Simple harmonic oscillator Definition and 114 Threads

Homework Statement A particle is in the ground state of a simple harmonic oscillator, potential → V(x)=\frac{1}{2}mω^{2}x^{2} Imagine that you are in the ground state |0⟩ of the 1DSHO, and you operate on it with the momentum operator p, in terms of the a and a† operators. What is the...

Homework Statement Consider a mass hanging from an ideal spring. Assume the mass is equal to 1 kg and the spring constant is 10 N/m. What is the characteristic frequency of this simple harmonic oscillator? Homework Equations No idea I think Hookes law F=-ky Some other relevant...

These are practice problems, not homework. Just wanting to check to see if my process and solutions are correct. 1. Given the following functions as solutions to a harmonic oscillator equation, find the frequency f correct to two significant figures: f(x) = e-3it f(x) = e-\frac{\pi}{2}it 2...

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...

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 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...

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 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 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)...

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...

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 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...

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...

Homework Statement To test the resiliency of its bumper during low-speed collisions, a 1000 kg automobile is driven into a brick wall. The car's bumper behaves like a spring with a force constant 5.00 x 106 N/m and compresses 3.16cm as the car is brought to rest. What was the speed of the car...

Find the expectation value of (px)2, keeping in mind that ψ0(x) = A0e−ax2 where A0 = (2mω0/h)^1/4, and <x2> = ∫x2|ψ|2dx = h_bar / 2mω0 <ψ(x)|px2|ψ(x)> = ∫ψ(x)(pop2)ψ(x) dx pop = [hbar / i] (\delta/\deltax) I'm not going to attempt to type out me solving the integral because it...

Homework Statement Compute the partition function Z = Tr(Exp(-βH)) and then the average number of particles in a quantum state <nα > for an assembly of identical simple harmonic oscillators. The Hamiltonian is: H = \sum _{k}[(nk+1/2)\hbar - \mu nk] with nk=ak+ak. Do the calculations once...

Hi I'm having problems with solving this question: a 90.0 kg skydiver hanging from a parachute bounces up and down with a period of 1.50 seconds. What is the new period of oscillation when a second skydiver, whose mass is 60.0 kg, hangs from the legs first? the answer is 1.94 seconds...

Homework Statement The period of a macroscopic pendulum made with a mass of 10 g suspended from a massless cord 50 cm long is 1.42 s. (a) Compute the ground state (zero-point) energy. (b) If the pendulum is set into motion so that the mass raises 0.1 mm above its equilibrium position, what will...

Homework Statement We know that a particle in SHM is in a state such that measurements of the energy will yield either E_0 or E_1 (and nothing else), each with equal probability. Show that the state must be of the form \psi = \frac{1}{\sqrt2} \psi_0 + \frac{e^{i \phi}}{\sqrt2} \psi_1 where...

Homework Statement Hi guys. I've been working on this problem for a while, it's starting to frustrate me. "Show that the function of Ѱ=e^(-bx^2) with b=mw/2ħ is a solution and that the corresponding energy is ħw/2." Homework Equations Schrodinger Eqn...

Pleae help me. a),b),c) was already solved. but question d) is not.

Homework Statement A simple harmonic oscillator of force constant 2*106 N/m and amplitude .01 m has total maechanical energy 160 J... Homework Equations The Attempt at a Solution Now this is not the question but what is the minimum potential energy...1/2kx^2 comes out to be...

The TISE can be written as -\frac{\hbar^{2}}{2m}\frac{d^{2}u}{dx^{2}} + \frac{1}{2}m\omega_{0}^{2}x^{2}u = Eu Now my lecture notes say that it is convenient to define scaled variables y = \sqrt{\frac{m\omega_{0}}{\hbar} x} and \alpha = \frac{2E}{\hbar\omega_{0}} Hence \frac{d}{dx} =...

Homework Statement What is the probability that a particle in the ground state of a simple harmonic oscillator 1-D potential will be found outside the region accessible classically Homework Equations ∫(between 1 and infinity) e^(-y^2 ) dy=0.08π^(1/2) I feel like it's quite a...

Homework Statement A particle is moving in a simple harmonic oscillator potential V(x)=1/2*K*x^2 for x\geq0, but with an infinite potential barrier at x=0 (the paddle ball potential). Calculate the allowed wave functions and corresponding energies.Homework Equations I am thinking that the...

For part of my lab write up on pendulum motion, my professor wanted us to find out why a pendulum was not a simple harmonic oscillator, and under what conditions it was. He also wanted to show this mathematically. So far what I have is that if there is no damping(friction?) and if the the...

Tuning forks are lightly damped SHO's. Consider a tuning fork who's natural frequency is f=392Hz. Angular frequency = w = 2(Pi)f = 2463 (rad/s) The damping of this tuning fork is such that, after 10 sec, it's amplitude is 10% of it's original amplitude. Here is my attempt to find the damping...

I need someone to please verify my work. Homework Statement A particle of mass m is suspended from the ceiling by a spring of constant k and initially relaxed length l_0. The particle is then let go from rest with the spring initially relaxed. Taking the z-axis as vertically oriented...

I've looked at a few introductory treatments of the quantum harmonic oscillator and they all show how one arrives at the discrete energy values E_n = ( \frac {1}{2} + n ) hf \hspace {10 mm} n=0,1,2... usually by setting up and then solving the Schrodinger equation for the system...

Homework Statement Hi, I'm currently studying for a quantum mechanics exam but I am stuck on a line in my notes: Ha\left|\Psi\right\rangle =\hbar\omega\left(a^{t}a a + \frac{a}{2}\right)\left|\Psi\right\rangleHa\left|\Psi\right\rangle =\hbar\omega\left(\left(a a^{t} - 1\right)a +...

Homework Statement A simple harmonic oscillator has amplitude 0.49 m and period 3.7 sec. What is the maximum acceleration? Homework Equations a(max)=Aw^2 w=angular frequency Vmax=Aw w= angular frequency The Attempt at a Solution I attempted to divide the Amplitude (.49m) by...

Homework Statement P4-1. The Method of Frobenius: Sines and Cosines. The solutions to the differential equation y"+ y = 0 can be expressed in terms of our familiar sine and cosine: y(x) = Acos(x) + Bsin(x) . Use the Method of Frobenius to solve the above differential equation for the even...

Homework Statement Calculate the ratio of the kinetic energy to the potential energy of a simple harmonic oscillator when its displacement is half its amplitude. Homework Equations KE=1/2mv2 = 1/2kA2sin2(wt) U=1/2kx2 = 1/2kA2cos2(wt) KEmax=1/2kA2 Umax=1/2KA2 The Attempt at a...

Homework Statement There is a mass attached to two springs on a table. Coefficients of static and sliding friction between the mass and table are equal with the value \mu. The particle is released at time t=0 with a positive displacement x0 from equilibrium. Given that 2kx0 > \mumg write...

Homework Statement One possible solution for the wave function ψn for the simple harmonic oscillator is ψn = A (2*αx2 -1 ) e-αx2/2 where A is a constant. What is the value of the energy level En? Homework Equations The time independent Schrodinger wave equation d2ψ / dx2 =...

Homework Statement Particle mass m is confined by a one dimensional simple harmonic oscillator potential V(x)=Cx2, where x is the displaecment from equilibrium and C is a constant By substitution into time-independant schrodingers with the potential show that \psi(x)=Axe-ax2 is a...

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}...

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 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 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 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]...

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?

If a mass that hangs suspended vertically from a spring is increased, then won't the period increase as a direct linear proportion? (Because the larger mass has a greater inertia and will require a larger force and longer time to change the direction of motion on each oscillation?) Some...

Homework Statement Show that, [a, \hat H] = \hbar\omega, [a^+, \hat H] = -\hbar\omega Homework EquationsFor the SHO Hamiltonian \hat H = \hbar\omega(a^+a - \frac{\ 1 }{2}) with [a^+, a] = 1 [a, b] = -[b, a] The Attempt at a Solution I have tried the following: [a, \hat H] = a\hat...

A 50.0-g mass connected to a spring with a force constant of 35.0 N/m oscillates on a horizontal, frictionless surface with an amplitude of 4.00 cm. Find the speed of the mass when the displacement is 1.00 cm. Can I use here something like : \frac{mv2}{2}=0,5kx2?

Homework Statement A particle moves along x-axis subject to a force toward the origin proportional to -kx. Find kinetic (K) and potential (P) energy as functions of time t, and show that total energy is contant. Homework Equations K = (1/2)m*v^2 P = (1/2)k*x^2 E = K+P x = Asin(wt...

Homework Statement A simple harmonic oscillator consists of a block of mass 2.30 kg attached to a spring of spring constant 440 N/m. When t = 1.70 s, the position and velocity of the block are x = 0.135 m and v = 3.130 m/s. (a) What is the amplitude of the oscillations? What were the (b)...

Homework Statement A particle is inside of a potential described by: H = p^2/2m + 1/2kx^2, x between -L/2 and L/2 H = infinity, otherwise. my task is to compute a first-order approximation to the energies of this potential. The Attempt at a Solution I attempted to use...