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.
I have a little doubt about Morse potential used for vibration levels of diatomic molecules. With regard to the image below, if the diatomic molecule is in the vibrational ground state, when the oscillation reaches the maximum amplitude for that state the velocity of the molecule must be zero so...
Hello there, I am trying to solve the above and I'm thinking that the solutions will be Hermite polynomials multiplied by a decaying exponential, much like the standard harmonic oscillator problem. The new Hamiltonian would be like so:
$$H = - \frac \hbar {2m} \frac {d^2}{dx^2}\psi + \frac...
I'm studying Ergodic Theory and I think I "got" the concept, but I need an example to verify it...
Let's take the simplest possible 2D classical harmonic oscillator whose kinetic energy is $$T=\frac{\dot x^2}{2}+\frac{\dot y^2}{2}$$ and potential energy is $$U=\frac{ x^2}{2}+\frac{y^2}{2}$$...
Hello everybody, new here. Sorry in advance if I didn't follow a specific guideline to ask this.
Anyways, I've got as a homework assignment two cannonical transformations (q,p)-->(Q,P). I have to obtain the hamiltonian of a harmonic oscillator, and then the new coordinates and the hamiltonian...
I know four different forms in which an SHM can be represented after solving the differential and taking the superposition
acos(wt+Ø)
asin(wt+Ø)
acos(wt-Ø)
asin(wt-Ø)
where a- amplitude
In the above image they took B as negative in order to arrive at acos(wt+e). If i already knew i wanted...
##\newcommand{\ket}[1]{|#1\rangle}##
##\newcommand{\bra}[1]{\langle#1|}##
I have a momentum-shifting operator ##e^{i\Delta p x/\hbar}## acting on the ground state ##\ket{0}## of the QHO, and I want to compute the overlap of this state with the n##^{th}## excited QHO state ##\ket{n}##. Given...
As far as I know we can express the position and momentum operators in terms of ladder operators in the following way
$${\begin{aligned}{ {x}}&={\sqrt {{\frac {\hbar }{2}}{\frac {1}{m\omega }}}}(a^{\dagger }+a)\\{{p}}&=i{\sqrt {{\frac {\hbar }{2}}m\omega }}(a^{\dagger }-a)~.\end{aligned}}.$$...
The wavefunction is Ψ(x,t) ----> Ψ(λx,t)
What are the effects on <T> (av Kinetic energy) and V (potential energy) in terms of λ?
From ## \frac {h^2}{2m} \frac {\partial^2\psi(x,t)}{\partial x^2} + V(x,t)\psi(x,t)=E\psi(x,t) ##
if we replace x by ## \lambda x ## then it becomes ## \frac...
Hey, I solved a problem about a double pendulum and got 2 euler-lagrange equations:
1) x''+y''+g/r*x=0
2) x''+y'' +g/r*y=0 (where x is actually a tetha and y=phi)
the '' stand for the 2nd derivation after t, so you can see the basic harmonic oscillator equation with a term x'' or y'' that...
I know that due to causality g(t-t')=0 for t<t' and I also know that for t>t', we should get
g(t-t')=\frac{sin(\omega_0(t-t'))}{\omega_0}
But I can't seem to get that to work out.
Using the Cauchy integral formula above, I take one pole at -w_0 and get
\frac{ie^{i\omega_0(t-t')}}{2\omega_0}
and...
I tried by taking the derivative of the potential to find the critic points and the I took the second derivative to find which of those points are minimum points. I found that the point is ##x=- a##. I don't understand how to calculate the period, since I haven't seen anything about the harmonic...
Homework Statement
Using the Schrödinger equation find the parameter \alpha of the Harmonic Oscillator solution \Psi(x)=A x e^{-\alpha x^2}
Homework Equations
-\frac{\hbar^2}{2m}\,\frac{\partial^2 \Psi(x)}{\partial x^2} + \frac{m \omega^2 x^2}{2}\Psi(x)=E\Psi(x)
E=\hbar\omega(n+\frac{1}{2})...
Hello everyone,
For weeks I have been struggling with this quantum mechanics homework involving writing a code to determine the energy spectrum and eigenvalues for the stationary Schrodinger equation for the harmonic oscillator. I can't find any resources anywhere. If anyone could help me get...
1. The problem statement
I want to write the angular momentum operator ##L## for a 2-dimensional harmonic oscillator, in terms of its ladder operators, ##a_x##, ##a_y##, ##a_x^\dagger## & ##a_y^\dagger##, and then prove that this commutes with its Hamiltonian.
The Attempt at a Solution
I get...
I have trouble with finding the eigenstates of a spherical pendulum (length $l$, mass $m$) under the small angle approximation. My intuition is that the final result should be some sort of combinations of a harmonic oscillator in $\theta$ and a free particle in $\phi$, but it's not obvious to...
Homework Statement
[/B]
A particle of mass 'm' is initially in a ground state of 1- D Harmonic oscillator potential V(x) = (1/2) kx2 . If the spring constant of the oscillator is suddenly doubled, then the probability of finding the particle in ground state of new potential will be?
(A)...
Hello,
in every book and on every website (e.g. here http://farside.ph.utexas.edu/teaching/315/Waves/node13.html) i found for driven harmonic osciallation the same solution for phase angle:θ=atan(ωb/(k−mω^2)) where ω is driven freq., m is mass, k is spring constant. I agree with it =it follows...
Homework Statement
Does the n = 2 state of a quantum harmonic oscillator violate the Heisenberg Uncertainty Principle?
Homework Equations
$$\sigma_x\sigma_p = \frac{\hbar}{2}$$
The Attempt at a Solution
[/B]
I worked out the solution for the second state of the harmonic oscillator...
Making use of the partition function, it is straight forward to show that the entropy of a single quantum harmonic oscillator is:
$$\sigma_{1} = \frac{\hbar\omega/\tau}{\exp(\hbar\omega/\tau) - 1} - \log[1 - \exp(-\hbar\omega/\tau)]$$
However, if we look at the partition function for a single...
I am confused about the difference between the two
In Griffith's 2.3 The Harmonic Oscillator, he superimposes the quantum distribution and classical distribution and says
What I understand for quantum case is that ##|\Psi_{100} (x)|^2## gives the probability we will measure the particle...
Homework Statement
I am considering the Klein Gordon Equation in a box with Dirichlet conditions (i.e., ##\hat{\phi}(x,t)|_{boundary} = 0 ##). 1-D functions that obey the Dirichlet condition on interval ##[0,L]## are of the form below (using the discrete Fourier sine transform)
$$f(x) =...
Homework Statement
Consider an inertial laboratory frame S with coordinates (##\lambda##; ##x##). The Lagrangian for the
relativistic harmonic oscillator in that frame is given by
##L =-mc\sqrt{\dot x^{\mu} \dot x_{\mu}} -\frac {1}{2} k(\Delta x)^2 \frac{\dot x^{0}}{c}## where ##x^0...
Suppose I have a 1-D harmonic oscilator with angular velocity ##\omega## and eigenstates ##|j>## and let the state at ##t=0## be given by ##|\Psi(0)>##. We write ##\Psi(t) = \hat{U}(t)\Psi(0)##. Write ##\hat{U}(t)## as sum over energy eigenstates.
I've previously shown that ##\hat{H} = \sum_j...
$$m_1 \ddot{x} - m_1 g + \frac{k(d-l)}{d}x=0$$
$$m_2 \ddot{y} - m_2 \omega^2 y + \frac{k(d-l)}{d}y=0$$
It is two masses connected by a spring. ##d=\sqrt{x^2 + y^2}## and ##l## is the length of the relaxed spring (a constant).
What is the strategy to solve such a system? I tried substituting...
Hi all,
at the moment I am doing my Master Thesis and have the following problem.
I am trying to measure Data and asign it a proper timestamp. My problem is, that the data is coming in bursts and the timestamps I assign with the software are wrong.
The controller I am using for monitoring the...
This is from *Statistical Physics An Introductory Course* by *Daniel J.Amit*
The text is calculating the energy of internal motions of a diatomic molecule.
The internal energies of a diatomic molecule, i.e. the vibrational energy and the rotational energy is given by...
Hi everyone, I have a great doubt in this problem:
Let a mass m with spin 1/2, subject to the following central potencial V(r):
V(r)=1/2mω2r2
Find the constants of motion and the CSCO to solve the Hamiltonian?
This is my doubt, I can't find the CSCO in this potencial. Is a problem in general...
Homework Statement
Find the first-order corrections to energy and the wavefunction, for a 1D harmonic oscillator which is linearly perturbed by ##H'=ax##.
Homework Equations
First-order correction to the energy is given by, ##E^{(1)}=\langle n|H'|n\rangle##, while first-order correction to the...
Homework Statement
I have ##H'=ax^3+bx^4##, and wish to find the general perturbed wave-functions.
Homework Equations
First-order correction to the wave-function is given by, $$\psi_n^{(1)}=\Sigma_{m\neq n}\frac{\langle\psi_m^{(0)}|H'|\psi_n^{(0)}\rangle}{n-m}|\psi_m^{(0)}\rangle.$$
The...
Hello, I'm studying the section 2.2 of "Introduction to Quantum Mechanics, 2nd edition" (Griffiths), and he shows this equation $$\frac{\partial^2\psi}{\partial x^2} = -k^2\psi , $$ where psi is a function only of x (this equation was derivated from the time-independent Schrödinger equation) and...
Homework Statement
I got an alpha particle (charge 2+) fixed at x=0 and an electron fixed at x=2. I then add a fluor ion (charge 1-) to the right of the electron and we note his position xeq. The question is to find the constant spring (k) relative to the harmonic oscillation made by the fluor...
A quantum mechanical oscillator with the Hamiltonian
H1=p^2/2m +(m(w1)^2 x^2)/2
is initially prepared in its ground state (zero number of oscillatory quanta). Then the
Hamiltonian changes abruptly (almost instantly):
H1→H2=p^2/2m +(m(w2)^2 x^2)/2
What is the mean number of oscillatory quanta...
Homework Statement
How to calculate the matrix elements of the quantum harmonic oscillator Hamiltonian with perturbation to potential of -2cos(\pi x)
The attempt at a solution
H=H_o +H' so H=\frac{p^2}{2m}+\frac{1}{2} m \omega x^2-2cos(\pi x)
I know how to find the matrix of the normal...
Hello, I encountered a mass on a string problem in which the mass, moved from the equilibrium, gets a harmonic motion. The catch, however, is that the mass of the string is not neglected. On the lecture, the prof. wanted to calculate, for some reason, the complete kinetic energy of the system...
Hi,
For a harmonic oscillator in 3D the energy level becomes En = hw(n+3/2) (Note: h = h_bar and n = nx+ny+nz) If I then want the 1st excited state it could be (1,0,0), (0,1,0) and (0,0,1) for x, y and z.
But what happens if for example y has a different value from the beginning? Like this...
Homework Statement
Experimental data for the heat capacity of N2 as a function of temperature are provided.
Estimate the frequency of vibration of the N2 molecule.
Homework Equations
Energy of harmonic oscillator = (n+1/2)ħω
C=7/2kB
Average molecular energy = C*T
But this is an expression...
Hi.
As far as I know, the movement of a harmonic oscillator normally is not considered to be chaotic. Why not? Since the angular frequency can never be known to absolute precision, an error in the phase builds up. I can see that this build-up is only linear in time (if we assume the angular...
Hi,
why there is only odd eigenfunctions for a 1/2 harmonic oscillator where V(x) does not equal infinity in the +ve x direction but for x<0 V(x) = infinity.
I understand that the "ground state" wave function would be 0 as when x is 0 V(x) is infinity and therefore the wavefunction is 0, and...
So this is something that troubled me a bit- in Shankar's PQM, there's an exercise that asks you to find the position expectation value for the harmonic oscillator in a state \psi such that
\psi=\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle)
Where |n\rangle is the n^{th} energy eigenstate of...
Homework Statement
I am trying to obtain the hermite polynomial from the schrødinger equation for a har monic oscillator. My attempt is shown below. Thank you! The derivation is based on this site:
http://www.physicspages.com/2011/02/08/harmonic-oscillator-series-solution/
The Attempt at a...
I studied this from Griffith Chapter 2, with the algebraic (raising and lowering operator) method, we reached the ground state by setting a_Ψ0 = 0 , then we got what the ground state is, and then plugged it in the Schrodinger equation to know the energy, and it turned out to be 0.5 ħω.
My...
Homework Statement
Show that for the one-dimensional linear harmonic oscillator the Hamiltonian is:
[; H = \frac{1}{2}[P^2+\omega ^2 X^2]-\frac{1}{2}\omega \hbar ;]
[; =\frac{1}{2}[P+i\omega X][P-i\omega X]+\frac{1}{2} \omega \hbar ;]
where P, X are the momentum and position operators...
Hello.
I'm studying quantization of electromagnetic field (to see photon!) and on the way to reach harmonic oscillator Hamiltonian as a final stage, sudden transition that the Fourier components of vector potential A become quantum operators is observed. (See...
Hi, I am trying to find the wavefunction of a coherent state of the harmonic oscillator ( potential mw2x2/2 ) with eigenvalue of the lowering operator: b.
I know you can do this is many ways, but I cannot figure out why this particular method does not work.
It can be shown (and you can find...
Homework Statement
"Show that the Hermite polynomials generated in the Taylor series expansion
e(2ξt - t2) = ∑(Hn(ξ)/n!)tn (starting from n=0 to ∞)
are the same as generated in 7.58*."
2. Homework Equations
*7.58 is an equation in the book "Introductory Quantum Mechanics" by...
Homework Statement
Hello,
I'm just curious as to whether I'm going about solving the following problem correctly...
Problem Statement:
A particle mass m and charge q is in the ground state of a one -dimensional harmonic oscillator, the oscillator frequency is ω_o.
An electric field ε_o is...