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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How can amplitudes be treated as vectors in Simple Harmonic motion?

The amplitudes of 2 SHM are scalors. When we combine the two SHM eq.(lying along the same line), the resultant expression becomes of amplitudes treated as vectors and the phase angle between them as ...
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21 views

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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77 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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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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27 views

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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67 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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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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29 views

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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108 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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44 views

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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1answer
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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1answer
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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49 views

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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35 views

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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1answer
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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176 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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2answers
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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24 views

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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1answer
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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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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1answer
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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1answer
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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5answers
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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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65 views

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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22 views

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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1answer
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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29 views

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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1answer
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 ...