Questions tagged [harmonic-oscillator]

The term "harmonic oscillator" is used to describe any system with a "linear" restoring force that tends to return the system to an 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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45 views

Does the number operator $N$ of the Quantum Harmonic Oscillator commute with $x$? [on hold]

Does the number operator $N$ of the Quantum Harmonic Oscillator commute with $x$?
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Can anybody tell me the difference between these two velocity?

In Waves i earlier studied that velocity of the wave is given $wa\sqrt{1-y^2}$ w= angular velocity a=amplitude of the wave y=post. of wave at any time and now when i am studying wave optics ...
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Velocity in SHM

By book defines SHM as Simple harmonic motion is defined as the projection on any diameter of a graph point moving in a circle with uniform speed. but in the next line it says - Moving back ...
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Frequency of oscillation in Simple U-Tube Manometer by Energy Method

Force Balance Method By balancing accelerating and restoring Force, it's easy to find the angular frequency $\omega$ Restoring Force= (Weight of the extra height of liquid column)=$$\rho*Area*2x*g$$ ...
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Relation between time period and amplitude of a pendulum [on hold]

What is the relation between time period and amplitude of a simple pendulum? Does time period increase with the increase in amplitude? What will be the graph of this relation?
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Brownian Harmonic Oscillator

I am trying to solve the below problem, but I am unsure about my attempted solution. Problem statement This problem comes from exercise 2 of the notes on Green's function. My attempted solution ...
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Thermal harmonic oscillator as bose factor

Let us consider a harmonic oscillator $V=\frac{1}{2}m\omega x^2$ in contact with a heat reservoir. Take the partition function $$Z=\sum e^{-\beta \hbar \omega (n+1/2)}=e^{-\beta \hbar \omega /2} \sum ...
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Discretizing the Action for Feynman Path Integral

I am trying to understand how to compute path integrals, given a Lagrangian. I understand how it is done for the free particle, but I am confused for other actions. I am having trouble understanding ...
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One spin-1/2 particle in B field in a 3D harmonic potential (Part III)

Consider a spin-1/2 particle in a magnetic field (say in z direction) and in a harmonic potential. For the 3D harmonic oscillator component, The Hamiltonian $H_1= \frac{p^2}{2m}+\frac{1}{2}m\omega ^2r^...
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One spin-1/2 particle in B field in a 3D harmonic potential (Part I)

Consider a spin-1/2 particle in a magnetic field (say in z direction) and in a harmonic potential. For the 3D harmonic oscillator component, The Hamiltonian $H_1= \frac{p^2}{2m}+\frac{1}{2}m\omega ^2r^...
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Central force and planar motion - what am I not getting [closed]

I am trying to solve the problem of planar motion for a body in the case of a radial force, $f(r)=-kr$, $k$ a constant. By considering polar coordinates $(r,\theta)$ for which the trajectory becomes ...
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What's the history behind the quantum harmonic oscillator?

The quantum harmonic oscillator has played central role of almost every field of physics: quantum field theory, particle physics, quantum mechanics, etc. I want to know what inspired people through ...
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Partition function of 2 particles connected by a spring

Consider a system composed of two, point-like particles connected by a linear spring, enclosed in a box. For simplicity, consider the system to be one dimensional. The energy of such a system is $E(...
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Symmetrical design of a tuning fork [duplicate]

I was reading Ch. 10 of Kleppner and Kolenkow and I came across an explanation which said "The energy loss in a tuning fork is primarily due to heating of the metal. Air friction and energy loss to ...
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Non-inertial analysis of the time-period of a pendulum in an elevator

Consider a Simple Pendulum attached to the roof of a elevator moving upwards with a constant acceleration A. Analysis of the pendulum from the elevator's frame shows that the effective downward ...
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Problem in Arnold's Mathematical Methods of Classical Mechanics regarding Lissajous figures

In Arnolds book Mathematical Methods of Classical Mechanics he defines the system $$\ddot{x}_1=-x_1,\,\,\,\ddot{x}_2=-\omega^2x_2^2.$$ The potential energy is $U(x_1,x_2)=\frac{1}{2}(x_1^2+\omega^...
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Is there a classical harmonic oscillator system with a discrete set of modes?

For a free rope arbitrary wavelengths are allowed. But if we fix a rope at two ends only a discrete set of wavelengths is allowed. Is there are similar way to modify a harmonic oscillator such that ...
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Factorising the supersymmetric oscillator Hamiltonian: what (anti)commutes?

In this paper on supersymmetry, the Hamiltonian for the supersymmetric oscillator is given: $$H = \frac12 p^2 + \frac12 \omega^2 x^2 + \omega\bar\psi\psi.$$ Furthermore, its factorisation is given as $...
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Damping of horizontal mass-spring system at high velocities [closed]

I am trying to study the damping of a horizontal mass-spring system at high velocities. The equation for damping due to a viscous fluid at low velocities is: $$ma+cv+kx=0$$ I changed the equation ...
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What are the energy eigenstates for a modified quantum harmonic oscillator?

Imagine a particle obeying Schrodinger's Equation with an harmonic oscillator potential modified with an additional linear potential and cut off with an infinite potential barrier at $x=0$. That is, $$...
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Energy eigenfunctions of a truncated harmonic oscillator-like potential

Assume a potential of the form \begin{eqnarray}V(x) &=& \frac{1}{2}m\omega^2x^2,~-x_0\leq x\leq x_0,\\&=& 0,\hspace{2.5cm}{\rm otherwise} \end{eqnarray} where $x_0$ is a finite ...
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Is there any proof that $F=-kx$?

How do you proof that $F = - kx $? And why is there (-) on the formula(?)
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Superposed simple harmonic oscillators

When deriving the equation for the superposed amplitude: $$A^2=A_1^2+A_2^2+2A_1A_2 \cos(\phi_2-\phi_1)$$ From $$x_1(t)=A_1 \cos(\omega t+\phi_1)$$and $$x_2(t)=A_2 \cos(\omega t+\phi_2)$$ How do you ...
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Question regarding radial raising/lowering operator for isotropic harmonic oscillator

I understand the symmetry structure of the 3D isotropic harmonic oscillator $H = \frac{\mathbf{P}^2}{2\mu} + \frac{1}{2}m\omega^2\mathbf{X}^2$ as follows. The energy levels are $E_N = \hslash \omega (...
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Do Hermite polynomials imply a weight for quantum harmonic oscillator wavefunctions?

I know that solutions of quantum harmonic oscillator can be expressed in the form of Hermite polynomials. But I recently came to know that Hermite polynomials are actually orthogonal polynomials ...
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Damped Forced Oscillator with initial conditions

The equation of motion of a damped forced oscillator is; $$\ddot{x}(t)+\gamma\dot{x}(t)+\omega_0^2x(t)=F(t),$$ $$F(t) = F_0 \cos(\omega_dt);$$ also for the purpose of this problem we may set $\...
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Complex number representation of a wave

There are some aspects to waves I am confused, for instance in Chapter 11. Fraunhofer Diffraction. The incoming electric fields can be partially expressed as $e^{i(kr-\omega t)}$. I have two ...
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Pendulum: Time period for effective length varying with angle

For a pendulum whose effective length changes with angle, my approach would be to write the length as a function of the angle: $\ell(\theta)$, then using the torque derivation of the time period I do ...
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Spring constant of harmonic oscillator [closed]

I got a task from my lecturer to solve a differential equation for a simple harmonic oscillator: $$m{d^2\vec{r} \over dt^2}=-k^2\vec{r}.$$ So far, I have managed to find this equation only in one book....
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Confusion on Quantum Harmonic Oscillator Eigenvalues

In standard PDE theory, one generates eigenvalues to Sturm-Liouville problems over a finite domain. So, for a wave equation, we have an infinite number of eigenvalues $\lambda_n$ for a Dirichlet ...
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Elongation of a simple pendulum

One of the questions on this weeks question sheet asks for the maximum elongation of a simple pendulum. The pendulum is set in motion on the moon with f = 0.5Hz. What is meant by the elongation of the ...
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Hermitian phase operator and quantum harmonic oscillator

I need to apply a hermitian phase operator $\dfrac{1}{\sqrt{1+\hat{a}^\dagger\hat{a}}}\hat{a}$ to the nth harmonic oscillator state, but I have no idea how to interpret the square root of a ...
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Average values of $\langle n|x_{op}|n\rangle$ and $\langle n|p_{op}|n\rangle$ [closed]

Let an harmonic oscillator described by the hamiltonian $H=p^2/2m+(1/2)mw^2x^2$. I have determined that the average values of the observables $x$ and $p$ in energy eigenstates , $\langle n|x|n\rangle$ ...
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Energy difference in the Hamiltonian $H_{b}=\hbar\omega(a^{\dagger}a+\frac{1}{2})+\hbar\omega (b-1/2)(a^{\dagger}a^{\dagger} +aa)$ [closed]

Given that a Hamiltonian is on the form $$H_{b}=\hbar\omega(a^{\dagger}a+\frac{1}{2})+\hbar\omega (b-\frac{1}{2})(a^{\dagger}a^{\dagger} +aa)$$ where $b$ is a dimensionless real number in the ...
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Show that the expectation value of the position with a Harmonic oscillator is $\langle x \rangle_{\psi(t)} = A\cos(\omega t + \phi)$

I am working on a harmonic oscillator problem I have not seen before. Given the position operator $$\hat{x} = \sqrt{\frac{\hbar}{2m\omega}}(a^{\dagger}+a)$$ and $$|{\psi}\rangle = \sum_{n=1}^{\infty}...
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Advantage of narrow cords in bifilar pendulums

I'm a high school student doing further reading for a project I'm starting and I came across this experiment. On slide eight it mentions that the fibres holding up the pendulum are 0.2 mm in ...
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Assume velocity in SHM [closed]

In this weeks problem set we have been given a 1D horizontal mass spring problem. The only initial conditions we have been given are: the system is released from displacement 4mm and that T = $\pi$ s. ...
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Two-coupled oscillator: Doubt in finding normal modes and natural frequency

I want to find the natural frequency of a two coupled oscillator system like this- My book does it this way but I don't really get it. The equations of motion for the pendula are- $$I\frac{d^...
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Finding general equation for motion of a radioactive particle performing SHM [closed]

Let us assume we have a particle of initial mass $m_{0}$ such that a general time $t$: $$ m(t) = m_{0} e^{- \lambda t} $$ Now, let us say this particle is attached to a spring of spring constant $k$,...
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Is it possible to construct a state for harmonic oscillator given the mean energy?

The harmonic oscillator is defined by the mean value energy $\langle E\rangle=\frac{2}{3} \hbar\omega$. Can we have a wavefunction which describes such a state? Any help is appreciated. Is it ...
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Harmonic oscillator with potential shifted by a constant

I've been thinking a lot about changes to the harmonic oscillator potential, and I was looking into the problem where $$V(x) = \frac{1}{2}m\omega ^2 x^2 + C$$ where $C$ is some positive real ...
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Does changing the angle of a pendulum also shift the coordinate plane w.r.t which we give rectangular components to the $mg$ vector?

So given a simple pendulum, which makes an angle of 0 with the vertical axis in it's resting position.Now the pendulum is moved to a side by an angle $\theta$ with the vertical axis. The components of ...
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Does resonance just depends upon the frequency of the external periodic force and the natural frequency of an object?

I am a little confused about the phenomenon of resonance, I read that it occurs when the frequency of an external force matches the natural frequency of an object. So, it was given that soldiers ...
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$Q$ factor of a pendulum

according to the definition of the Q-factor of damping, it is given by: $Q = 2\pi\frac{Energy \; Stored}{ Energy \;Dissipated \; per \;cycle }$ Q = 1⁄2 --> Critical damping Q > ​1⁄2 --> Over ...
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Canonical quantisation harmonic oscillator

I have a question on the canonical quantisation as described at the linked wiki page: https://en.wikipedia.org/wiki/Quantum_field_theory#Canonical_quantisation we take the displacement of a classical ...
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Why is resultant displacement in an composition of simple harmonic motion the sum of individual displacements?

I recently came across the concept of the composition in simple harmonic motion. A paragraph says that: If $$x_1 = A_1sin(\omega t)$$ $$x_2 = A_1sin(\omega t + \phi)$$ Then, the resultant ...
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Why is the force of gravity positive for an oscillating spring?

When analyzing the movement of a weight attached to a spring, many sources set up the force equation using newton’s second law as follows. $$mg-k(L+x)=ma$$ where $L$ is the length that the mass $m$ ...
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Quantum harmonic oscillator hamiltonian in terms of the parity operator

Can you write the quantum harmonic oscillator hamiltonian $$H = -\dfrac{\hbar^2}{2m}\dfrac{d^2}{dx^2}+\dfrac{1}{2}m\omega^2x^2$$ in terms of the parity operator $P$?
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Oscillator integral for frequency

If, for a (not necessarily simple harmonic) oscillator I have that $$\frac{dx}{dt} = G(x)$$ then I can express the period of motion as $$\int_{0}^{T/4} dt = \int_{0}^{X_{max}} \frac{dx}{G(x)}.$$ What ...
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Period of Small Oscillations for Perturbation on SHO

I am trying to find the period of small oscillations of the potential $$ V(x) = \frac{1}{2}m\omega_0^2(x^2-bx^4) $$ It is given that the particle oscillates between $-a$ and $a$ for some $a < \...