Oscillator Definition and 1000 Threads

Homework Statement I have a similar problem to this one on Physicsforum from a few years ago. Homework Equations Cleggy has finished part a) saying he gets the answer as Ψ(x, t) = (1/√2) (ψ1(x)exp(-3iwt/2+ iψ3(x)exp(-7iwt/2) OK classical angular frequency ω0 = √C/m for period of...

Homework Statement I must calculate the probability that the position of a harmonic oscillator in the fundamental state has a greater value that the amplitude of a classical harmonic oscillator of the same energy.Homework Equations ##\psi _0 (x)=\left ( \frac{m \omega}{\pi h } \right ) ^{1/4}...

Homework Statement Calculate the expectation value for a harmonic oscillator in the ground state when operated on by the operator: $$AAAA\dagger A\dagger - AA\dagger A A\dagger + A\dagger A A A\dagger)$$ Homework Equations $$AA\dagger - A\dagger A = 1$$ I also know that an unequal number of...

Hi, I am reading through the book "Quantum Mechanics and Path Integrals" by Feynman and Hibbs and am having a bit of trouble with problem 3-12. The question is (all Planck constants are the reduced Planck constant and all integrals are from -infinity to infinity): The wavefunction for a...

Homework Statement I'm having some trouble calculating the 2nd order energy shift in a problem. I am given the pertubation: \hat{H}'=\alpha \hat{p}, where $\alpha$ is a constant, and \hat{p} is given by: p=i\sqrt{\frac{\hbar m\omega }{2}}\left( {{a}_{+}}-{{a}_{-}} \right), where {a}_{+} and...

Homework Statement I have a wavefunction Cxe^{-ax^2} and I have to find the expected value of x. Homework Equations ∫_{-∞}^{∞} x^3 e^{-Ax^2} dx = 1/A^2 for A>0 The Attempt at a Solution I get an integral like this: <x>=|C|^2 ∫_{-∞}^{∞} x^3 e^{-Ax^2} dx After trying integration by parts...

From page 91 of "Modern Quantum Mechanics, revised edition", by J. J. Sakurai. Some operators used below are, a = \sqrt{\frac{m \omega}{2 \hbar}} \left(x + \frac{ip}{m \omega} \right)\\ a^{\dagger} = \sqrt{\frac{m \omega}{2 \hbar}} \left(x - \frac{ip}{m \omega} \right)\\ N = a^{\dagger}...

1. A torsional oscillator of rotational inertia 2.1 kg·m2 and torsional constant 3.4 N·m/rad has a total energy of 5.4 J. What is its maximum angular displacement? What is its maximum angular speed? Homework Equations θ(t)=Acosωt The Attempt at a Solution still trying to...

Homework Statement Hi guys, I don't really know how to solve the first part of a problem which goes like this: Consider a 1 dimensional harmonic oscillator of mass m, Hooke's constant k and angular frequency ##\omega = \sqrt{\frac{k}{m} }##. Remembering the classical solutions, solve the...

Homework Statement If both k of the spring and m are doubled while the damping constant b and driving force magnitude F0 are kept unchanged, what happens to the curve, which shows average power P(ω)? Does the curve: a) The curve becomes narrower (smaller ω) at the same frequency; b) The curve...

Hi guys, is there a reason why the energy of the harmonic oscillator is always written as:$$ E_{n} = \hbar \omega (n + \frac{1}{2})$$ instead of : $$ E_{n} = h \nu (n + \frac{1}{2})$$ ? THX Abby

Homework Statement Hi guys, The title says it all pretty much. I need to know a handful of practical uses for each of the following, in the context of oscillatory motion (springs, pendulums etc): 1) light damping 2) critical damping 3) heavy damping Homework Equations Light...

The Wigner function, W(x,p)\equiv\frac{1}{\pi\hbar}\int_{-\infty}^{\infty} \psi^*(x+y)\psi(x-y)e^{2ipy/\hbar}\, dy\; , of the quantum harmonic oscillator eigenstates is given by, W(x,p) = \frac{1}{\pi\hbar}\exp(-2\epsilon)(-1)^nL_n(4\epsilon)\; , where \epsilon =...

http://www1.gantep.edu.tr/~physics/media/kunena/attachments/382/chapter2.pdf On page 9 and 10 of the above PDF the method for deriving the fractional energy loss per cycle in a lightly damped oscillator is described. I understand and follow this derivation. What would the derivation...

Hi all, this is my first time on PF. I do not know English but I have a problem of a harmonic oscillator. I have rather large head, help me please , I do not know what else to do ... I have this problem: Consider the harmonic oscillator with an additional repulsive cubic force...

Homework Statement Find the uncertainty of the kinetic energy of a quantum harmonic oscillator in the ground state, using \left\langle p^2_x \right\rangle = \displaystyle\frac{\hbar^2}{2a^2} and \left\langle p^4_x \right\rangle = \displaystyle\frac{3\hbar^2}{4a^2} Homework Equations...

Hi, i regard a single harmonic oszillator: $$H_{1}=\frac{p^{2}}{2m} + \frac{m \omega^{2}}{2} x^{2}$$ I know the partition function of the oszillator is: $$Z=\frac{kT}{\hbar \omega}$$ so the probability function is: $$F_{1}(x,p)=\frac{1}{Z}\exp{\frac{-H_{1}(x,p)}{kT}}$$ Now I want to...

Homework Statement What is the effect of the sequence of ladder operators acting on the ground eigenfunction \psi_0 Homework Equations \hat{A}^\dagger\hat{A}\hat{A}\hat{A}^\dagger\psi_0The Attempt at a Solution I'm not sure if I'm right but wouldn't this sequence of opperators on the ground...

Homework Statement Homework Equations The Attempt at a Solution

Okay, so if a harmonic oscillator has a restoring force given by Hooke's Law such that Fs = -kx and its integral gives the potential energy associated with the restoring force: PE = -(1/2)kx2 Then for the total energy of a harmonic oscillator, why is the TE: TE = Evibration +...

Homework Statement (A) A damped oscillator is described by the equation m x′′ = −b x′− kx . What is the condition for critical damping? Assume this condition is satisfied. (B) For t < 0 the mass is at rest at x = 0. The mass is set in motion by a sharp impulsive force at t = 0, so...

Homework Statement For the n = 1 harmonic oscillator wave function, find the probability p that, in an experiment which measures position, the particle will be found within a distance d = (mk)-1/4√ħ/2 of the origin. (Hint: Assume that the value of the integral α = ∫01/2 x2e-x2/2 dx is known...

Homework Statement I am trying to show that for a duffing oscillator described by x''+2g x'+ax+bx^3=0 with a<0, b>0 the equilibria at x=+- \sqrt{-a/b} are stable Homework Equations I used coupled equations, and the characteristic equation of a linear system The Attempt at a Solution...

Hi all Homework Statement I have the first three states of the harmonic oscillator, and I need to know the amplitudes for the states after the potential is dropped.Homework Equations u_{0}=(\frac{1}{\pi a^{2}})^{\frac{1}{4}} e^{{\frac{-x^2}{2a^2}}} u_{1}=(\frac{4}{\pi})^{\frac{1}{4}}...

Homework Statement Hello everyone. I need to demonstrate that with a damped free oscillator, which is linear, the total energy is a function of the time, and that the time derivative of the total energy is negative, without saying if the motion is underdamped, critically damped or overdamped...

Problem: In a harmonic oscillator \left\langle V \right\rangle=\left\langle K \right\rangle=\frac{E_{0}}{2} How does this result compare with the classical values of K and V? Solution: For a classical harmonic oscillator V=1/2kx^2 K=1/2mv^2 I don't really know where to begin. Is it safe...

$$ -\sum_{n = 0}^{\infty}n^2\omega^2C_ne^{in\omega t} + 2\beta\sum_{n = 0}^{\infty}in\omega C_ne^{in\omega t} + \omega_0^2\sum_{n = 0}^{\infty}C_ne^{in\omega t} = \sum_{n = 0}^{\infty}f_ne^{in\omega t} $$ How can I justify removing the summations and solving for $C_n$? $$...

Homework Statement the problem and a possible solution(obtained from a book) is attached as a pdf to the post.However Iam unable to understand it.Please download the attachment. Homework Equations equation no (2) in the pdf.Is there any use of space translation operator in here.The Attempt at...

Homework Statement A particl of mass m in the potential V(x) (1/2)*mω^{2}x^{2} has the initial wave function ψ(x,0) = Ae^{-αε^2}. a) Find out A. b) Determine the probability that E_{0} = hω/2 turns up, when a measuremen of energy is performed. Same for E_{1} = 3hω/2 c) What energy...

Homework Statement 1)Consider a particle subject to the following force ##F = 4/x^2 - 1## for x>0. What is the particle's maximal velocity and where is it attained? 2)A particle of unit mass moves along positive x-axis under the force ##F=36/x^3 - 9/x^2## a)Given that E<0 find the turning...

Homework Statement Write down the v=1 eigenfunction for the harmonic oscillator. Substitute this eigenfunction into the Schrodinger equation and show that the eigenvalue is (3/2)hν. Homework Equations The Attempt at a Solution I'm not really sure on how to to this, but here's...

Homework Statement The amplitude of an underdamped oscillator decreases to 1/e of its initial value after m complete oscillations. Find an approximate value for the ratio ω/ω0.Homework Equations x''+2βx'+ω02x = 0 where β=b/2m and ω0=√(k/m) x(t) = Ae-βtcos(ω1t-δ) where ω1 has been defined as...

Homework Statement I am unsure as to a step in Griffiths's derivation of the quantum harmonic oscillator. In particular, I am wondering how he arrived at the equations at the top of the second attached photo, from the last equation (at the bottom) of the first photo (which is the recursion...

Hi, Consider a block of mass M connected to a spring of mass m and stiffness k horizontally on a frictionless table. We elongate the block some distance, and then release it so that it now oscillates. According to the theoretical study using energy methods, we see that the mass of the...

Homework Statement I have a ball of 20 kg describing a damped harmonic movement, ie, m*∂^2(x)+R*∂x+K*x=0, with m=mass, R=resistance, K=spring constant. The initial position is x(0)=1, the initial velocity is v(0)=0. Knowing that v(1)=0.5, v(2)=0.3, I have to calculate K and R...

So I am trying to model a harmonic oscillator floating on the oceans surface. I treated this as a harmonic oscillator within a harmonic oscillator and I am not sure if I am heading in the correct direction. Just to be clear this isn't a homework problem just something I am working on. The...

Homework Statement Show that the following is an eigenfunction of \hat{H}_{QHO} and hence find the corresponding eigenvalue: u(q)=A (1-2q^2) e^\frac{-q^2} {2} Homework Equations Hamiltonian for 1D QHO of mass m \hat{H}_{QHO} = \frac{\hat{p}^2}{2m} + \frac{1}{2} m \omega^2 x^2...

I am investigating the mathematics behind driven damped oscillators, I will then simulate it in MATLAB and observe the unpredictable long term behavior of the system. In order to create non-linearity in a oscillating spring I can no longer use hookes law but a form of it by introducing a...

Homework Statement Particle of mass m undergoes simple harmonic motion along the x axis Normalised eigenfunctions of the particle correspond to the energy levels E_n = (n+ 1/2)\hbar\omega\ \ \ \ (n=0,1,2,3...) For the two lowest energy levels the eigenfunctions expressed in natural...

Homework Statement How long will it take until the mass is within 10% of its equilibrium? I already solved most of what is needed in previous parts of the question. I just need help solving for t because it is in two exponents in the equation. Homework Equations This is the equation...

Homework Statement Consider as an unperturbed system H0 a simple harmonic oscillator with mass m, spring constant k and natural frequency w = sqrt(k/m), and a perturbation H1 = k′x = k′sqrt(hbar/2m)(a+ + a−) Determine the exact ground state energy and wave function of the perturbed system...

Hey, My question is on determing the expectation value of position of the Harmonic Oscillator using raising and lowering operators, the question is part d) below: I have determined the position operator to be: \hat{x}=\sqrt{\frac{\hbar}{2m\omega}}(a+a^{\dagger}) and so the...

Homework Statement Prove that that the power given by \bar{P} = \frac{1}{2} \gamma m \omega_r^2 A_{(\omega)}^2 is at a maximum for \omega_r = \omega_0 Only variable is \omega_r \omega_r is the resonant frequency of the external force while \omega_0 is the eigen frequency of the...

Homework Statement The position of a mass that is oscillating on a Slinky (which acts as a simple harmonic oscillator) is given by 18.5 cm cos[ 18.0 s-1t]. What is the speed of the mass when t = 0.360 s? Homework Equations x(t)=Acos(ωt+θ) v(t)=-Aωsin(ωt+θ) The Attempt at a Solution...

We usually only consider the first order term for an oscillation, are there any papers on extending that model and including third and fifth order terms (since only odd power terms would cause a periodic motion)? The ODE would look like x''=-αx-βx^3+O(x^5)

I am building a HAM radio transmitter. I have noticed most crystal oscillators above 100mhz are very hard to find. Is there any way to multiply an oscillator's output, say, four times?

Homework Statement Homework Equations The Attempt at a Solution for part a I do not know how to write it in power series form ? for part b : I chose the perturbed H' is v(x)= (1+ε )K x^2 /2 then I started integrate E_1 = ∫ H' ψ^2 dx the problem was , the result equals to ∞ ! shall I...

[b]1. The motion of a forced harmonic oscillator is determined by d^2x/dt^2 + (w^2)x = 2cos t. Determine the general solution in the two cases w = 2 and w is not equal to 2. To be honest I've no idea where to start!

I was reading Strogatz's book on nonlinear dynamics and chaos and in Example 7.2.2, he stated the energy function of the nonlinear oscillator \ddot{x} + (\dot{x})^3 + x = 0 as E(x, \dot{x}) = \frac{1}{2} (x^2 + \dot{x}^2) But isn't this the energy function for the harmonic...

Homework Statement The 3-dimensional harmonic oscillator potential holds N identical non-reacting spin 1/2 particles a)How many particles are needed to fill the low lying states through E=(3+3/2)\bar{h}ω b)What is the total energy of the system c)what is the fermi energyHomework Equations...