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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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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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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Do states with infinite average energy make sense?
Do states with infinite average energy make sense?
For the sake of concreteness consider a harmonic oscillator with the Hamiltonian $H=a^\dagger a$ and eigenstates $H|n\rangle=n|n\rangle$, $\langle 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 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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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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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 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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"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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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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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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