Harmonic oscillator Definition and 743 Threads

Homework Statement Use the following trial function: \Psi=e^{-(\alpha)r} to estimate the ground state energy of the central potential: V(r)=(\frac{1}{2})m(\omega^{2})r^{2} The Attempt at a Solution Normalizing the trial wave function (separating the radial and spherical part)...

Homework Statement I need to find the momentum space wavefuntion Phi(p,t) for a particle in the first excited state of the harmonic oscillator using a raising operator. Homework Equations Phi_1(p,t)= "raising operator" * Phi_0 (p,t)The Attempt at a Solution In position space, psi_1 (x) =...

When people talk about harmonic oscillators it seems to me that they always assume either that the relationship of force and displacement is linear, or that it behaves in some sinusoidal fashion. Do you always have to assume one to be able to arrive at the other? Or is there something I'm...

Find under what conditions the transformation from (x,p) to (Q,P) is canonical when the transformation equations are: Q = ap/x , P=bx2 And apply the transformation to the harmonic oscillator. I did the first part and found a = -1/2b I am unsure about the next part tho: We have the...

Homework Statement A 3D harmonic oscillator has the following potential: V(x,y,z) = \frac{1}{2}m( \varpi_{x}^2x^2 + \varpi_{y}^2y^2 + \varpi_{z}^2z^2) Find the energy eigenstates and energy eigenvalues for this system. The Attempt at a Solution I found the energy eigenvalue to...

Homework Statement Find the energy eigenvalue. Homework Equations H = (p^2)/2m + 1/2m(w^2)(x^2) + λ(x^2) Hψ=Eψ The Attempt at a Solution So this is what I got so far: ((-h/2m)(∂^2/∂x^2)+(m(w^2)/2 - λ)(x^2))ψ=Eψ I'm not sure if I should solve this using a differential...

Homework Statement (See attachment) Homework Equations x = \sqrt{\frac{\hbar}{2m \omega}} ( a + a^{\dagger} ) x = i \sqrt{\frac{\hbar m \omega}{2}} ( a^{\dagger} - a ) The Attempt at a Solution In part a) I was able to construct a separable Hamiltonian for the harmonic...

Homework Statement Consider an underdamped harmonic oscillator (Q > 1/2) with a sinusoidal driving force Focos(ωdt). (a) (5 pts) By using differential calculus find ωd that maximizes the displacement amplitude. (b) (7 pts) By using differential calculus find ωd that maximizes the velocity...

Ground State Wave Equation: ψ0=(a/∏)(1/4)e(-ax2/2) Prove the Heisenberg Uncertainty principle ≥h(bar)/2 by way of expectation values. First I found <x>=0 because it was an odd function then I found <Px>=0 because it was an odd function Then <x2>=∫(a/∏)(1/2)x2e(-ax2)/2dx=1/2a by way of...

I believe this is pretty standard. Given a mass m on a spring with spring constant k, a solution to the second order differential equation of motion m\ddot{x} = -kx, is x = cos ωot, and ωo = \sqrt{k/m}. If that same oscillator is driven with a force F(t) = Fo cos ωt the equation of motion...

Homework Statement A simple harmonic oscillator has an amplitude of 0.1 m. At what displacement will its kinetic and potential energies be equal? Homework Equations The Attempt at a Solution I'm trying to figure out how to solve this problem but I'm totally stuck and even don't...

Homework Statement The problem wants me to calculate (Δx)^2 and (Δp)^2 to find the uncertainty principle. Delta x is the variance and the problem gives the formula as.. Δx= <n|x^{2}|n>-<n|x|n>^{2}Homework Equations x=\sqrt{\frac{\hbar}{2m \omega}}(A^{-}+A^{+}) Where A+ and A- are the raising...

This is a problem I've been trying to solve for quite some time now. Any help would be appreciated. Homework Statement When a person with the mass of 105kg sits in a car, the body of the car descends by 2,5cm in total. In the car there are four shock absorbers filled with oil and a spring...

Homework Statement The displacement amplitude of a lightly damped oscillator with m=0.250kg and k=6400N/m is observed to decrease by 15% in exactly five minutes a) Calculate the fraction (in%0 of the initial mechanical energy of the oscillator that has been converted to other forms of energy...

Homework Statement A harmonic oscillator has angular frequency ω and amplitude A. What is the magnitude of the displacement when the elastic potential energy is equal to the kinetic energy? (Assume that U = 0 at equilibrium.) Express your answer in terms of the variables ω and A...

hello, new here and confused about Newton second Law. given: vertical mass damper system, position of the mass: x(t)=sin(t) velocity is: v(t)=cos(t) acceleration is: a(t)=-sin(t) function x(t): above x-axis describes position of the mass below the vertical equilibrium point, which (below) is...

I need to find the value σ for which: ψ0(x) = (2πσ)-1/4 exp(-x2/4σ) is a solution for the Schrodinger equation I know the equation for the QHO is: Eψ = (P2/2m)ψ + 1/2*mw2x2ψ I've tried normalizing the wavefunction but I end up with a σ/σ term :( Any help would be greatly...

Over which interval do the wave functions of a harmonic oscillator form a complete and orthogonal system? Is it (-inf,+inf)? The case with particle in a box is rather clear(system is complete and orthogonal only for the interval of the well), however the harmonic oscillator is a bit less intuitive.

here is a link to the pdf file with my question and answershttp://dl.dropbox.com/u/2399196/harmonic%20osc.pdf i'm not sure where to start, because i don't want to assume anything that i haven't been given. i'm stuck on part (iv) where i have to derive explicit expressions for 2 wave functions...

I've heard before that it's because when you expand around a minimum point in the potential energy you get a quadratic function, but I can't recall where I read this. Can anyone point me in the right direction, or give their own explanation? I only ask because I just solved a problem in my...

Just have a few questions regarding the method of solving the damped-driven harmonic oscillator. Once we have rewritten the differential equation in terms of z and it's derivatives, we try a solution z(t) = Ce^{i \omega t}. When we sub in z and it's derivatives we then rewrite the complex...

Homework Statement A physical system is designed having the following equation of motion md2x/dt2 + c(dx/dt) - kx = 0. (a) From the corresponding subsidiary equation, find the solution to this equation of motion. (HINT: use the solution of the damped harmonic oscillator as a guide)...

the problem is attached as an image. im having troubles with the question. I'm assuming this is an induction question? i can prove it for the basis step n=0. but I am having trouble as to what i have to do for n+1 (inductive step). any help or hints would be great!thanks

Homework Statement trying to write a program in C++ to calculate the solution of a damped harmonic oscillator and compare with the exact analytic solution. i am using the classic 4th order Runge-Kutta, which I'm fairly sure is programmed right. Homework Equations m\ddot{x} + c\dot{x}...

Homework Statement Consider a quantum mechanical particle moving in a potential V(x) = 1/2mω2x2. When this particle is in the state of lowest energy, A: it has zero energy B: is located at x = 0 C: has a vanishing wavefunction D: none of the above Homework Equations The...

Homework Statement I need to transform the Hamiltonian of a coupled Harmonic Oscillator into the sum of two decoupled Hamiltonians (non-interacting oscillators). Homework Equations H = H1 + H2 + qxy, where H1=0.5*m*omega^2*x^2+0.5m^-1P_x^2 and H2=0.5*m*omega^2*y^2+0.5m^-1P_y^2, and q is...

Homework Statement Use a series expansion ψ=A0x0+A1x1+A2x2+... to determine the three lowest-order wave functions for a harmonic oscillator with spring constant k and mass m, and show that the engergies are the expected values. Homework Equations Series expansion given above Time...

Hello everyone, i'm looking for anypaper or such kind of thing that explain the resolution of the harmonic oscillator in the Dirac Theory. I have worked with the exact spin symmetry. I feel like a fish out the water and I'm sure that there are lot of bibliography about this area, but i...

Homework Statement Given an initial (t=-∞) Fock state , \left|n\right\rangle, and a function f(t), where f(±∞)=0, show that for a Harmonic Oscillator perturbed by f(t)\hat{x} the difference \left\langle H(+∞) \right\rangle - \left\langle H(-∞) \right\rangle is always positive.Homework Equations...

Homework Statement Consider a steel spring with the property that it extends by 10cm (0.1m) in equilibrium when you attach the upper end of the spring to a fixed support and hang a weight of 100g (0.1kg) at the springs lower end. 1) Use the equation for the harmonic oscillator to determine...

Homework Statement A mass m is attached to a spring of stiffness k. The spring is attached to the ceiling and the mass hangs freely from the spring under the force of gravity. (a) Derive the equation of motion for this system. (b) Find an expression for the equilibrium position of the...

At classical harmonic oscillator, total energy is proportional to square of frequency, but at quantum harmonic oscillator, total energy is proportional to frequency. Are those two frequencies the same? How it is with transition from quantum harmonic oscillator to classical harmonic oscillator...

Hello, I have this problem with deriving the formule from de definition of potential energy Picture show a mass-spring system in rest position: In general potential energy can be written as dot product: \frac{dE_{P}}{d\overrightarrow{y}}=-\overrightarrow{F}. Potential energy wil...

I have a question regarding an oscillator design from a controls perspective. An ideal harmonic oscillator has just 2 poles, both on the imaginary axis, and their location along the axis determines the frequency of oscillation as well as the amplitude. Now, please correct me if this is...

Homework Statement I'm looking at the 1d harmonic oscillator \begin{equation} V(x)=\frac{1}{2}kx^2 \end{equation} with eigenstates n and the time dependent perturbation \begin{equation} H'(t)=qx^3\frac{(\tau^2}{t^2+\tau^2} \end{equation} For t=-∞ the oscillator is in the groundstate...

So I was just thinking about regenerative braking, piezoelectric sensors/strain gauges, magnetic-induced currents etc. and I thought of a question that would make a simple/decent discussion/practice in general engineer/physics (lots of /'s) Suppose you have a simple harmonic oscillator :: WALL...

Let's say I have a 2D harmonic oscillator: Homework Statement The potential is of course defined by: V = 1/2m(Omegax)x^2 + 1/2m(Omegay)y^2 Homework Equations Generally when doing a harmonic oscillator we find that in two dimensions the energy is just: (Nx+Ny+1)hbarOmega is the energy. How...

Hello there, lets say i have a harmonic oscillator equation d^2x/dt^2 = -w^2 x = -Asin(wt) w=frequency, A=amplitude..how can i plot this equation for w^2=1, x(0)=1? and what if the equation contains random number d^2x/dt^2 = -w^2x+Bn, n=gaussian random number with mean value equal to zero...

Homework Statement I showed earlier this semester that in the presence of a "constant force", F_{o}, i.e. V=-Fx, that the eigenvalues for the Harmonic oscillator are shifted by \frac{F^{2}}{2m\omega^{2}} from the "unperturbed" case. It was also discussed that x\rightarrow...

Homework Statement Using the normalization constant A and the value of a, evaluate the probability to find an oscillator in the ground state beyond the classical turning points ±x0. Assume an electron bound to an atomic-sized region (x0 = 0.1 nm) with an effective force constant of 1.0...

I'm not understanding the following formula. I'm a computer programmer and was given a set of formulas to have an application to solve; however I'm not completely understanding how this works. I'm just looking for a step by step way to solve this and an explanation on why there are 3 assignment...

Ok here's the question: A body m is attached to a spring with spring constant k. While the body executes oscillations it also experiences a damping force F = -βv where 'v' is time derivative of displacement of the body from its equilibrium position. I believe equation of motion is F =...

Hey guys, For a particular problem I have to determine the total degeneracy across N 3-D Quantum Harmonic oscillators. Given that the degree of degeneracy for a 3-D harmonic oscillator is given by: (n+1)(n+2)/2 and the Total energy of N 3d quantum harmonic oscillators is given by...

For a harmonic oscillator with mass M, spring of stiffness k and displacement the force equation is: -kx = Md2x/dt2 How do you handle the situation and work out a solution for x(t) when the mass has an initial velocity. E.g. a mass dropped onto the spring?

Homework Statement Relate the frequency of a harmonic oscillator (spring) to that of a simple harmonic oscillator (pendulum) Show all derivations. Homework Equations pendulum: f=(1/(2∏))√(g/L) The Attempt at a Solution Not exactly sure how to go about this...is it saying...

Problem: Consider a harmonic oscillator of mass m undergoing harmonic motion in two dimensions x and y. The potential energy is given by V(x,y) = (1/2)kxx2 + (1/2)kyy2. (a) Write down the expression for the Hamiltonian operator for such a system. (b) What is the general expression for...

Hello everybody, recently in my quantum mechanical course we were introduced to the concept of the quantum harmonic oscillator. My question is: is there a physical significance attached to the fact that the classical turning points overlap with the sign change of the second derivative of the...

Hi, In one of my advanced quantum mechanics classes, the instructor posed a problem, namely to show that the ground state of a one dimensional quantum harmonic oscillator is unique, without getting into differential equations. I know that the equation a\left|0\right\rangle = 0 when...

Homework Statement An underdamped harmonic oscillator with mass m, spring constant k, and damping resistance c is subject to an applied force F0cosωt. (a) [analytical] If, at t = 0, x = x0 and v = v0, what is x(t)? Homework Equations Ωinitial = √(k/m) The Attempt at a...

What is the normalized ground-state energy eigenfunction for the three-dimensional harmonic oscillator V(r) = 1/2 m* ω^2 * r^2 Use separation of varaibles strategy. Express the wave function in spherical coordinates. What is the orbital angualar momentum of the ground state? Explain? I...