New answers tagged harmonic-oscillator
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Numeric Solution vs Analytic Solution (1D Quantum Harmonic Oscillator)
To elaborate on @Javi's answer, yes this is a normalization issue, but there's some detail that might be helpful. An eigenvector can be scaled by any complex number and is still an eigenvector (with ...
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Numeric Solution vs Analytic Solution (1D Quantum Harmonic Oscillator)
I think that the numerical scheme is incomplete because it lacks the proper normalization of the Eigenfunction.
Say that $\Psi_n$ for $n\in[0,N]$ is the numerical solution that you find applying ...
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Non-linear spring systems
If motion isn't constrained to one dimension, Philip Wood's answer nicely illustrates that the force is not necessarily proportional to displacement. If motion is constrained to one dimension only, it ...
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Non-linear spring systems
Piano wires exhibit this sort of behaviour- specifically, the larger ones for low notes, that are made of a longitudinal core with a transverse outer coil. The friction between these components ...
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Non-linear spring systems
Here's a set-up that will give a cube law response for small displacements. Join 2 springs (each of natural length $l$ and obeying $T=kx$) end-to-end and anchor the far end of each, so that the ...
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Difference between harmonic motion and simple harmonic motion?
Simple Harmonic motion is defined as motion where the restoring force is proportional to the displacement, and pointed towards equilibrium.
The example of the simple pendulum is a good one, though a ...
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Does the formula $ V =\omega r$ holds in angular frequency
Angular frequency is the magnitude of angular velocity, so yes. Although as far as I'm aware, we typically only speak of angular frequency when the angular velocity (or speed) is constant.
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Why does the eigenvalues of an angular frequency matrix are the natural frequency? (INTUITION)
The acceleration is the second derivative of the position i.e. $a \equiv \ddot{x}$.
This means that the you have managed to decompose the couple system of oscillators into independent modes (to state ...
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Is simple harmonic potential only example which is invariant under gauge transform (in SUSY/QM context)?
There are multiple theories which are invariant to various kinds of gauge transforms, the simplest gauge transform is simply the SU(1) transformation of a field like
$$A_\mu(x) \rightarrow A_\mu(x) - \...
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Applying the position operator several times to a harmonic oscillator state $\hat x^m |n\rangle =$ ______?
A foolproof approach is doing this in position representation (where it is just multiplication by $x^m$) using your favorite methods of dealing with Hermit polynomials (e.g., Schiff has a clear ...
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Integral arising from the proof that the quantum harmonic oscillator satisfies Heisenberg's uncertainty principle
You are missing the forest for the trees, and tobogganing into error. Set $\sqrt{m\omega/\hbar}=1, \implies x=\xi$.
Recall the basic recursions,
$$ \bbox[yellow]{
\xi H_n(\xi)= \tfrac{1}{2} H_{n+1}(\...
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What does $\omega$ mean in SHM?
Simple harmonic motion and circular motion are mathematically closely related.
On a unit circle (circle with radius 1), every point has the coordinates $(\cos\theta,\sin\theta)$ for some angle $\theta$...
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What does $\omega$ mean in SHM?
It is just an analogy with the circular motion. If we define the frequency $f$ of a harmonic oscillator as the number of oscillations ("back and forth") in one second, then we can define the ...
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What does $\omega$ mean in SHM?
$\omega$ is the "angular frequency." If you plot the particle's velocity and position as a function of time, the particle moves around a circle (see the comments for a discussion of the ...
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Ground eigenstate of the quantum harmonic oscillator with the interacting vacuum $| \Omega \rangle$
There's a little bit of confusion here between notions from quantum mechanics and notions from quantum field theory, but this is a very interesting question that leads in interesting directions!
The ...
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Integral arising from the proof that the quantum harmonic oscillator satisfies Heisenberg's uncertainty principle
Calculating integrals with polynomials is probably not the easiest way to find $\langle x^2\rangle$ and $\langle p^2\rangle$. In my opinion, it is better to use ladder operators. Or an even simper ...
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Difference between Displacement from Equilibrum and Amplitude of SHM
Amplitude of a particle performing SHM is the maximum displacement from the equilibrium position it can achieve during its motion while displacement from mean/equilibrium position suggests its set off ...
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Difference between Displacement from Equilibrum and Amplitude of SHM
The maximum displacement of a particle from its mean position in SHM is defined as the amplitude of the SHM, whereas displacement is just the shift or movement of the particle from its equilibrium ...
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Confusion regarding simple pendulum
$$T~=~2π\sqrt{\frac{Lθ}{g\sinθ}}$$
doesn't make sense, the period shouldn't depend on the displacement, which is changing over the course of the motion. The usual expression for small displacements is
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