All Questions
Tagged with harmonic-oscillator hamiltonian-formalism
54
questions
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Action-angle variables for three-dimensional harmonic oscillator using cylindrical coordinates
I am solving problem 19 of ch 10 of Goldstein mechanics. The problem is:
A three-dimensional harmonic oscillator has the force constant k1 in the x- and y- directions and k3 in the z-direction. Using ...
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Symmetry group of a two-dimensional isotropic harmonic oscillator
The Lagrangian and the Hamiltonian of a two-dimensional isotropic oscillator (with $m=\omega=1$) are
$$L=\frac{1}{2}(v^2_1+v_2^2-q_1^2-q_2^2)\tag{1}$$ and $$H=\frac{1}{2}(p^2_1+p_2^2+q_1^2+q_2^2),\tag{...
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Order $\epsilon^2$ mean rate of change in action variable for adiabatic oscillator
In adiabatic theory of classical mechanics, considering a linear oscillator with a slowly changing frequency $\omega=\omega(t)$, Percival and Richards's nice book (pp. 144-147) discusses how it is ...
- 632
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Interpretation of this Hamiltonian
I'm studying a system with the following Hamiltonian $$ H = \frac{1}{2}P^TAP + \frac{1}{2}Q^T B Q$$ where $P,Q$ are canonical variables (4-vectors) and $A,B$ matrices such that $A = A^\dagger$ and $B =...
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Meaning of "the symmetry group of an $N$-dimensional quantum isotropic oscillator is $U(N)$"
Symmetry of this system has been discussed here but I'm still confused.
Consider a $N$-dimensional isotropic harmonic oscillator, with hamiltonian
$$H = \hbar \omega \left(a^\dagger_i a_i + \frac{N}{2}...
2
votes
1
answer
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Using the EoM in the canonical quantization of EM field
Starting from the classical electromagnetic field, there two approaches to quantization that I want to compare. The problem arises when I write the classical fields in terms of $a$ and $a^*$, which ...
- 1,989
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Almost all Liouville torus is preserved for small oscillation problems even if we don't use second-order approximation to potential energy, right?
In small oscillation problems, we use a second-order approximation to the potential energy function (suppose the oscillation is around the point $(0,\cdots, 0)$),
$$
V(x) = V(0) + \frac{\partial^2 V(0)...
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1
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1
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Separation of Hamilton's Principal Function
In Goldstein's "Classical Mechanics", at page 437, a two-dimensional anisotropic harmonic oscillator is studied by means of Hamilton-Jacobi formalism. In particular, Goldstein claims that:
...
- 402
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Hamiltonian constraint
I am having some difficulties regarding this wikipedia article. I can't understand why the action (for the harmonic oscilator) is written as it is.
That is, why is the Hamiltonian followed by a $\...
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Hamilton's equations of motion for damped oscillator
Consider a parallel RLC oscillator. Kirchhoff's equations of motion are
$$
\ddot{\Phi} + \frac{1}{\tau}\dot{\Phi} + \omega_0^2 \Phi = 0
$$
where $\tau = RC$ and $\omega_0 = 1 / \sqrt{LC}$.
What is ...
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Canonical transformation of the harmonic oscillator‘s Hamiltonian [closed]
I could deduce the Hamiltonian of the damped harmonic oscillator:
$$
H=\frac{p^2}{2m}e^{-2 \gamma t}+\frac{m \omega_0^2 q^2}{2}e^{2 \gamma t}
$$
Using the canonical transformation $Q=e^{\gamma t}q, P=...
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Existence of a unitary transform $(q,p) \rightarrow (-q, p)$
If $q$ and $p$ are the canonical position and momentum operators of a quantum harmonic oscillator, is there a unitary that transforms $(q,p)$ into $(-q, p)$?
For instance, denoting the annihilation ...
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Symmetry associated to a part of a separable Hamiltonian
The harmonic oscillator in 3D is:
$$H=\frac{p_x^2+p_y^2+p_z^2}{2m}+ \frac{k}{2} (x^2+y^2+z^2) = H_x + H_y + H_z,$$
where $H_x$, $H_y$ and $H_z$ are all constants of motion (alongside $\vec{L}$).
Time ...
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Why does metaplectic correction fix the vacuum energy?
In geometric quantization we want to go from a symplectic manifold $\left( M, \omega \right)$ to a Hilbert space $H$. If $M$ is prequantizable, we find a prequantum bundle $L \rightarrow M$ with ...
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Simple (classical, non-damped) 1D harmonic oscillator - how to represent phase space elliptical trajectory in polar form for an ellipse?
Considering a classical, non-damped 1D harmonic oscillator (e.g. mass $m$ oscillating along $x$-axis attached to spring with constant $k$) -- described by Hamiltonian (for constant energy $E$) $$E=p^2/...
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