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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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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60 views

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

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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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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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 ...
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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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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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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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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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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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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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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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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
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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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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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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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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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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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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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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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30 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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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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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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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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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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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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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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31 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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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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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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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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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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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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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
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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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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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 ...
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Harmonic oscillator and cyclic coordinates

I am reading goldstein there is some comment I don't understand. Consider the following hamiltonian $$H = \frac{p^2}{2m} + \frac{kq^2}{2}$$, which can be rewritten as follows $$H = \frac{1}{2m}(p^2 ...
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Period of small oscillations of liquid in a bottle

I have seen that water in a bottle of water when perturbed invariably starts oscillation back and forth, with the shape of the water surface remaining intact. I was wondering if there was any way to ...
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Deriving an equation for the mass of a pendulum (Follow up)?

Following this question: Deriving mass from simple pendulum which is summarized below Some mass $m$ is release from rest at a horizontal position. $m$ reaches the bottom of its path (so directly ...
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Why does my teacher say “the simple harmonic oscillator passes through the amplitude four times in one cycle”?

I asked him to explain it. But he simply told that where the sinusoidal graph the x axis counts as two times. I don't get why. I thought the answer should be three.
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Fluid filled harmonic oscillator

A vessel (preferably circular) filled with water is accelerating unidirectionally such that the level of water is higher on one end than the other. What I want to know is that if the vessel is ...
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1answer
152 views

Combination of Simple Harmonic Motions

Will the combination of 2 Simple Harmonic motions will be an SHM in itself? For example for simple functions such as $$\ f(t)=\sin\omega t-\cos\omega t$$ I can use trigonometry to show that it can ...
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1answer
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AFM cantilevers driven below resonance?

Is there a physical reason why AFM cantilevers are driven below their resonance frequencies? In all of the AFMs I have used, once you measure the resonance frequency of the cantilever, it is set up ...