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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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38 views

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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Trouble with Damped Pendulum

For a lab report, I am measuring the impact of initial angular displacement on the stopping time of a pendulum. It is not ideal to swing a pendulum and wait for it stop completely, so I decided to ...
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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 [on hold]

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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Damped oscillation [closed]

If in an damped system, the damping force varies as $x^2$, what would the displacement would look like with initial conditions, $x(0)= A$ and $\dot x(0)=0$?
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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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Are there stationary state solutions for semi-infinite, quantum harmonic oscillator potentials?

Suppose a quantum system obeying Schrodinger's Equation has the following potential: $$ V(x)=\frac{1}{2}m\omega^2x^2 \ \ \ \ \ \ \text{for} \ \ x\ge-a \\ V(x)=\infty \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \...
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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

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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2answers
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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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52 views

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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69 views

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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43 views

$Q$ 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 < \...
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Quantum Harmonic Oscillator- Solving the Differential Equation at a Limit

The eigenvalue equation for the quantum harmonic oscillator is $$\langle y | E\rangle '' +(2\epsilon-y^2)\langle y| E \rangle=0$$ where $\epsilon = \frac{E}{\hbar\omega}$ and $y=\sqrt{\frac{\hbar}{m\...
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Difference between the equation of SHM and a travelling wave

The displacement of an oscillator is written as $y ( t ) = a \sin (ω t ± φ)$ whereas the equation of a wave is written as $y ( x , t ) = a \sin (ω t − kx )$. The differences I can enumerate are- The ...
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Functions of ladder operators - Identities for or methods of solution

I would like to find the eigenstates of a potential with terms like $$ \left(\frac{1}{4}A^{4}x^{4}-A^{2}x^{2}+1\right)^{2} $$ and am planning to use ladder operators to find the solution in the basis ...
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Harmonic oscillator eq. for complex amplitude ---field quantization

I am new to quantum optics and going through "Introductory quantum optics" by C. Gerry and P. Knight. In chapter 2 they are writing eq. (2.81), the harmonic oscillator eq. for complex amplitude A ...
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Why is the classical solution of an harmonic oscillator more like a coherent state than an eigenstate of the Hamiltonian?

In general the classical harmonic oscillator should be a superposition of eigenstates of Hamiltonian. Why does it always turn out to be a coherent state than any other kind of superposition? Note: A ...
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Why is water in the asymmetric cylinder not capable of Simple Harmonic Motion?

I'm learning physics in a high school. I'm curious why water in the asymmetric cylinder is not capable of SHM. I've learned that water in a symmetric cylinder can make a Simple Harmonic Motion. But ...
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Prove that group velocity is the velocity of energy transport in wave

Generally, the group velocity $v_g = \dfrac{\partial \omega}{\partial k}$ of a wave is the velocity of energy transport. In "Introduction to Solid State Physics", Kittel following is stated: The ...
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Path integral for the harmonic oscillator

I would like to check the derivation for the harmonic oscillator in here using the Gateaux derivative explicitly. (I know that the same result can be achieved using this but I want to gain practice ...
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Is the statement that $U(x)$ is quadratic for simple harmonic motion equally strong as the statement that $F(x)$ is linear?

Is the statement "If the potential energy of a particle under oscillatory motion is directly proportional to the second power of displacement from the mean position, the particle performs a ...
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What is the quantum mechanical turning point of the $n^{th}$ energy eigenstate of an oscillator? [closed]

I am looking for an analytical expression for the most likely position for a quantum harmonic oscillator (which I refer to as the quantum mechanical "turning points"), in terms of $n$. For the quantum ...
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4answers
49 views

Why is acceleration intuitively greatest at endpoints of simple harmonic motion? [closed]

In simple harmonic motion (for example a spring moving horizontally), acceleration is greatest when the mass reaches either end of the spring. Using the formula $F = ma = kx$ and then $a = \frac{kx}{m}...
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What's wrong with my approach?

Let's say you have a pendulum moving in SHM with a length $L$ and an amplitude $\theta$. Suppose you wanted to find the linear velocity $v$ at it's lowest point. The way that gets the right answer ...
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1answer
44 views

Finding momentum amplitude of a wave packet when initial wave form is given

At time $t = 0$, a one-dimensional free wave packet for a particle of mass $m$ takes the form: $$ \Psi(x,0) = \begin{cases} \frac{1}{\sqrt{L}}e^{i\alpha x} & \text{for } -L/2 < x < +L/2 \\ ...
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1answer
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Simple harmonic motion qsn [closed]

Have been stuck with this question from classical mechanics under the simple harmonic motion the question is saying that if $$y=a\cos(\omega t)+b\sin(\omega t)$$ show it represents simple harmonic ...
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Reading energy Eigenvalues from a Hamiltonian matrix for 1D harmonic oscillator

After a perturbation $V(x)$ added to the system, a matrix element $H_{nn}$ calculated in unperturbed Eigenstates for one-dimensional harmonic oscillator is given as: $$\epsilon \hbar \omega_0\begin{...
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Extracting solution from driven SHM

I guess maybe I should rather ask at the math stack exchange? I have a simple harmonic undamped oscillator driven by a cosinusoidal force: $$ \ddot{x}+\omega_o^2x=f\cos(\Omega t).$$ I've managed to ...