The term "harmonic oscillator" is used to describe any system with a "linear" restoring force that tends to return the system to a equilibrium state. There is both a classical harmonic oscillator and a quantum harmonic oscillator. Both are used to as toy problems that describe many physical systems.

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Angular momentum for elliptic path in 2D isotropic oscillator

Assume a 2D isotropic oscillator, i.e $$U = \frac{1}{2}m\omega^2(x^2+y^2),$$ and assume for simplicity that the oscillator performs elliptical motion, with major axis $A$ and semi-major axis $B$. My ...
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76 views

Complex resonant frequency not resonant without imaginary part. So can I still just take real part as solution?

I am working with a matrix on a harmonic oscillator problem and the lowest (absolute) frequency $\omega_0$ where the matrix becomes singular is the resonant frequency. Now I obtained this frequency ...
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68 views

Are there any conditions under which the Christoffel symbols can be treated as a damping term in a harmonic oscillator?

(Mathjax did not seem to be working as I composed this question. Hopefully it will kick into action once I post.) Note I am a novice at tensor notation. I am working with the following Lagrangian ...
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78 views

Harmonic oscillator: if $E=\frac{1}{2} q \dot{\theta}^2+\frac{1}{2} s \theta^2$ then $\omega=\sqrt{\frac{s}{q}}$?

Consider an harmonic oscillator. Suppose that I manage to write the mechanical energy as a function of a quantity, like the angle $\theta$ in this way $$E=\frac{1}{2} q \dot{\theta}^2+\frac{1}{2} s ...
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Resonant frequency(s) of system of resonant objects?

I am trying to gain a better understanding of resonance when we are dealing with a coupled system of resonating objects. For example, say we have a single gas bubble in liquid and suppose it ...
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29 views

Simple harmonic motion in a horizontally accelerating lift [duplicate]

How would the time period of a pendulum in a lift be if the lift was accelerating horizontally to the right ? My teacher did it by putting a pseudo force "ma" to the left, then getting the resultant ...
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66 views

Period on the phase plane (small oscillations)

I have this formula to calculate the period of a motion in the phase space (plan, in this case) along a phase curve. \begin{equation} T(E)=\int_{x_1}^{x_2}\frac{dx}{\sqrt{2(E-U(x))}} \end{equation} ...
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65 views

Why doesn't amplitude affect frequency for an object in simple harmonic motion even though we have an equation relating the two?

I understand conceptually why amplitude doesn't affect frequency (especially when relating to sound - singing louder shouldn't change the frequency of the note I sing). However, the equation for the ...
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27 views

Initial value in rescaling differential equation

I've re scaled the simple harmonic oscillator differential equation as below: original equation: $d^2x/dt^2+\omega^2x=0$ re scaling factor: $\omega t\to t'$ re scaled (dimensionless) equation: ...
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103 views

Why do trees sway?

Resonance can also occur in three dimensions (such as wind induced swaying) I tried to make a free body diagram (I know it is terribly wrong) to find the forces that causes the tree to undergo ...
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31 views

Confusion on Time and Ensemble Averages of Classical Harmonic Oscillator

Assume we have a classical harmonic oscillator $$ \ddot{x} = -k^2x.$$ Then the general solutions are of the form $x(t) = x_0cos(kt) + \frac{v_0}{k}sin(kt)$ where $x_0$ and $v_0$ are initial ...
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SHM with acceleration at mean position

Suppose we have an equation of motion as $$\frac{d^2x}{dt^2} = -kx + c,$$ then can it be called a SHM? Since acceleration is still proportional to displacement. But then, how will we define the mean ...
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Does everything in this world vibrate? Why?

I was introduced with something called natural frequency of a body. It was taught us in the chapter Simple Harmonic Motion. Our teacher said that everything in this world has its natural frequency of ...
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33 views

Is there no rotational mechanics for non rigid body? How do we deal with such situations in real life?

I am in 12th grade right now. We have a chapter on Rotational dynamics in which it is clearly stated that it is for rigid bodies. I understand that, Moment of Inertia will remain constant only for ...
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59 views

A problem about the harmonic oscillator in Quantum mechanics

When I learned quantum mechanics by reading Griffith's book called Introduction to quantum mechanics, I was confused by his description. In Page 53 of the 2ed edition book, after got the recursion ...
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Can I determine the maximum acceleration with frequency and magnitude?

Consider a mass oscillating up and down on a spring with negligible energy loss. If only the frequency and magnitude of the oscillation is known, how can one determine the maximum acceleration of the ...
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106 views

Sakurai Exercise 2.17 (Harmonic Oscillator, Ladder operators) [closed]

I the field of the harmonic oscillator and ladder operators I am trying to solve exercise 2.17 from Sakurai and want to proof the following relation $$ \langle x^{2n} \rangle = (2n - 1)!! \langle ...
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Simple harmonic oscillators [closed]

A block whose mass m is $680$ g is fastened to a spring whose constant k is $65$ N/m . The block is pulled a distnce $x=11$ cm from its equilibrium postion at $x=0$ cm on a frictionless surface and ...
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63 views

Damped Pendulum (generalised)

I know the differential equation for the swinging of a simple pendulum: $\displaystyle\frac{\partial^2\theta}{\partial t^2} + \left(\frac{g}{L}\right)\sin\theta = 0$ where: $L$ is the length of ...
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48 views

Simple Harmonic Motion given velocity and acceleration

I am trying to understand how to relate velocity and acceleration of an object to it's amplitude, period, and frequency given only the following: An object of mass m=20kg moves with SHM along the ...
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82 views

Time period of a pendulum when a constant horizontal force acts

The time period of a pendulum is given by $$T=2\pi\sqrt{\frac{l}{g}}$$ Will the time period change if a constant horizontal force acts on the pendulum? For example, if a force $F$ acts on the Bob ...
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1D Harmonic Oscillator: Eigenstate (|x=0>) at position x=0

Given an harmonic oscillator I need to calculate the eigenvector $|x=0\rangle$. Knowing that $$x|x=0\rangle = 0 \quad \Rightarrow \quad (a + a^\dagger) | x = 0 \rangle = 0 $$ I started to plug in the ...
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55 views

Phase angle in simple harmonic motion

I know the phase constant depends upon the choice of the instant $t=0$. Is it compulsory that the phase constant must be between $[0,2 \pi]$? I know that after $2\pi$ the motion will repeat itself so ...
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What is meant by taking the partial derivative of the Hamiltonian in this situation?

I'm doing a computation involving the quantum mechanical harmonic oscillator, and I have an expression of the form $\frac{\partial}{\partial \omega} \hat{H}$ where $$\hat{H} = \frac{1}{2m} \left( - ...
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Damping Coefficient of SHM

In lab for my physics of digital systems class, we were told to find the damping coefficient of a spring experiencing simple harmonic oscillation. We were given the formula $$x = A e^{\left( ...
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Confusion regarding sinusodial function in SHM [closed]

A block is connected to a spring. The block is pulled from the initial position $t=0$ and $x=0$ to lets say Zcm and released. Now if I have to write the SHM equation when the body is Z/2 distance away ...
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90 views

Evaluating path integral

I am having some trouble remembering how to evaluate path integrals involving multiple particles. Suppose that I have two interacting particles with Lagrangian $$L= \frac{1}{2}m \dot y^2-\frac{1}{2}m ...
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173 views

Understanding transverse oscillation in 1 mass, 2 spring systems

Lately I have been working through some nice problems on mass-spring systems. There are tons of different configurations - multiple masses, multiple springs, parallel/series, etc. A few possible ...
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49 views

Force applied by a spring stretched to a different direction

This may be a bit basic but I am unsure of the answer. Assume the following simple setup: a spring with a spring constant k and of length L, connected to mass m. What is the force applied by the ...
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51 views

Q factor of driven oscillator

In driven oscillator it can be explained by the following differential equation $$\ddot{x} + 2\beta \dot {x} + \omega_0 ^2x = A \cos(\omega t)$$ where the $2\beta$ is coefficient of friction, the ...
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70 views

Motion of $n$ bodies connected with springs

Let's consider $n$ cuboids moving without friction, each of mass $m_i$. Each wo neighboring cuboids are connected with a spring of the coefficient $k$. ...
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What does it mean for a particle to be subjected to 'more than one' simple harmonic motion? [closed]

Also what can we say now about its --> Resulting Energy? -> Resulting Amplitude? -> Maximum Velocity? Please help as I am not able to understand the process going on. I also tried to represent this ...
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47 views

Tension in a vibrating loop

Consider a super basic 1D vibrating string, with standing waves on it. The string has length $L$, and the wave propagates at a velocity $v$. The fundamental frequency $f_1$ is given by $$f_1 = ...
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69 views

Quantum mechanics: SHM expectation of $x^2$ time independent for one state but not superposition of 2 states?

my answers for the first bits $$\langle H\rangle =n\hbar\omega$$ $$\langle x\rangle =\sqrt\frac{\hbar n}{2m\omega}\cos(\omega t)$$ $$\langle p\rangle =-\sqrt\frac{\hbar m\omega n}{2}\sin(\omega ...
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75 views

Hamiltonian of a quantum harmonic oscillator

On page 286-287 of Nielsen Chuang's Quantum Information and Quantum Computation (10th edition) book, the Hamiltonian for a quantum harmonic oscillator is approximated as $H=a^\dagger a.$ What are the ...
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35 views

Frequency of driven damped oscillation and the driven force

For a driven damped oscillation, if the driven force $F = F_0 \cos(\omega t)$, then the solution to the motion is $$x = A \cos(\omega t+\varphi ) \, .$$ Why must the the oscillation and the driven ...
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Slowest mode of two string with different masses

I was watching MIT OCW recitation course. For most slowest mode, we have frequency with $\lambda= 4L$ when, the total lenght of string is $2L$. But for second slowest mode, why we do not take $\lambda ...
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A conceptual doubt regarding Longitudinal Waves

I was recently studying about Longitudinal Waves and I have a little trouble understanding the Displacement versus distance graph for these waves. Firstly, how exactly does one come up with such a ...
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37 views

How can I derivate the solution of the under-damped harmonic oscillator?

The equation is $$ m\ddot x =-k x -\gamma x$$ Multiply by $1/m$ we get: $$ \ddot x=-\omega_0^2x - \beta x $$ We use the ansatz $x(t)=e^{\lambda t}$ So for the $\lambda_{1,2}$ we get: $$ ...
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A conceptual doubt regarding Forced Oscillations and Resonance

While studying about the Resonance and Forced Oscillations, I came across a graph in my textbook that is given below:- Now, the author writes As the amount of damping increases, the peak shifts ...
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34 views

Conceptual Doubt regarding Simple Harmonic Motion

While studying about a simple pendulum, I came across the following line. At the bottom of the swing the tension will actually be greater than the weight, causing the bob to move in a circle. ...
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Damped oscillator undergoing viscous and coulomb damping

I am asking this for a damped oscillator in this situation: What if there are two damping forces present, one being friction and the other being air resistance? When the damping force is dependent on ...
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Obtaining the correlator of harmonic oscillator

The two point function for harmonic oscillator can be written as $$\langle \langle x(t_1)x(t_2) \rangle \rangle =\frac{\int Dx(t) x(t_1)x(t_2) e^{-S(x)}}{\int Dx(t) e^{-S(x)}} \tag{21} \, .$$ In ...
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Calculating the RMS angle of a driven oscillator

We are given that the oscillator obeys the following DE: $\ddot{\theta} + \omega_0^2\theta = \text{cos}(\omega t)$. The solution is $\theta = Ae^{i\omega_0t} + Be^{-i\omega_0t} + ...
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Are there exact expressions for the Floquet states of a periodically-forced, undamped harmonic oscillator?

For this question I was looking for the Floquet states of a quantum harmonic oscillator driven by a non-resonant harmonic force, and I had a rather harder time finding it than the simplicity of the ...
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Harmonic oscillator identity : show $ \sum_{k = 0}^{n-1} \phi_k(x)^2 = \phi_n'(x)^2 + (n - \frac{x^2}{4})\phi_n(x)^2 $ [closed]

I am reading about Hermite polynomials in a math textbook and I am sure they are working too hard. Let $H = p^2 + x^2$ be the quantum mechanical harmonic oscillator. Or perhaps $H = \frac{1}{2m}p^2 ...
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83 views

Harmonic oscillator with heat bath

I need to calculate the expectation value for a harmonic oscillator coupled to a heat bath using the trace method. I know that the density operator looks like: $$\rho = \frac{e^{-H / k_B ...
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Equations of motion for normal modes

I really need some help understanding how to find normal modes. So I brought the euler-lagrange equation of my probelm to this form: $X'' = -AX.$ Where $X$ is the coordinates vector. So I found the ...
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94 views

Approximating taylor series for a harmonic oscillator

The elastic potential energy is defined as $V\left ( x \right )=\frac{1}{2}Kx^{2}$ Then suppose the point $x=x_{0}$ is the point of a local minimum. We know that any potential about a local ...
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Wave oscillation [closed]

A 2 kg block is attached to a spring for which $k=200N/m$ . It is held at an extension of 5 cm and then released at t=0 , Find a, the displacement as a function of time and b, the velocity when ...