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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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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Are there pure sine waves in nature or are they a mathematical construct that helps us understand more complex phenomena?

I've studied a bit of frequency analysis with FFT and optimal phase binning and was taught that we can represent any composite waveform as the sum of its component frequencies. I understand the maths ...
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Hilbert space of harmonic oscillator: Countable vs uncountable?

Hm, this just occurred to me while answering another question: If I write the Hamiltonian for a harmonic oscillator as $$H = \frac{p^2}{2m} + \frac{1}{2} m \omega^2 x^2$$ then wouldn't one set of ...
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Why does a simple pendulum or a spring-mass system show simple harmonic motion only for small amplitudes?

I've been taught that in a simple pendulum, for small $x$, $\sin x \approx x$. We then derive the formula for the time period of the pendulum. But I still don't understand the Physics behind it. Also, ...
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Evolution operator for time-dependent Hamiltonian

When I studied QM I'm only working with time independent Hamiltonians. In this case the unitary evolution operator has the form $$\hat{U}=e^{-\frac{i}{\hbar}Ht}$$ that follows from this equation $$ i\...
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Why doesn't my kitchen clock violate thermodynamics?

My kitchen clock has a pendulum, which is just for decoration and is not powering the clock. The pendulum's arm has a magnet that is repelled by a second magnet that is fixed to the clocks body. The ...
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Origin of Ladder Operator methods

Ladder operators are found in various contexts (such as calculating the spectra of the harmonic oscillator and angular momentum) in almost all introductory Quantum Mechanics textbooks. And every book ...
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“QFT is simple harmonic motion taken to increasing levels of abstraction”

"QFT is simple harmonic motion taken to increasing levels of abstraction." This is my memory of a quote from Sidney Coleman, which is probably in many textbooks. What does it refer to, if he meant ...
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Intuition - why does the period not depend on the amplitude in a pendulum?

I'm looking for an intuition on the relationship between time period and amplitude (for a small pertubation) of pendulums. Why does the period not depend on the amplitude? I know the math of the ...
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Proof that the One-Dimensional Simple Harmonic Oscillator is Non-Degenerate?

The standard treatment of the one-dimensional quantum simple harmonic oscillator (SHO) using the raising and lowering operators arrives at the countable basis of eigenstates $\{\vert n \rangle\}_{n = ...
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In what sense is a quantum field an infinite set of harmonic oscillators?

In what sense is a quantum field an infinite set of harmonic oscillators, one at each space-time point? When is it useful to think of a quantum field this way? The book I'm reading now, QFT by ...
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Why not drop $\hbar\omega/2$ from the quantum harmonic oscillator energy?

Since energy can always be shifted by a constant value without changing anything, why do books on quantum mechanics bother carrying the term $\hbar\omega/2$ around? To be precise, why do we write $H =...
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Why don't tuning forks have three prongs?

I was reading Why tuning forks have two prongs?. The top answer said the reason was to reduce oscillation through the hand holding the other prong. So if having 2 prongs will reduce oscillation loss, ...
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Why do coherent states have Poisson number distribution?

In quantum mechanics, a coherent state of a quantum harmonic oscillator (QHO) is an eigenstate of the lowering operator. Expanding in the number basis, we find that the number of photons in a ...
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Why do electromagnetic waves oscillate?

I've been considering this question, and found many people asking the same (or something similar) online, but none of the answers seemed to address the core point or at least I wasn't able to make ...
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What symmetry is responsible for the amplitude independence of the period of a simple harmonic oscillator?

In the ICTP lectures of Y. Grossman: Standard Model 1, in about minute 54:00, he leaves an informal homework for the students. He ask to find the symmetry related to the conservation of the amplitude ...
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Definition of the $Q$ factor?

According to Wikipedia, the $Q$ factor is defined as: $$Q=2\pi\frac{\mathrm{energy \, \, stored}}{\mathrm{energy \, \,dissipated \, \, per \, \, cycle}}.$$ Here are my questions: Does the energy ...
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Is the harmonic oscillator potential unique in having equally spaced discrete energy levels?

I was wondering if the good old quadratic potential was the only potential with equally spaced eigenvalues. Obviously you can construct others, such as a potential that is infinite in some places and ...
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Is there an analog to the Runge-Lenz vector for a 3D spherically symmetric harmonic potential?

The Runge-Lenz vector is an "extra" conserved quantity for Keplerian $\frac{1}{r}$ potentials, which is in addition to the usual energy and angular momentum conservation present in all central force ...
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Why is the simple harmonic motion idealization inaccurate?

While in my physics classes, I've always heard that the simple harmonic motion formulas are inaccurate e.g. In a pendulum, we should use them only when the angles are small; in springs, only when the ...
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Proof that energy states of a harmonic oscillator given by ladder operator include all states

In quantum mechanics, while studying the harmonic oscillator, I learnt about ladder operators. And I realised that if you are able to find or determine any energy state of the quantum harmonic ...
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Is the quantization of the harmonic oscillator unique?

To put it a little better: Is there more than one quantum system, which ends up in the classical harmonic oscillator in the classial limit? I'm specifically, but not only, interested in an ...
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How does Ehrenfest's theorem apply to the quantum harmonic oscillator?

Ehrenfest's theorem, to my level of understanding, says that expectation values for quantum mechanical observables obey their Newtonian mechanics counterparts, which means that we can use newton's ...
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Better method to measure the time period of a pendulum

My physics textbook states that in measuring the time period of a pendulum it is advised to measure the time between consecutive passage though the mean position in the same direction. This results in ...
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How do we know that we have captured the entire spectrum of the Harmonic Oscillator by using ladder operators?

Consider standard quantum harmonic oscillator, $H = \frac{1}{2m}P^2 + \frac{1}{2}m\omega^2Q^2$. We can solve this problem by defining the ladder operators $a$ and $a^{\dagger}$. One can show that ...
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Why is the wave equation so pervasive?

The homogenous wave equation can be expressed in covariant form as $$ \Box^2 \varphi = 0 $$ where $\Box^2$ is the D'Alembert operator and $\varphi$ is some physical field. The acoustic wave ...
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Why can all solutions to the simple harmonic motion equation be written in terms of sines and cosines?

The defining property of SHM (simple harmonic motion) is that the force experienced at any value of displacement from the mean position is directly proportional to it and is directed towards the mean ...
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Explanation: Simple Harmonic Motion

I am a Math Grad student with a little bit of interest in physics. Recently I looked into the Wikipedia page for Simple Harmonic Motion. Guess, I am too bad at physics to understand it. Considering ...
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Why Don't the Ladder Operators Commute?

I have two problems with ladder operators. The first is that I feel they should somehow result in measurable things. The asymmetry of applying the plus operator versus the minus operator is very ...
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Why are the energy levels of a simple harmonic oscillator equally spaced?

The energy level of a simple harmonic oscillator is $E_n=(n+\frac{1}{2})\hbar\omega$. Is there any physical explanation why these levels are equally spaced ($= \hbar\omega$)? Maybe this link can be ...
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How to compute expectation value $\langle e^{iH}\rangle$ for quadratic Hamiltonians?

I have a rather basic, but actually non-trivial question: We consider a bosonic system with creation operators $\hat{a}_i^\dagger$ and annihilation operators $\hat{a}_j$ and vacuum state $|0\rangle$ ...
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Consistent, complete, and generalized description of the quantum harmonic oscillator

Textbooks often introduce the quantum harmonic oscillator with the example of a mass on a spring, giving the Hamiltonian $$H = \frac{1}{2} k x^2 + \frac{1}{2m}p^2 \qquad [x,p] = i \hbar \, .$$ The ...
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Book recommendations for second quantization

I am trying to familiarize myself with the ideas of second quantization. However, the literature that I can find online seems only to outline the tools of this formalism of quantum mechanics. ...
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Constant magnetic field applied to a quantum harmonic oscillator

I have a spinless particle of mass $m$ and charge $q$ which is an isotropic harmonic oscillator of frequency $\omega_0$, then I apply a constant magnetic field in the $z$ direction. We can show the ...
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Exact energies of spherical harmonic oscillator in Dirac equation

The potential is given by: $$ V(r) = {1\over 2} \omega^2 r^2 $$ and we are solving the radial Dirac equation (in atomic units): $$ c{d P(r)\over d r} + c {\kappa\over r} P(r) + Q(r) (V(r)-2mc^2) = E Q(...
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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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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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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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Dilation operator in CFT viewed as 'hamiltonian'?

From the commutation relations for the conformal Lie algebra, we may infer that the dilation operator plays the same role as the Hamiltonian in CFTs. The appropriate commutation relations are $[D,P_{...
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Can the equivalence principle be safely used in non-relativistic mechanics?

Imagine an ideal pendulum in a train. While the train is in uniform motion, Newton's laws apply within the train, and we can easily write down the equations of motion for the pendulum. Now assume the ...
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Eigenstates of a shifted harmonic oscillator

Let's say I have a quantum harmonic oscillator $H = \omega a^\dagger a$, where $a^\dagger$ is the raising operator and $a$ is the lowering operator and $H |n\rangle = \omega n |n\rangle$. Now assume ...
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Why $3$-dim isotropic harmonic oscillator's symmetry is not $O(6)$ but $U(3)$?

3D harmonic oscillator's Hamiltonian is $$H=\sum_{i=1}^3p_i^2+q_i^2$$ Why all textbooks say that its symmetry is $U(3)$. But I think it's $O(6)$. Because the rotation of $6$ coordinates in phase ...
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Why does the acceleration $g$ due to gravity not affect the period of a vertically mounted spring?

For a vertically mounted spring, I was looking at the formula $ T= 2\pi \sqrt{m/k}$ for a period. Why doesn't the gravitational acceleration $g$ factor in?
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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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Will a damped harmonic oscillator, with no initial amplitude, oscillate if there was background “noise”?

Suppose I have a damped harmonic oscillator which is at rest, sitting comfortably with no initial amplitude, obeying the equation $$\ddot{x} + \frac{1}{Q}\dot{x} + x = 0$$ where x is the vertical ...
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Angular momentum for 3D harmonic oscillator in two different bases

I know that the energy eigenstates of the 3D quantum harmonic oscillator can be characterized by three quantum numbers: $$ | n_1,n_2,n_3\rangle$$ or, if solved in the spherical coordinate system: $$|...
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Can we guess the periodic/aperiodic nature of motion from the equation of motion?

The equation of motion of a pendulum with a bob of mass $m$, and hanging by means of a massless thread of length $T$ is given by $$\ddot{\theta}+\frac{g}{l}\sin\theta=0,$$ and that of a damped one-...
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Proof that $\exp(-\beta H)$ is a trace-class operator for the harmonic oscillator

Let $H=\frac{p^2}{2}+\frac{x^2}{2}\, : D(H) \to L^2(\mathbb{R})$ be the Hamiltonian of the harmonic oscillator with $m=\hbar=\omega=1$. Prove that $\exp(-\beta H)$ is a trace-class operator if $\beta&...
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Why do we nondimensionalize the Schrödinger equation when solving the quantum harmonic oscillator?

I read about how to solve the Schrödinger equation for the quantum harmonic oscillator in one dimension. It started with the Schrödinger equation, $$ \frac{p^2}{2m}\psi(x, t)+\frac{1}{2}m\omega^2x^2\...
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Is the vacuum state a coherent state?

I'm asking because I got introduced to the state $|0\rangle$ as a fock-state. Nevertheless: $$ \hat{a} |0\rangle = 0 |0 \rangle $$ It is an eigenstate of $\hat{a}$ with eigenvalue $0$, and it can be ...

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