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.
Hi,
First of all, I'm wondering if a beaded string is the right term?
I have to find the amplitude of the modes 2 and 3 for a string with 5 beads.
In my book I have $$A_n = sin(\kappa p)$$ or $$A_n = cos(\kappa p) $$ it depends if the string is fixed or not I guess. where $$\kappa = \frac{n\pi...
What I already know
In general, power gain is desirable for an oscillator in order to make up for the losses and then feedback that gain (amplified signal) into the oscillator for it to keep oscillating. Voltage gain is not generally used for oscillators.
What I want to know
Since power gain is...
Hello everyone.
I am having some trouble with an RC phase shift oscillator that I built as a hobby project. I am completely stuck on this and I just cannot figure it out. My oscillator is not oscillating.
Here is the circuit that I am trying to get to work. Taken from...
According to Vacuum Electronic High Power Terahertz Sources book by John H Booske,
"In solid state electronic devices, the electron stream is a conduction (ohmic, collisional) current whereas in vacuum electronic devices (VEDs) the current is a convection (ballistic, collisionless) current...
Hello everyone.
I'm currently trying to solve the damped harmonic oscillator with a pseudospectral method using a Rational Chebyshev basis
$$
\frac{d^2x}{dt^2}+3\frac{dx}{dt}+x=0, \\
x(t)=\sum_{n=0}^N TL_n(t), \\
x(0)=3, \\
\frac{dx}{dt}=0.
$$
I'm using for reference the book "Chebyshev and...
As we see in this Phet simulator, this is only the real part of the wave function, the frequency decreases with the potential, so lose energy as moves away the center.
we se this real-imaginary animation in Wikipedia, wave C,D,E,F. Because with less energy, the frequency of quantum wave...
Homework Statement
The harmonic oscillator's equation of motion is:
x'' + 2βx' + ω02x = f
with the forcing of the form f(t) = f0sin(ωt)The Attempt at a Solution
So I got:
X1 = x
X1' = x' = X2
X2 = x'
X2' = x''
∴ X2' = -2βX2 - ω02X1 + sin(ωt)
The function f(t) is making me doubt this answer...
Homework Statement
Using paper, pencil and the Virial theorem, calculate the position uncertainty (an estimate of the vibration amplitude) of the H atom in its ground state C-H stretching mode. In more precise language, calculate the bond length uncertainty in a C-H bond due to the C-H...
Homework Statement
A simple harmonic oscillator consists of a block of mass 45 g attached to a spring of spring constant 240 N/m, oscillating on a frictionless surface. If the block is displaced 3.5 cm from its equilibrium position and released with an initial velocity of 2.5 m/s, what is its...
Homework Statement
A forced damped oscilator of mass ##m## has a displacement varying with time of ##x=Asin(\omega t) ## The restive force is ## -bv##. For a driving frequency ##\omega## that is less than the natural frequency ## \omega_{0}##, sketch graphs of potential energy, kinetic energy...
Homework Statement
A 1-d harmonic oscillator of charge ##q## is acted upon by a uniform electric field which may be considered to be a perturbation and which has time dependence of the form ##E(t) = \frac{K }{\sqrt{\pi} \tau} \exp (−(t/\tau)^2) ##. Assuming that when ##t = -\infty##, the...
When i take a coherent state ##|\alpha>## if ##\alpha -> 0## then the limit is the Fock state for n = 0. so ##|n = 0> = |\alpha = 0>##
The problem is that they seem to have different http://www.iqst.ca/quantech/wiggalery.php:
Where is the error?
Thanks.
Edit sorry, in the link the W function is...
Homework Statement
A test was made with a basket and 20 gram weights. they were put in the basket which hang on a spring, the basket was raised and released. the period was measured for a few number of weights in the basket. the results are as follows. the first of every pair is the number of...
Homework Statement
A system consists of a spring with force constant k = 1250 N/m, length L = 1.50m, and an object of mass m = 5.00kg attached to the end. The object is placed at the level of the point of attachment with the spring unstretched, at position yi= L, an then is released so that it...
Homework Statement
## H=\frac{p^2}{2m}+\frac{1}{2}m\omega^2x^2##
Show that
##[H,[H,x^2]]=(2\hbar\omega)^2x^2-\frac{4\hbar^2}{m}H##
Homework Equations
##[x,p]=i\hbar##
The Attempt at a Solution
I get
##[H,x^2]=-\frac{i\hbar}{m}(px+xp)##
what is easiest way to solve this problem?
Homework Statement
I have small block of mass m=1kg on top of a bigger block mass M=10kg
The friction coefficient between the blocks μ=0.40
No fricton between the big block and the ground.
There is a spring with k=200N/m attached to the bigger block.
The problem asks what is the maximum...
Hello! I am new here...
I am a physics student, seeking for help. There is something I just can't seem to understand...
Your help will be much appreciated...
In my exercise, I have the following Hamiltonian:
H=\frac{p_x^2}{2m}+\frac{p_y^2}{2m}+\frac{1}{2}m \omega^2 (x^2+y^2)+\lambda xy
I was...
Dimensionless equation for quantum harmonic oscilator in the lowest energy state is:
d2u/dx2=(x2-1)u
u means wave function and solution is:
u = exp(-x2/2)
As we can see, solution is the Gauss curve.
But, what is special in the above equation that it give the Gauss curve?
Maybe...
Homework Statement
An Oscillator with free oscillation period \tau is critically damped and subjected to a periodic force with the 'saw-tooth' form:
F(t)=c(t-n\tau) , (n-1/2)\tau<t<(n+1/2)\tau
for integer n with c constant. Find the ratios of the amplitudes of oscillations at the...
1. Driven Harmonic Oscillator with an arbitrary driving force:
f(t)=x"+2bx'+w^2 x
Let x(t) be expressed by x(t)= g(t)*exp(a1*t), where a1 is a solution to the characteristic equation a^2 + 2ba+w=0 for the above second order differential equation. Find the ordinary differential equation that is...
hi,
i am supposed to solve this excerise and i don't even know where to start.
A mass M is suspended from a spring and oscillates with a period of 0.880 s. Each complete oscillation results in an amplitude reduction of a factor of 0.96 due to a small velocity dependent frictional effect...
Homework Statement
hello;
i have this past paper question relevant to a oscillator
they have given the following schematic and question is
"identify the type of oscillator giving reasons"
Homework Equations
The Attempt at a Solution
i tried lecture notes,some books but couldn't...
Homework Statement
The Hamiltonian for the one-dimensional harmonic oscillator is given by:
H = \frac{p^2}{2m}+ \frac{mw^2q^2}{2} Homework Equations
(a) Express H in terms of the following coordinates:
a = \sqrt{\frac{mw}{2}} (q+i\frac{p}{mw})
a^* = \sqrt{\frac{mw}{2}} (q-i\frac{p}{mw})...
Hallo,
Does the velocity i simple harmonic oscillator is zero in equilibrium points? if it's true
how does it make sense with the fact that i suppose to get a maximum kinetic
Energy in those points (stable ones)
i would really appreciate if someone could clear this issue for me...
Hi, Finally! I reached harmonic oscilator! Congratulation!
Most of all QM textbook introduced this formula :
Time independent energy eigenstate equation is
( - \frac{\hbar^2}{2m} \frac{\partial}{\partial x) + \frac{Kx^2}{2} )\varphi = E\varphi
(1) \varphi_{xx} = -k^2 \varphi...
Hi, I am preparing for a quantum mechanics exam, and I have this problem that I can`t solve:
I have to find the complete energy eigenvalue spectrum of a hamiltonean of the form:
H = H0 + c
and also another of the form
H = H0 + \lambdax^{2}
Where in both cases, H0 is the...
Hello, can someone please help me out here, i want to step up the voltage of a 1.5V battery to 300V, i know that to do so you need an oscilatro circuit consisting of a transformer a small capacitor and a transistor. so my question is, does the type of transistor/capacitor i use matter?
[SOLVED] Kinetic energy of harmonic oscilator
Homework Statement
Find the expectation value of the kinetic energy of the nth state of a Harmonic oscillator
Homework Equations
<T> = \frac{<p^2>}{2m}
p_{x} = \frac{1}{i} \sqrt{\frac{m\hbar\omega}{2}} (\hat{a} -\hat{a}^\dagger)...
Just a quickie:
A particle is in the first excited Eigenstate of energy E corresponding to the one dimensional potential V(x) = \frac{Kx^2}{2}. Draw the wavefunction of this state, marking where the particles KE is negative.
Now my question.
The first excited state will be n=1 correct...
I'm currently taking graduate classes toward my phD in physics... when I was undergraduate I learn the harmonic oscillator just solving the schrodinger equation with such potential can be derive that: E=(n+1/2)hw, the wave functions (with hermite polynomial *e^-x2). that take to pages of...