In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force F proportional to the displacement x:
F
→
=
−
k
x
→
,
{\displaystyle {\vec {F}}=-k{\vec {x}},}
where k is a positive constant.
If F is the only force acting on the system, the system is called a simple harmonic oscillator, and it undergoes simple harmonic motion: sinusoidal oscillations about the equilibrium point, with a constant amplitude and a constant frequency (which does not depend on the amplitude).
If a frictional force (damping) proportional to the velocity is also present, the harmonic oscillator is described as a damped oscillator. Depending on the friction coefficient, the system can:
Oscillate with a frequency lower than in the undamped case, and an amplitude decreasing with time (underdamped oscillator).
Decay to the equilibrium position, without oscillations (overdamped oscillator).The boundary solution between an underdamped oscillator and an overdamped oscillator occurs at a particular value of the friction coefficient and is called critically damped.
If an external time-dependent force is present, the harmonic oscillator is described as a driven oscillator.
Mechanical examples include pendulums (with small angles of displacement), masses connected to springs, and acoustical systems. Other analogous systems include electrical harmonic oscillators such as RLC circuits. The harmonic oscillator model is very important in physics, because any mass subject to a force in stable equilibrium acts as a harmonic oscillator for small vibrations. Harmonic oscillators occur widely in nature and are exploited in many manmade devices, such as clocks and radio circuits. They are the source of virtually all sinusoidal vibrations and waves.
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
Hi,
Could someone give me a tip or two for how to calculate the mean square position ( <x^2>) for a linear quantum harmonic oscillator?
Homework Equations
I think I'm supposed to use the following recursion relation for Hermite polynomials:
yHv=vHv-1 + .5Hv+1
the...
1. Homework Statement and the attempt at a solution
Please see attached.
I'm not so sure if my problem lies with the physics or the mathematics. I have the distinct feeling that it's the latter and that I'm missing something elementary, but truly have no idea how to proceed.
Any...
Homework Statement
This is supposed to be a question for high school seniors who've had instruction in introductory concepts of special relativity and non-relativistic QM. According to my TA, he isn't too certain if it can be done within these confines but nonetheless I've attempted the...
Thanks in advance for reading/replying
Homework Statement
In the interval (t+dt, t) the Hamiltonian H of some system varies in such a way that |H|psi>| remains finite. Show that under these circumstances |psi> is a continuous function of time.
A harmonic oscillator with frequency w is in...
I have the following equation subject to y(0)=0 and y'(0) = 0
my'' + b y' + k y = C
I have done an experiment where I measured force at given depth for a dampened harmonic oscillator. Is it possible to use the force I measured to solve for displacement and then back out coefficient b for...
Aplication of the harmonic oscillator to an arbitrary number of free bosons
Homework Statement
I will like to know if my answer to this problem is correct and If not, what I am missing.
Given a system with an arbitrary number of free bosons, where the hamiltonian for one particle with...
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?
Many quantum physics/chemistry books use Schrodinger's equation to derive a differential equation which describes the possible wavefunctions of the system. One form of it is this:
\frac{d^{2}\psi}{dx^{2}} + (\lambda - a^{2}x^{2})\psi = 0
"a" and lambda are constants. Most books solve this...
How to describe a harmonic oscillator defined by
H(q,p) = \frac{p^2}{2m} + \frac{1}{2}kq^2
in a heat bath with some fixed temperature T?
I suppose this question alone is not quite well defined, because it mixes classical and statistical mechanics in confusing manner, but I thought that one...
I have trouble understanding how damping affects the period (of a torsion pendulum). I know that damping affects the amplitude of the oscillator, however how would damping change the period then?
I have a feeling this has to do with angular frequency, w, given by:
w = sqrt( (k/m) -...
Homework Statement
Given a particle is confined in a one dimensional harmonic oscillator potential, find the matrix representation of the momentum operator in the basis of the eigenvectors of the Hamiltonian.
Homework Equations
Potential: V(x) = 0.5 m w^2 x^2 where m is the mass of...
I am looking into the calculations of a harmonic oscillator potential for nuclei single-particles. The information I am looking at is at:
http://en.wikipedia.org/wiki/Shell_model the specific section “Deformed harmonic oscillator approximated model”
The specific question is, I don’t...
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...
Does anyone know why a harmonic potential gives rise to coherent states? In other words, what is special about a quadratic potential that causes the shifted ground state to oscillate like a classical particle without dispersing so as to saturate the uncertainty principle? Any help or insight...
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...
Homework Statement
A damped harmonic oscillator originally at rest and in its equilibrium position is subject to a periodic driving force over one period by F(t)=-\tau^2+4t^2 for -\tau/2<t<\tau/2 where \tau =n\pi/\omega
a.) Obtain the Fourier expansion of the function in the integral...
Hi, does anyone know of a good resource that focuses on the history of the harmonic oscillator and other classical systems like the vibrating string and vibrating drum? I need to talk about the historical aspect in a project and was having a hard time finding some good material. anyway let me...
Homework Statement
An un-damped driven harmonic oscillator satisfies the equation of motion: ma+kx=F(t) where we may write the un-damped angular frequency w-naught^2=k/m. The driving force F(t)=F-naught*sin(wt) is switched on at t=0. Find x(t) for t>0 for initial conditions x=0, v=0,at t=0...
Homework Statement
A car is moving along a hill at constant speed on an undulating road with profile h(x) where h'(x) is small. The car is represented by a chassis which keeps contact with the road , connected to an upper mass m by a spring and a damper. At time t, the upper mas has...
Homework Statement
A driven oscillator satisfies the equation
x'' + omega2=F0cos(omega(1+episilon)t]
where episilon is a positive constant. Show that the solution that satisfies the iniitial conditions x=0 and x'=0 when t=0 is
x= (F0*sin(.5episilon*omega * t)...
Homework Statement
H = p^2/2m + (kx^2)/2 - qAx (THis is a harmonic potentional with external electric force in 1D)
Braket:
Definitions:
|0, A=0 > = |0>_0 for t=0 (ground state)
|0, A not 0 > = |0> for t=0 (ground state)
2. Question
1. Find the probability of being in the state |0, A...
I have been given at t=1.00 a position and velocity. And the spring constant and mass.
I have found the maximum amplitude.
The question is, where was the block at time t=0? And apparently this can be done without solving for the phase constant and making an equation.
The question doesn't...
Hi all,
I have to determine the potential energy of a hanging spring with a mass m in the end and spring constant k. I try to write down the force in the system
F = m*g + k*x
and integrate the force in order to get the potential energy
E_p = m*g*x+0.5*k*x*x
Does this look correct...
Homework Statement
Predict the wavenumber (cm-1) position of infrared absorption due to fundamental vibration from v=0 to v=1 and 2nd overtone from v=0 to v=3. For a harmonic occilator whose frequency=8.00x1013 s.
Homework Equations
Energy expression for harmonic oscilator:
Ev=...
I am working with the following harmonic oscillator formula.
\psi_n \left( y \right) = \left( \frac {\alpha}{{\pi}} \right) ^ \frac{1}{{4}} \frac{1}{{\sqrt{2^nn!}}}H_n\left(y\right)e^{\frac{-y^2}{{2}}}
Where
y = \sqrt{\alpha} x
And
\alpha = \frac{m\omega}{{\hbar}}
I...
We have a potential that is (1/2)kx^2 for x>0 and is infinity for x<0 ( half harmonic oscillator.
Now i want to calculate the bound states of the system for given E. My question is this:
Do we apply
1. \int p(x) dx = (n - \frac{1}{4} ) h ( Since there is only one turning point that can...
The displacement of a harmonic oscillator is given by
x = A Cos(wt + Phi)
The phase angle phi is equally likely to have any value in the range 0 < Phi < 2Pi, so the probability W(Phi) that Phi lies between Phi and Phi + delta-Phi is delta-Phi/(2Pi). For a fixed time t, find the probability...
Homework Statement
Hello. I am attempting to evaluate the classical action of a harmonic oscillator by using the Euler-Lagrange equations.
Homework Equations
The Lagrangian for such an oscillator is
$$ L=(1/2)m(\dot{x}^2-\omega^2 x^2) $$
This is easy enough to solve for. The classical action...
I wish to graph a couple of the waveforms of a harmonic oscillator. I have consulted several resources and have found two that I like but the final equation differs even though they are both labeled normalized harmonic oscillator wavefunction.
The first reference explains how the harmonic...
1. At a certain time the wavefunction of a one-dimensional harmonic oscillator is
\psi(x) = 3\phi0(x) + 4\phi1(x)
where \phi0(x) and \phi1(x) are normalized energy eigenfunctions of the ground and first excited states respectively. Normalize the wavefunction and determine the probability...
Alright, I'm sure I'm missing something extremely simple, but in Griffiths (and another text I'm reading) coherent states are mentioned as eigenfunctions of the annihilation operator.
I just don't understand:
a) how you can have an eigenfunction of the annihilation operator (other than |0>) if...
I'm reading Griffiths', and I had a question about the harmonic oscillator.
Griffiths solves the Schrodinger equation using ladder operators, and he then says that there must be a "lowest rung," or \psi_{0}, such that a_\psi_{0} = 0. I'm guessing this also means that E = 0 for a_\psi_{0}...
Homework Statement
A particle oscillates between the points x = 40mm and x = 160mm with an acceleration a = k(100-x) where k is a constant. The velocity of the particle is 18mm/s when x=100 and zero at x = 40mm and x = 160mm. Determine a) the value of hte constant k, b) the velocity when x =...
Problem
Show that in the nth state of the harmonic oscillator
\langle x^2 \rangle = (\Delta x)^2
\langle p^2 \rangle = (\Delta p)^2
Solution
This seems too simple... I'm not sure if it's correct...
It is obvious that \langle x \rangle = 0... this is true because the parity of the...
Problem
A harmonic oscillator consists of a mass of 1 g on a spring. Its frequency is 1 Hz and the mass passes through the equilibrium position with a velocity of 10 cm/s. What is the order of magnitude of the quantum number associated with the energy of the system?
Solution?
Okay, so the...
Homework Statement
Two masses are connected via a spring. Write the equations of motion, solve them (x(t)), and find the normal mode frequencies.
Homework Equations
F=ma
F=-kx
The Attempt at a Solution
I set up two equations:
F_{1}=ma_{1} , F_{2}=ma_{2}...
[SOLVED] Identical Particles in a 1-D Harmonic Oscillator
Homework Statement
Three particles are confined in a 1-D harmonic oscillator potential. Determine the energy and the degeneracy of the ground state for the following three cases.
(a) The particles are identical bosons (say, spin 0)...
Hi everyone,
Could someone please help me with this problem?
Homework Statement
A simple harmonic oscillator takes 11.5s to undergo four complete vibrations.
a. Find the period of its motion
b. Find the frequency in Hertz
c. Find the angular frequency in radians per second...
Homework Statement
I'm trying to follow through a derivation involving the equation of motion for the displacement x(t) of a damped driven harmonic oscillator.
m\frac{d^{2}x}{dt^{2}}+\gamma x + \beta \frac{dx}{dt}=F_{0}cos(\omega t)
Where
cos(\omega t) = \frac{1}{2}\left( e^{i \omega...
Homework Statement
Consider the SHO inside a square well, which looks like a soda can cut in half inside of a box.
\[
V(x,y) = \left\{ \begin{array}{l}
\frac{1}{2}kx^2 ,{\rm{ }}y{\rm{ }} < {\rm{ }}|a| \\
\infty ,{\rm{ }}y{\rm{ }} \ge {\rm{ }}|a|{\rm{ }} \\
\end{array} \right\}
\]...
[SOLVED] Simple Harmonic Oscillator
Homework Statement
The equation of motion of a simple harmonic oscillator is (second derivative of x wrt t) d2x/dt2 = -9x, where x is displacement and t is time. The period of oscillation is?
Homework Equations
2 pi f = omega
f = 1/T...
Homework Statement
consider V(x,y) = inf, |y| > a; 1/2kx^2, |y|<=a.
find the energies of this potential.
my initial idea was to just look for solutions of the form X(x)*Y(y), and solve for the separation constant, which should give me the energy, right?
Homework Statement
\psi(x,0) = N exp[-\alpha(x-a)^2]
(1):This wavefunction is a solution to the time dependent schrödinger equation for a harmonic oscillator, but not to the time independent one. How is that possible?
(2):Explain without calculating how would you find the time...
Homework Statement
Two identical spin 1/2 particles of mass m exist in a one dimensional harmonic oscillator potential \frac{kx^{2}}{2} where x is the position coordinate and k is a constant. The particles interact with eact other with a potential W\delta(x-x'), where \delta(x-x') is a Diract...
\psi(x,0) = N exp[-\alpha(x-a)^2] is a solution to the time-independent SE at time t = 0 for the potential
\ V(x) = (1/2)*m\omega^2x^2
where N is a constant and \alpha = m\omega/(2\hbar).
I'm asked to show that the solution is valid only if a = 0.
I'm a little at loss as to...