Harmonic oscillator Definition and 699 Threads

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 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 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...

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...

Homework Statement A harmonic oscillator with a vertical mass on a string has a hanging mass of 2m and a spring constant of K. It oscillates with an amplitude of Z. When its position is at a distance Z/2 of the equilibrium point, its potential energy is Ui. What is the maximum kinetic energy...

Homework Statement Show that the underdamped oscillator solution can be expressed as x(t)=x_{0}e^{-γt}[cos(Ω't+((v_{o}+γx_{o})/(x_{o}Ω')sinΩ't] and demonstrate by direct calculation that x(0)=x_{o} and \dot{x}(0)=v_{o} Homework Equations The underdamped oscillator solution is...

In the context of normal modes, what is a free mode? When the whole system is in motion?

Homework Statement I was wondering if there was a general method for finding a function that fits a set of data for a damped harmonic oscillator I'm currently writing up a presentation on the experiment for the gravitational constant and the way i did the experiment was to use a torsion...

Homework Statement A simple harmonic oscillator with mass m = 1/2 and k = 2 is initially at the point x = √3 when it is projected towards the origin with speed 2. Find the equation of motion describing x(t). Homework Equations x=Asin(ωt+θ) The Attempt at a Solution At t=0...

I have an interesting problem I have come across in my research. It results in the differential equation as follows: x''+2γ(x')^\nu+\omega_{o}^2x=g(t) Primes indicate the derivative with respect to t. \gamma and \omega are constants. The non-linearity comes from the first derivative x'...

Homework Statement A particle is in a region with the potential V(x) = κ(x2-l2)2 What is the approximate ground state energy approximation for small oscillations about the location of the potential's stable equilibrium? Homework Equations ground state harmonic oscillator ~ AeC*x2...

Homework Statement Consider an harmonic oscillator with time-dependent frequency as: \omega (t)=\omega_0 * \exp^{- \lambda t} Find the time dependence of the ground state energy of this oscillator for \lambda << 1 situation. Homework Equations H=H_{0} + V(t) H_{0} = \frac{p^2}{2m} +...

Hello fellow computer physics nerds, I'm trying to write a program to plot the positions of the three particles connected by two springs (one dimensional) in Fortran 90. I have a main program block and a module that calls a PGPLOT. My problem is that the positions of the second and third...

I REALLY need help with this one guys! As of right now I believe I only need help with just the set up of the problem. The rest is just solving a differential equation and I assume the frequencies they want will just pop out. Homework Statement Two identical springs and two identical...

Homework Statement A particle with mass m moves in 3-dimensions in the potential V(x,y,z)=\frac{1}{2}m\omega^{2}x^{2}. What are the allowed energy eigenvalues?Homework Equations The Attempt at a Solution The Hamiltonian is given by H=\frac{P^{2}}{2m}+\frac{1}{2}m\omega^{2}X^{2} where P is the...

Hi everybody, This is my first post in this forum although I started following it some time ago. My question is related to rotational properties involving harmonic oscillator model. Homework Statement We are told to evaluate the expectation value of the rotational constant of a...