Harmonic oscillator Definition and 743 Threads

The problem is: Two damped harmonic oscillator are coupled. Both oscillators has same natural frequency \omega_0 and damping constant \beta. 1st oscillator is damped by 2nd oscillator. Damping force is proportional to velocity of 2nd oscillator. And, vice versa, 2nd oscillator is...

Homework Statement Show simple harmonic motion starting from Hooke's Law. The Attempt at a Solution F=-kx =m\frac{d^2x}{dt^2}=-kx \frac{1}{x}\frac{d^2x}{dt^2}=-\frac{k}{m} =\frac{1}{x}\frac{d}{dt}\frac{dx}{dt}=-\frac{k}{m}...

a) Show that the Hamiltonian for the quantum harmonic oscillator in 3D is separable, b) calculate the energy levels.----a) If it's separable H = H_x + H_y + H_z, so do I just re-arrange the kinetic and potential terms of the Hamiltonian in this case? that seems kind of trivial, as if I'm...

Homework Statement A charged harmonic oscillator is placed in an external electric field \epsilon i.e. its hamiltonian is H = \frac{p^2}{2m} + \frac{1}{2}m \omega ^2 x^2 - q \epsilon x Find the eigenvalues and eigenstates of energy Homework Equations The Attempt at a Solution...

Homework Statement Prove that a 1-d harmonic oscillator in ground state obeys the HUP by computing delta P sub x and delta X Homework Equations delta x = sqrt(<x^2>-<x>^2) delta px = sqrt(<px^2>-<px>^2) The Attempt at a Solution I have absolutely no idea where to start with...

Homework Statement A particle of mass m moves (in the region x>0) under a force F = -kx + c/x, where k and c are positive constants. Find the corresponding potential energy function. Determine the position of equilibrium, and the frequency of small oscillations about it. The Attempt at a...

Homework Statement Determine the normalization constants for the harmonic oscillator wavefunctions with v=0, and v=1 by evaluating their normalization integrals and show that they correspond to N=\frac{1}{\pi^{.5} * 2^v * v!}Homework Equations The Attempt at a Solution \int \psi^{2}d\tau=1...

Homework Statement A particle with with the mass of m is attached to a spring (with no mass, spring constant k, length l) which is attached to a wall. The particle is moving with no friction along the x-axis. a) Write the particles motion equation, and find the general solution to the motion...

Homework Statement The ground state wave function of a one-dimensional simple harmonic oscillator is \varphi_0(x) \propto e^(-x^2/x_0^2), where x_0 is a constant. Given that the wave function of this system at a fixed instant of time is \phi\phi \propto e^(-x^2/y^2) where y is another...

Homework Statement This is a 3 part problem, mass M on a spring of length l with mass m. The first part was to derive the Kinetic Energy of one segment dy, second part was to Integrate this and get the Kinetic Energy of (1/6)m(V^2) where V is the velocity of the Mass M at the end of the...

Homework Statement Find the eigenvalues and eigenfunctions of H\hat{} for a 1D harmonic oscillator system with V(x) = infinity for x<0, V(x) = 1/2kx^2 for x > or equal to 0. Homework Equations The Attempt at a Solution I think the hamiltonian is equal to the potential + kinetic...

Homework Statement The harmonic oscillator problem may be used to describe the vibrations of molecules. For example, the hydrogen molecule H2 is found to have equally spaced vibrational energy levels separated by 8.7 × 10-20 J. What value of the force constant of the spring would be needed to...

Homework Statement The state of Harmonic Oscillator with spin half is |\psi>=\frac{1}{\sqrt{2}}( |n=0,\uparrow> + |n=1, \downarrow>) a, say which one is the possible outcome for a measure of ^{^}S_{x} and find the probability of measuring each possible outcome. b, find the state of the...

Hello everybody, I noticed these questions are lengthy. If you want to skip my introduction, just scroll down to the questions. I put *** next to each one. I just started Quantum Theory I this semester and I have a question (actually two questions) regarding the quantum harmonic...

http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hosc4.html#c1 I know the energy should be E = \frac{{{p^2}}}{{2m}} + \frac{1}{2}m{\omega ^2}{x^2} But I can't figure out why the minimum energy is related to \Delta p

Homework Statement "Show that the ratio of two successive maxima in the displacement of a damped harmonic oscillator is constant."Homework Equations x = a e^(-\upsilont/2) cos (\omegat - \vartheta)The Attempt at a Solution So I want to find when this beast has its maximum values, so I take the...

Hello PF members, Is there some good book, which contain the derivation of average energy of a harmonic oscillator at temperature T. I want to derive from Planck's distribution (PD) function (<n>=(exp(##\hbar\omega/kT##)-1)##^{-1}##)...to get the following relation: energy E=...

Homework Statement Calculate the quantized energy levels of a linear harmonic oscillator of angular frequency $\omega$ in the old quantum theory. Homework Equations \[ \oint p_i dq_i = n h \] The Attempt at a Solution This is supposed to be a simple "exercise" (the first in...

Homework Statement use the hamiltonian equation H=H_x+H_y+H_z to show that wave functions of the form \varphi(r)=\phii(x)\phij(y)\phik(z) where the functions phi_i(x) are the energy eigenfunctions for a 1-d SHM , satisfy H*phi=E*phi , and find the followed values of E for the 3-d...

Homework Statement I wonder if someone could help me to arrive at equation 2.56 by performing the substitutions. Please see the attachment Homework Equations Please see the attachment for this part. and also for the attempt of a solution.

I'm trying to read through Griffiths' QM book, and right now I'm on the series solution to the harmonic oscillator (ch 2). I'm having a hard time following the math (especially after equation 2.81) in this section, so if anyone has read this book, please help. My first question is about the...

I've followed this: http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hosc3.html#c1, up to the part where it gets to here: . The guide says: "Then setting the constant terms equal gives the energy"? Am I being stupid? I really can't see where that equations come from.

I am not really asking how to solve the problem but just for explanation of what I know to be true from the problems solution. Basically the original problem statement is this: A particle in a harmonic oscillator potential starts out in the state |psi(x,0)>=1/5 * [3|0> + 4|1>] and it asks to...

Hi, I'm trying to learn quantum physics (chemistry) on my own so that my work with Gaussian and Q-Chem for electronic structural modeling is less of a black box for me. I've reached the harmonic oscillator point in McQuarrie's Quantum Chemistry book and I'm having trouble justifying a step in...

Homework Statement I have to find the minimum and maximum values of the uncertainty of \Deltax and specify the times after t=0 when these uncertainties apply. Homework Equations The wave function is Ψ(x, 0) = (1/√2) (ψ1(x)+ iψ3(x)) and for all t is Ψ(x, t) = (1/√2)...

1. Explain why any term (such as AA†A†A†)with unequal numbers of raising and lowering operators has zero expectation value in the ground state of a harmonic oscillator. Explain why any term (such as AA†A†A) with a lowering operator on the extreme right has zero expectation value in the...

Homework Statement Calculate the expected value of the kinetic energy being \varphi(x,0)=\frac{1}{\sqrt{3}}\Phi_0+\frac{1}{\sqrt{3}}\Phi_2-\frac{1}{\sqrt{3}}\Phi_3 Homework Equations K=\frac{P^2}{2m} The Attempt at a Solution I tried to solve it using two diffrent methods and...

Homework Statement Consider a particle of mass m moving in a 3D potential V(\vec{r}) = 1/2m\omega^2z^2,~0<x<a,~0<y<a. V(\vec{r}) = \inf, elsewhere. 2. The attempt at a solution Given that I know the solutions already for a 1D harmonic oscillator and 1D infinite potential well I'm going to...

Why does in QM the electron does not fall toward the nucleus? After all, the only force between nucleus and electron is attractive. It seems that the electron can and does indeed fall toward the center in <simple harmonic oscillator>? My question is what's so different in these two systems...

Homework Statement Trying to normalize the first excited state. I have, 1 = |A_1|^2(i\omega\sqrt{2m}) \int_{-\inf}^{\inf} x \exp(-m\omega x^2/2\hbar) How do I do the integral so I don't get zero since it's an odd funciton?

My question is pretty easy (i think). I have a wavefcn PSI defined at t=0. The PSI is a mix of several eigenstates. Let's say PHI(x,0)=C1phi1 + C2 phi3. Now C1 and C2 are given to me, so I am wondering is this wavefcn. already normalized, or do i have to normalize it despite definite C1 and C2...

Hi! Would anyone be able to point me toward a detailed explanation of determining the Hamiltonian of a polyatomic quantum oscillator? My current text does not explain the change of coordinates ("using normal coordinates or normal modes") in detail. All I can find is material on a diatomic...

Can someone tell me if there is a difference in the moving motion between a quantum harmonic oscillator and a simple harmoic oscillator. Also, does anoyone know a good site where i could learn more on quantum harmonic oscillator. ty

The frequency of a harmonic oscillator is (as you know) f=\frac{1}{2\pi}\sqrt{\frac{k}{m}} I am wondering if this equation only applies for massless harmonic oscillators (or oscillators oscillating sideways)? The proof for the equation above is \sum {F=ma} -kx=ma...

The question is as follows: Suppose that, in a particular oscillator, the angular frequency w is so large that its kinetic energy is comparable to mc2. Obtain the relativistic expression for the energy, En of the state of quantum number n. I don't know how to begin solving this question. I...

Homework Statement "Two non-interacting particles are placed in a one-dimensional harmonic oscillator potential. What are the degeneracies of the two lowest energy states of the system if the particles are a)identical spinless bosons b)identical spin-1/2 fermions? Homework Equations...

Homework Statement 3-dimensional harmonic oscillator has a potetnial energy of U(x,y,z)=\frac{1}{2}k'(x^2+y^2+z^2). a) Determine the energy levels of the oscillator as a function of angular velocity. b) Calculate the value for the ground state energy and the separation between adjacent...

Hi, Why is it that the WKB approximation produces the correct eigenvalues for the Harmonic Oscillator problem, but the wrong wavefunctions, whereas for the square well, we get correct wavefunctions and the wrong eigenvalues? I've been trying to dig through the approximations we make in...

The one dimensional harmonic oscillator is associated with the group U(1) and the three dimensional harmonic oscillator is associated with the group SU(3). Is their a group associated with the two dimensional harmonic oscillator? Thank you for any thoughts.

The 3-dimensional harmonic oscillator has SU(3) symmetry. This is stated in many papers. It seems to be due to the spherical symmetry of the system. (After all, the idea of a 3d harmonic oscillator is that a mass is attached to the origin with a spring, and that the mass can move in 3...

Homework Statement The wave function \Psi(x,t) ofr the lowest energy state of simple harmonic oscillator, consisting of a particle mass m acted on by a linear restoring force F=Cx, where C is the force constant, can be expressed as.. \Psi(x,t)=Aexp[-(\sqrt{}Cm/2h)x^{}2-(i/2)(\sqrt{}C/m)t]...

Homework Statement Suppose there is a square plate, of side a and mass M, whose corners are supported by massless springs, with spring constants K, K, K, and k <= K (the faulty one). The springs are confined so that they stretch and compress vertically, with unperturbed length L. The...

Homework Statement Hi all. At time t<0 a particle is in the stationary state \left| {\psi _0 } \right\rangle of the harmonic oscillator with frequency omega1 (i.e. the ground state of the H.O.). At t=0 the Hamiltonian changes in such a way that the new angular frequency is omega =...

Homework Statement A damped harmonic oscillator is driven by a force F external= F sin (omega * t) where F is a constant, and t is time. Show that the steady state solution is given by x(t)= A sin (omega * t - phi) where A is really A of (omega), the expression for the amplitude...

How to find the probability density function of a simple harmonic oscillator? I know that for one normal node is should be a parabola but what is the formula and how do we derive it?

Thanks for all the help on the first question but now I have to solve for <T>. I have no idea how to do this, and I could use some help for a kick start. thanks!

first post! but for bad reasons lol Im trying to find <x> and <p> for the nth stationary state of the harmonic potential: V(x)=(1/2)mw^2x^2 i solved for x: x=sqrt(h/2mw)((a+)+(a-)) so <x> integral of si x ((a+)+(a-)) x si. therefor the integral of si(n+1) x si + si(n-1) x si. si(n+1)...

In quantum mechanics, one of the major concerns is the propagator determination of the system. The propagator is completely expressed in terms of its classical in the Van-Vleck Pauli Formula. In a harmonic oscillator in a time dependent force, the Lagrangian is given by...

Homework Statement When trying to generate solutions to the harmonic oscillator, I'm trying to use hermite polynomials. I understand that there's a recursive relationship between the hermite polynomials but I'm confused in how each hermite polynomial is generated. Homework Equations...

damped harmonic oscillator, urgent help needed! Homework Statement for distinct roots (k1, k2) of the equation k^2 + 2Bk + w^2 show that x(t) = Ae^(k1t) + Be^(k2t) is a solution of the following differential equation: (d^2)x/dt^2 + 2B(dx/dt) + (w^2)x = 0 Homework Equations The...