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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Very confused about effective spring constant

I know that for springs in parallel, the effective spring constant is $k_1+k_2$ and for springs in series the constant is $1/(1/k_1+1/k_2)$. But there are some weird problems where finding the ...
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22 views

Finding the amplitude of a pendulum [on hold]

I'm tring to simulate the behaviour of a pendulum. I have it in the equilibrium position, then I apply on it an initial velocity $\vec{v_{0}}$ Knowing $\vec{v_{0}}$ and its mass m, how can I find the ...
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108 views

Understanding an equation in quantum mechanics (J. J. Sakurai, “Modern QM”, eq. 2.3.13) [closed]

Hello, It is from quantum mechanics book of sakurai. You can see equation 2.3.12b implies 2.3.13. But my question is, how? Could you please show me how can i bring equation 2.3.13 from 2.3.12b ...
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1answer
138 views

Different hamiltonians for quantum harmonic oscillator?

The Hamiltonian for a classical simple harmonic oscillator is $$ H = \frac{p^2}{2m} + \frac{1}{2}m\omega^2x^2$$ With the usual choice of the ladder operators $$a = ...
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81 views

Hamiltonian related to Riemann zeta function [closed]

using the eigenvstates of the Harmonic oscillator could we give a meaning to the Hamiltonian $$ H=\log(a.a^{+}+1) $$ here $ a$ and $ a^{+}$ are the creation/anihilation operators with commutation ...
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207 views

Strange Behavior in Spring Computer Model

To learn more about oscillatory motion which I am learning about in my high school physics class, I have created a computer model of a damped spring where the damping force is proportional to ...
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2answers
444 views

Pendulum Wave Period

Recently I've seen various videos showing the pendulum wave effect. All of the videos which I have found have a pattern which repeats every $60\mathrm{s}$. I am trying to work out the relationship ...
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74 views

Creating an arbitrary state of the quantum simple harmonic oscillator

Suppose $\mathcal{B}=\{|0\rangle, |1\rangle, |2\rangle, ... \}$ is the energy eigen-basis of a quantum simple harmonic oscillator. I want to create the state \begin{equation} |\Psi\rangle = ...
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121 views

Derive Equation For a Cantilever in SHM

I am currently investigating how a hacksaw blade's time period of oscillation changes when I add mass to the end of it or when I change the length it is clamped at. I found the following equation ...
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1answer
56 views

Finding the wave function of a quantum harmonic oscillator [duplicate]

How can I find the wave function of a quantum harmonic oscillator? If I measure its energy several times, my measurements will change the state of a system. All I know are the possible states, given ...
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4answers
6k views

Why doesn't mass of bob affect time period?

Please correct me if I'm going wrong - By the gravitation formula: $F = \frac{G m_1 m_2}{r^2} $, So if the mass of a bob is greater then the torque on it should increase because the Force increased ...
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28 views

The classical correspondence in the harmonic oscillator

In the harmonic oscillator, the ground state is the one with minimum uncertainty (and also other squeezed states that satisfy the 'equal to' in the Heisenberg inequality). This should mean that these ...
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1answer
25 views

Undamped Resonance of a Classical Harmonic Oscillator

Consider an undamped harmonic oscillator. It may be driven at it's natural frequency, $\omega_0^2 = \frac{k}{m}$. According to Feynman, and other sources, were this to happen, the amplitude of the ...
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1answer
34 views

Creation and annhilation operator in the Heisenberg picture

I am trying to calculate the time evolution of the creation/anni. operator in the Heisenber picture. On this webpage http://quantummechanics.ucsd.edu/ph130a/130_notes/node191.html, they used the ...
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1answer
42 views

General solution of a mass spring system

This is the differential equation that describes small amplitude vertical oscillations of a mass $m$ that is hanging from a spring $$\frac{d^2x}{d t^{2}} + \frac{b}{m}\frac{dx}{dt} + \frac{k}{m} x = ...
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26 views

Periodically connected QHO's

I've recently been thinking about what happens when you connect quantum harmonic oscillators in a periodic way. I'm actually thinking about when you take a mass-spring system (which can easily be put ...
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41 views

Constructing a dispersion relation from the Hamiltonian

I'll begin by saying that I'm not entirely clear on if this is possible. I have a Hamiltonian of the form $$ \left( \begin{array}{cccc} \text{$\omega $1} & \text{J12} & 0 & \text{J14} \\ ...
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29 views

Occurance and disappearance of degeneneracies in a periodic structure of (quantum) LC circuits

Introductory part I'm currently studying an analytical model of coupled LC circuits, in preparation for actually performing measurements on such structures. While the final goal will struggle with a ...
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32 views

Oscillation frequency of a dipole

So I have found the potential to be: $U(x) = \frac{\mu_0 m_2 m}{4 \pi} (\frac{1}{(d+x)^3}+\frac{1}{(d-x)^3})$ Afterwards I have found the position which minimized the energy to be $x=0$. (By doing ...
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3answers
174 views

Quantum simple harmonic oscillator interpretation

I am just wondering what does the SHO system from quantum mechanics actually physically represent? Is it just a SHO of a quantum particle, seems a little too obvious for quantum theory? I'm from a ...
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2answers
67 views

What is the physical reason that the undamped driven oscillator has mean power zero?

The instantaneous power absorbed by an undamped driven oscillator is given by:$$\mathbf{P} =-\omega\dfrac{F_o/m}{{\omega_0}^2 -{\omega}^2} F_0 \sin{\omega t}\cos{\omega t}$$. But my book says the mean ...
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129 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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180 views

Spring Stiffness Calculation Problems

This may sound like a trivial question but please hear me out. I am trying to model a 1 DoF electromagnetic vibration sensor (geophone) analytically and with finite elements. A geophone consists of ...
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2answers
57 views

Physical reason behind having greater amplitude when driving frequency$ < $ natural frequency than that when driving frequency $>$ natural frequency

This is quoted from A.P.French's Vibrations & Waves: If the driving force is of low frequency relative to the natural frequency, we would expect the particle to move essentially with the ...
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49 views

How to find the period of a simple pendulum in an elevator going up with an acceleration?

How to find the period of a simple pendulum in an elevator going up with an acceleration of $a$. Don't just say, $T=2 \pi$ $\sqrt{ \frac l {g+a}}$ I want to know how the above equation is formed.
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Is a constant on the RHS of the equation of simple harmonic motion allowed? [closed]

I read at a STEP booklet that we have to know how to bring a simple harmonic motion's equation to the form: $$\frac{\mathrm{d}^2x}{\mathrm{d}t^2} + \omega^2x= c$$ where $c$ is a constant. We also ...
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14 views

The spring of air: Why does a piston undergo SHM when it is displaced lengthening the air-column inside a cylinder filled with air?

Let there be a cylindrical tube, closed at one-end, with a well-fitting but freely moving piston of mass $m$. [. . .] The piston has certain equilibrium position. If the piston is moved a distance ...
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29 views

Period of a spring in SHM (simple harmonic motion)

An object with unknown mass M is hanged on a vertical spring with unknown spring constant K, the spring is in rest and is 14 cm from its normal point (if it didn't had the mass hanged it had less 14 ...
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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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1answer
59 views

Density of states of 3D harmonic oscillator

for the first red box, shouldn't be $\epsilon^2 =\epsilon_{n_x}^2 +\epsilon_{n_y}^2 + \epsilon_{n_z}^2 + 2\epsilon_{n_x}\epsilon_{n_y} + 2 \epsilon_{n_x}\epsilon_{n_z} + ...
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123 views

Is it possible to write the fermionic quantum harmonic oscillator using $P$ and $X$?

The Hamiltonian of the quantum harmonic oscillator is $$\mathcal{H}=\frac{P^2}{2m}+\frac{1}{2}m\omega^2X^2$$ and we can define creation and annihilation operators ...
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169 views

Simple harmonic oscillator, calculate the trajectory in real space

Potential of a simple harmonic oscillator: $$U=\frac{1}{2}k x^2$$ I'm asked to calculate the trajectory of a particle moving in this potential, with initial conditions $x(t=0) = 0$ and $v(t=0)=v_0$. ...
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74 views

Pendulum on the moon, (Highschool)

A simple pendulum used as a clock, set with the correct time at earth, was sent to moon, it was noticed that it is late 36mins for each hour on earth. Calculate the ratio between acceleration of ...
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50 views

Lax representation of the harmonic oscillator

Peter Lax showed that the differential operators $$L=-6\partial_x^2-u,\quad B=-4\partial_x^3-u\partial_x-(1/2)u_x$$ fulfilling the Lax equation $$\dot{L}+[L,B]=0$$ is equivalent to the KdV equation ...
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22 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 ...
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1answer
40 views

Trick for reformulating in terms of centre of mass and relative variables

I am working through a problem that has caused me difficulties in the past. I have the Hamiltonian $$\mathcal{H}=\frac{p_1^2}{2m_1}+\frac{p_2^2}{2m_2}+\frac{k}{2}(q_1-q_2)^2$$ I want to express the ...
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412 views

Coupled Oscillators

This is an exercise of my last exam. Since I couldn't find anybody who solved it or knows how to, it would be really nice if somebody could tell me if my thoughts on it go into the right direction. ...
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46 views

Finding the phase space density of $N$ harmonic oscillators

Consider a system of $N$ identical harmonic oscillators in 1d. The Hamiltonian will be given by $$\mathcal{H}_N=\sum_{i=1}^N \frac{p_i^2}{2m}+\frac{m\omega^2}{2}q_i^2$$ Now supposedly the Hamiltonian ...
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Where is there posted dampening coefficients for viscous dampers?

Depending on the size of the viscous dampener the coefficient will change, however i am trying to model a "physical" system with calculating everything out, so that way i can test failure before it ...
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4answers
1k views

Simple Harmonic Motion - What are the units for $\omega_0$?

I'm trying to understand the units in: $$mx''+kx=0$$ And the general solution is $$x(t)=A \cos(\omega_0 t)+B \sin(\omega_0 t).$$ Let $\omega_0 =\sqrt{\frac{k}{m}}$ - the unit for the spring ...
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1answer
45 views

What are the units of the creation and annihilation operators?

The creation and annihilation operators - also known as ladder operators are; $ \hat{a}^\dagger$ and $\hat{a}$ respectively. Using the equation $\hat{H} = \hbarω\left(\hat{a}^\dagger \hat{a} + ...
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1answer
33 views

Pendulum at an angle

A pendulum has a period of $T$ when swinging from a string. The pendulum is now placed on a frictionless incline at a 30 degree angle. What is the new period of the pendulum?
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1answer
37 views

How to find the spring coefficient of a simply supported beam?

So I've been searching wikipedia and google but nothing can show how to find the spring coefficient of a simply supported beam with a uniformly distributed load. The spring coefficient, $k$, is ...
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12 views

Usage of concept of static deflection on classical mechanics (ex. SHM based problems)

Can anyone explain how the concept of static deflection (static displacement) is used in problems of SHM? Explanation by/with an illustration would be even the more helpful. Thank you
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2answers
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Using fluid mechanics to show that force is directly proportional to velocity

So I am writing a paper about viscous dampers in harmonic oscilators, however I was looking at some old fluid mechanic notes and I thought I had come across what I needed although I have gotten stuck. ...
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1answer
2k views

3D Quantum harmonic oscillator

For an isotropic 3D QHO in a potential $$V(x,y,z)={1\over 2}m\omega^2(x^2+y^2+z^2).$$ I can see by independence of the potential in the $x,y,z$ coordinates that the solution to the Schrodinger ...
4
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2answers
91 views

“Equidistant” spectra in quantum mechanics [duplicate]

In one-dimensional quantum mechanics, it seems that the only kind of potential able to produce an "equidistant" spectrum, i.e. with $E_{n+1}-E_{n}=\text{constant}$, is the harmonic oscillator. Why is ...
38
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5answers
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Why is the harmonic oscillator so important?

I've been wondering what makes the harmonic oscillator such an important model. What I came up with: It is a (relatively) simple system, making it a perfect example for physics students to learn ...
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2answers
36 views

Is it possible to derive the angular frequency of a simple harmonic oscillator using total energy?

I want to show that $$\omega=\sqrt{\frac{k}{m}}$$ using the fact that $$E=K+U=\frac{1}{2}mv_x^2+\frac{1}{2}kx^2=\frac{1}{2}kA^2.$$ The issue is that I have derived a formula that isn't ...
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179 views

Is it possible to write explicitly the exact solution for forced damped harmonic oscillator?

Preamble Consider a damped harmonic oscillator, with his well know differential equation \begin{equation*} m \ddot{x} + c \dot{x} + kx=0 \end{equation*} and let's find the solution that satisfies ...