All Questions
Tagged with harmonic-oscillator wavefunction
81
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Regarding to the asymptotic solution of quantum harmonic oscillator
In quantum mechanics, the radial equation of the SHO takes the form
\begin{align}
\frac{d^2 u}{dx^2}+\left(\epsilon-x^2-\frac{l(l+1)}{x^2}\right)u=0,
\end{align}
where $x=\sqrt{\frac{m\omega}{\hbar}}r$...
- 27
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Closed expression of eigenfunctions of a two dimensional isotropic harmonic oscillator
Where can one find the closed expression of the eigenfunctions of the 2d isotropic harmonic oscillator?
I saw something like this:
$$ \psi_{n_r m }(r, \theta) \propto e^{im\theta} r^{|m|} e^{-r^2/2} F(...
- 1,957
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Doubt obtaining the expected value of $x^2$ of a bidimensional harmonic oscillator
Just for the sake of context I'll add a little bit of introductory of the theory we were doing:
Say we are in the context of a bidimensional isotropic harmonic oscillator with an energy found of $2\...
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How do you determine that the series solution to the hermite differential equation is not square integrable?
When solving the Schrodinger equation of the harmonic oscillator in one dimension you encounter the hermite differential equation:
\begin{equation}
\left[\frac{d^{2}H}{d\xi^{2}}-2\xi\frac{d H}{d \xi }+...
3
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Wave amplitude as a complex number?
In section 1-3 An experiment with waves of The Feynman Lectures on Physics (https://www.feynmanlectures.caltech.edu/III_01.html) it says:
"The instantaneous height of the water wave at the ...
- 167
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Bra-Ket Notation vs Wavefunction Notation [duplicate]
We know that the rule for creating excited states for a Quantum Harmonic Oscillator is $|n\rangle=\frac{(a^\dagger)^n(|0\rangle)}{\sqrt{n!}}$. I wanted to derive from this the familiar rule in terms ...
- 442
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Finding the wavefunction of coherent state in 2D oscillator
Suppose I have a two-dimensional harmonic oscillator, $H= \hbar\omega(a_x^{\dagger}a_x+a_y^{\dagger}a_y)$. We define the operator $b=\frac{1}{\sqrt{2}}(a_x+ia_y)$.
If eigenkets of the hamiltonian are $...
- 81
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What are the eigenstate and energy eigenvalues of shifted Harmonic oscillators?
Suppose I have the potential of the shifted harmonic oscillators as
$$H=\frac{1}{2}m\omega^2(x\pm a)^2.$$
Then the energy eigenvalues will be $\hbar\omega(n+\frac{1}{2})$ and eigenfunctions simply as $...
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Why the Gaussian wavepacket only spreads in the free Schrodinger equation? It doesn't spread in the case where you have a harmonic oscillator
Why the Gaussian wavepacket only spreads in the free Schrodinger equation? It doesn't spread in the case where you have a harmonic oscillator. How to prove the situation in a harmonic oscillator?
Your ...
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Understanding how to terminate recurrence relations in quantum SHO
In the coordinate representation solution to the quantum SHO (the solution via differential equations rather than Dirac's "trick") we ultimately work out that our eigenfunction solutions are ...
- 1,095
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Can we think of spontaneous emission of a photon from an excited atom as a driven harmonic oscillator problem?
This is a kind of strange question, but I'm wondering, in the context of a fully quantum field theoretic treatment of spontaneous emission, if there is any model or way of calculating the process that ...
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Basis representation for isotropic 2D quantum harmonic oscillator
The basis functions of the 2D isotropic quantum harmonic oscillator are of the form
$$ \psi_{n,\ell} (r,\varphi) = A_{n\ell}(r)e^{i\ell\varphi}$$
where $A_{n\ell}(r) = \frac{\sqrt{2 \times p!}}{\sqrt{(...
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$x$-representation of half (truncated) harmonic oscillator?
The problem I'm struggling with has asked me to find the $x$-representation of the half harmonic oscillator wave function with a potential of $\frac12kx^2$. Our setup started with the WKB ...
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Does a non-stationary quantum particle in a potential well approach a stationary solution of the Schrödinger Equation?
I have come across this video on Youtube where someone simulated the wave function of a moving particle in an unspecified harmonic potential well. (Link: https://www.youtube.com/watch?v=hHAxLE181sk , ...
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A doubt regarding Quantum Harmonic Oscillator
Classically when we solve Newton's equation for $V=\frac{1}{2}m\omega^2x^2$ we get two linearly independent solutions (for $\omega\not=0$): $Ae^{\omega t}$ & $Be^{-\omega t}$, their linear ...
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