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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Find the period of a pendulum system [on hold]

How to solve the following problem, A mathematical pendulum is formed by a cable with length $l$ and a sphere at it's end with mass $m$. While the pendulum swings at it's position of ...
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1answer
39 views

Simple harmonic oscillators [on hold]

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 ...
3
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49 views

Harmonic oscillator with squared damping term [migrated]

Does a solution exist for a harmonic oscillator with a squared damping term? $$m\ddot{u}+c\dot{u}^2+ku=0$$
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2answers
33 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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3answers
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Using $\sin()$ or $\cos()$ for computing SHM?

In simple harmonic motion, you can use either the sin or cos form of the equation but my question is which one do you use when and why? I am having a tough time understanding this, so any help would ...
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26 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
25 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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2answers
32 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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24 views

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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1answer
46 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( - ...
3
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1answer
40 views

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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2answers
32 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
82 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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1answer
109 views

Cause of SHM of liquid-column in V-tube

Suppose there is a v-shaped tube filled with water. The left limb is at $\theta_1$ & the right limb is at $\theta_2$ with the horizontal base. Initially, the level of water in both the columns are ...
4
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3answers
7k views

Can someone please derive $T=2\pi\sqrt{l/g}$ or prove it without using calculus?

I don't know much calculus, but I want to know that how one derives the formula to find time period $T$ of a simple pendulum.
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56 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
35 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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2answers
2k views

Coupled quantum harmonic oscillator

Given the following Hamiltonian for two identical linear oscillators with spring constant $k$ and interaction potential $\alpha x_1x_2$; I was asked to find the expectation value $\langle ...
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2answers
40 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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1answer
64 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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1answer
23 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
28 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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1answer
59 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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1answer
31 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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2answers
68 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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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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5answers
99 views

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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1answer
31 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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1answer
185 views

$N$ coupled quantum harmonic oscillators

I want to find the wave functions of $N$ coupled quantum harmonic oscillators having the following hamiltonian: \begin{eqnarray} H &=& \sum_{i=1}^N \left(\frac{p^2_i}{2m_i} + ...
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1answer
30 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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30 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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19 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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1answer
72 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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2answers
140 views

Does spatial coupling prohibit resonances due to an external source field?

The harmonic oscillator coupled to a sinodial external source $$\frac{\partial^2 x(t)}{\partial t^2}+\omega_0^2 x(t)=F_0\sin(\omega_\text{ext}\ t),$$ has the solution $$x(t)=x(0)\cos(\omega_0 t)+C ...
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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 ...
2
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2answers
95 views

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
13 views

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} + ...
2
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1answer
86 views

Evolution operator in driven harmonic oscillator

I have a question about a problem I was set recently. The exercise reads: The Hamiltonian of an harmonic oscillator driven by a classical force is $H=H_0+H_1$ with $$H_0=\hbar \omega \left( ...
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0answers
18 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 ...
3
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1answer
51 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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1answer
34 views

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 ...
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1answer
37 views

Can a uniform circular motion be considered as simple harmonic motion? [duplicate]

The acceleration in a circular motion is directed towards the centre and is directly proportional to the radius of circle if it has uniform angular velocity. Is circular motion with uniform angular ...
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1answer
30 views

Oscillating block amplitude change when 2nd mass added [closed]

There is a oscillating block with amplitude $A$ and mass $M$. We add a mass $m$ with zero velocity and vertically.when the block is in this two conditions: ...
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1answer
7k views

Evolution operator for time-dependent Hamiltonian

When I studied QM I'm only working with time independent Hamiltonians. In this case the unitary evolution operator has the form $$\hat{U}=e^{-\frac{i}{\hbar}Ht}$$ that follows from this equation $$ ...
4
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268 views

Are Black Holes set to take over the Harmonic Oscillator in the 21st century? [closed]

A few years ago I attended a talk given by Andy Strominger entitled Black Holes- The Harmonic Oscillators of the 21st Century. This talk, ...
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1answer
43 views

Birkhoff Method for Harmonic Oscillator Perturbation

Problem: Given Hamiltonian $$H = \frac12 (p^{2}+q^{2})+q^{3}-3qp^{2}$$ make a perturbative canonical transformation $(q,p) \rightarrow (Q,P)$ such that the new Hamiltonian, apart from terms of degree ...
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3answers
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Hilbert space of harmonic oscillator: Countable vs uncountable?

Hm, this just occurred to me while answering another question: If I write the Hamiltonian for a harmonic oscillator as $$H = \frac{p^2}{2m} + \frac{1}{2} m \omega^2 x^2$$ then wouldn't one set of ...
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1answer
20 views

Simple Harmonic Motion Derivatives, and the equation

If the velocity time graph of a SHM is the derivative of the Distance time graph, and the kinetic energy of the mass in the SHM is maximum when the displacement is 0, how can the maximum velocity be ...
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27 views

The “general uncertainty” of the harmonic oscillator defies the correspondence principle?

If you use the definition of $(\Delta x)^2 = \langle n | x^2 | n \rangle - \langle n | x | n \rangle^2$ and the same for $(\Delta p)^2$ to calculate $\Delta x \Delta p$ for the state $|n\rangle$ of a ...