $$E=\frac{\frac{\hbar...

Homework Statement The Hamiltonian is given by: H = \frac{1}{2} \sum_{i=1,2}[p_i^2 + q_i^2] We define the following operators: J = \frac{1}{2} (a_1^+ a_1 + a_2^+ a_2) J_1 = \frac{1}{2} (a_2^+ a_1 + a_1^+ a_2) J = \frac{i}{2} (a_2^+ a_1 - a_1^+ a_2) J = \frac{1}{2} (a_1^+ a_1 - a_2^+...

I have a question that if we use external quartz crystal in Microcontroller and connect it to battery... Crystal gives regular and very accurate electrical frequency... But it should get mechanical vibration from somewhere to give electrical pulses... From where it receives mechanical vibrations?

The ground state wave-function of a 1-D harmonic oscillator is $$ \psi(x) = \sqrt\frac{a}{\sqrt\pi} * exp(-\frac{a^2*x^2}{2}\frac{i\omega t}{2}). $$ a) find Average potential energy ? $$ \overline{V} = \frac{1}{2} \mu\omega^2\overline{x^2} $$ b) find Average kinetic energy ? $$ \overline{T} =...

Homework Statement Find the driving frequencies at which the mechanical energy of the forced oscillation is 64 % of its maximum value. (Do not assume weak damping.) Homework Equations E∝A2ω2, where A is amplitude & ω is the angular frequency. The Attempt at a Solution Of course this...

Homework Statement Homework Equations ∑F=ma F=-kx x = Ae-γt/2cos(ωt + ∅) The Attempt at a Solution I have looked up this old thread but don't understand how to get the equation of motion from : ma1=-kx1-k(x1-x2) - bv ma2=-kx2-k(x2-x1) - bv...

Homework Statement A particle in SHM is subject to a driving force F(t)= ma*e^(-jt). Initial position and speed equal 0. Find x(t). Homework Equations F = -kxdx = mvdv F(t) = F(0)*e^(iωt) x(t) = Acos (ωt +φ) The Attempt at a Solution I have no idea how to deal with the exponential term. I...

Homework Statement The problem asked me to derive an expression for the stationary wave function of the 3d harmonic oscillator which I have done. It then tells me a particle is in the stationary state $$\psi_{n_x,n_y,n_z}(x,y,z)=\psi_{100}(x,y,z)$$ and to express this in spherical coordinates...

I was wondering how to derive the sinusoidal equation for the simple armonic oscillator. But I am currently trying to understand this step in this webpage: I don't get where do P and Q come from and why it is summing pe^iwt + qe^-iwt. please I need some help. The rest of it pretty much makes...

Hi everyone I was wondering, why is vacuum energy related to the zero point energy of an harmonic oscillator? The hamiltonian of an harmonic oscillator is $H= \frac{p^{2}}{2m} + \frac{1}{2} \omega x^{2}$. Where does the harmonic potential term come from in the vacuum?

Homework Statement We have the lagragian L = \frac{m}{2} \dot{x}^2 - \frac{m \omega x^2}{2} + f(t) x(t) where f(t) = f_0 for 0 \le t \le T 0 otherwise. The only diagram that survives in the s -matrix expansion when calculating <0|S|0> is D = \int dt dt' f(t)f(t') <0|T x(t)x(t')|0>...

Hi, I have a question about LC series oscillators. Specifically when a DC supply is applied. Solving the differential equation for such circuit using as initial conditions at t=0 that Vc=0 and I=0 I get as a solution that the circuit oscillates and the voltage across the capacitor has an...

Hi, I am trying to find the wavefunction of a coherent state of the harmonic oscillator ( potential mw2x2/2 ) with eigenvalue of the lowering operator: b. I know you can do this is many ways, but I cannot figure out why this particular method does not work. It can be shown (and you can find...

1. A bidimensional oscillator have energies: With C and K constants a) show by a transform of coordinates that this oscillator is equivalent to two isotropic harmonic oscillators. b) then find two independent constants of motion and verify this using: with "a" the constant. I tried to do...

So I've read you can get the corresponding wave function of a quantum harmonic oscillator in momentum space from position space by making the substitution ##x \to k## and ##m \omega \to 1/m \omega##. However in deriving the TISE for momentum space, I seem to be making a mistake. In momentum...