Harmonic oscillator Definition and 743 Threads

The model of damped harmonic oscillator is given by the composite system with the hamiltonians ##H_S\equiv\hbar \omega_0 a^\dagger a##, ##H_R\equiv\sum_j\hbar\omega_jr_j^\dagger r_j##, and ##H_{SR}\equiv\sum_j\hbar(\kappa_j^*ar_j^\dagger+\kappa_ja^\dagger...

Homework Statement Considering the Hamiltonian for a harmonic oscillator: H=\frac{p^2}{2m}+\frac{mw^2}{2}q^2 We have seen that the equations of motion are significantly simplified using the canonical transformation defined by F_1(q,Q)=\frac{m}{2}wq^2cot(Q) Show explicitly that between both...

I was reading through my Principles of Quantum Mechanics textbook and arrived at the section that discusses the quantum harmonic oscillator. In this discussion the equation ψ"-(y^2)ψ=0 presents itself and a solution is given as ψ=(y^m)*e^((-y^2)/2), similar to a gaussian function i assume. My...

The Hamiltonian is ##H=\hbar \omega (a^\dagger a+b^\dagger b)+\hbar\kappa(a^\dagger b+ab^\dagger)## with commutation relations ##[a,a^\dagger]=1 \hspace{1 mm} and \hspace{1 mm}[b,b^\dagger]=1##. I want to calculate the Heisenberg equations of motion for a and b. Beginning with ##\dot...

This isn't homework. I'm reviewing calculus and basic physics after many years of neglect. I want to show that a damped harmonic oscillator in one dimension is nonconservative. Given F = -kx - \small\muv, if F were conservative then there would exist P(x) such that \small -\frac{dP}{dx} = F...

Homework Statement Hey! I got this problem about 3D harmonic oscillator, here it goes: A particle can move in three dimensions in a harmonic oscillator potential ##V(x,y,z)=\frac{1}{2}m\omega^2(x^2+y^2+z^2)##. Determine the ground state wave function. Check by explicitly counting that it is...

If the ladder operator ##a=\sqrt {\frac{m\omega}{2\hbar}}x+\frac{ip}{\sqrt{2m\hbar \omega}}## and ##a^\dagger=\sqrt {\frac{m\omega}{2\hbar}}x-\frac{ip}{\sqrt{2m\hbar \omega}}## then I get that the number operator N, defined as ##a^\dagger a## is worth ##\frac{m \omega...

Homework Statement Consider an electron confined by a 1 dimensional harmonic potential given by ## V(x) = \dfrac{1}{2} m \omega^2 x^2##. At time t=0 the electron is prepared in the state \Psi (x,0) = \dfrac{1}{\sqrt{2}} \psi_0 (x) + \dfrac{1}{\sqrt{2}} \psi_4 (x) with ## \psi_n (x) = \left(...

Anyone know if there are any graphical simulations online for the field of a charged harmonic oscillator, or better yet maybe some kind of paper on it?

Homework Statement consider a harmonic oscillator of mass m and angular frequency ω, at time t=0 the state if this oscillator is given by |ψ(0)>=c1|Y0> + c2|Y1> where |Y1> , |Y2> states are the ground state and the first state respectively find the normalization condition for |ψ(0)> and the...

Homework Statement i need to calculate the orbital angular momentum for 3D isotropic harmonic oscillator is the first excited state The Attempt at a Solution for the first excited state...

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...

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}...

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...

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

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 =...

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...

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 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...

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}}...

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...

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 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...

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 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...

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!

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...

Hi All, If there is something fundamentally wrong in my understanding of quantum mechanics, pardon me for I have just started learning it. We know that if we can come up with a solution for Schrodinger Equation of a Harmonic Oscillator, then we can generate further solutions by acting on it...

I've been looking around and trying to figure it out, but I can't seem to figure out how the cosine function get's into the solution to the HO equation d2x/dt2=-kx/m. I know this is extremely basic, but could someone indulge me?

A particle has its wave function as the ground state of the harmonic oscillator. Suddenly the spring constant doubles (so the angular frequence dobules). Find the propability that the particle is afterwards in the new ground state. I did solve this, it was quite easy. But doing so I encountered...

Homework Statement At time t < 0 there is an infinite potential for x<0 and for x>0 the potential is 1/2m*w^2*x^2 (harmonic oscillator potential. Then at time t = 0 the potential is 1/2*m*w^2*x^2 for all x. The particle is in the ground state. Assume t = 0+ = 0- a) what is the probability that...

So, this has been bothering me for a while. Lets say we have the wavefunction of a harmonic oscillator as a general superposition of energy eigenstates: \Psi = \sum c_{n} \psi _{n} exp(i(E_{n}-E_{m})t/h) Is it true in this case that <V> =(1/2) <E> . I tried calculating this but i...