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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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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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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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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 ...
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 $a = \frac{kx}{m}... 2answers 60 views What's wrong with my approach? Let's say you have a pendulum moving in SHM with a length$L$and an amplitude$\theta$. Suppose you wanted to find the linear velocity$v$at it's lowest point. The way that gets the right answer ... 1answer 44 views Finding momentum amplitude of a wave packet when initial wave form is given At time$t = 0$, a one-dimensional free wave packet for a particle of mass$m\$ takes the form: $$\Psi(x,0) = \begin{cases} \frac{1}{\sqrt{L}}e^{i\alpha x} & \text{for } -L/2 < x < +L/2 \\ ... 1answer 41 views Simple harmonic motion qsn [closed] Have been stuck with this question from classical mechanics under the simple harmonic motion the question is saying that if$$y=a\cos(\omega t)+b\sin(\omega t)$$show it represents simple harmonic ... 1answer 43 views Reading energy Eigenvalues from a Hamiltonian matrix for 1D harmonic oscillator After a perturbation V(x) added to the system, a matrix element H_{nn} calculated in unperturbed Eigenstates for one-dimensional harmonic oscillator is given as:$$\epsilon \hbar \omega_0\begin{...
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 ...