What is Quantum harmonic oscillator: Definition and 108 Discussions

The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary smooth potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics. Furthermore, it is one of the few quantum-mechanical systems for which an exact, analytical solution is known.

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  1. A

    Coherent States of the Quantum Harmonic Oscillator

    Does anyone know why a harmonic potential gives rise to coherent states? In other words, what is special about a quadratic potential that causes the shifted ground state to oscillate like a classical particle without dispersing so as to saturate the uncertainty principle? Any help or insight...
  2. M

    How Does an External Electric Force Affect Quantum Harmonic Oscillator States?

    Homework Statement H = p^2/2m + (kx^2)/2 - qAx (THis is a harmonic potentional with external electric force in 1D) Braket: Definitions: |0, A=0 > = |0>_0 for t=0 (ground state) |0, A not 0 > = |0> for t=0 (ground state) 2. Question 1. Find the probability of being in the state |0, A...
  3. B

    What is the solution for the quantum harmonic oscillator formula for nucleons?

    I am working with the following harmonic oscillator formula. \psi_n \left( y \right) = \left( \frac {\alpha}{{\pi}} \right) ^ \frac{1}{{4}} \frac{1}{{\sqrt{2^nn!}}}H_n\left(y\right)e^{\frac{-y^2}{{2}}} Where y = \sqrt{\alpha} x And \alpha = \frac{m\omega}{{\hbar}} I...
  4. N

    Quantum harmonic oscillator problem

    Homework Statement Is there any way to find <\varphi_{n}(x)|x|\varphi_{m}(x)|> (where phi_n(x) , phi_m(x) are eigenfunction of harmonic oscillator) without doing integral ? Homework Equations perhaps orthonormality of hermite polynomials ...
  5. T

    A general math theory on the quantum harmonic oscillator?

    looking at the quantum mechanical harmonic oscillator, one has the differential equation in the form: \frac{d^2\psi}{du^2}+(\alpha-u^2)\psi=0 when a person who doesn't know any physics sees the equation, he will try a serial solution for psi, and he will find a solution with some recursive...
  6. 6

    How Does the Quantum Harmonic Oscillator Transition from U(x) to E?

    Can someone explain to me how they went from U(x)=(1/2)kx^2 to E=(n+1/2)(h/2pi)w? http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hosc.html
  7. quasar987

    Explicit formula the nth eigenfunctions of the quantum harmonic oscillator?

    Hi, Is there an explicit formula for the eigenfunctions of the harmonic oscillator? By explicit, I mean "not written as the nth power of the operator (ax-d/dx) acting on the ground state". Thanks.
  8. R

    Quantum harmonic oscillator system

    Hi, I am wondering how i would go about calculating the canonical partition function for a system of N quantum harmonic oscillators. The idea of the question is that we are treating photons as oscillators with a discrete energy spectrum. I'm confused as whether to use Maxwell-Boltmann...
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