Timeline for Quantum Harmonic Oscillator Virial theorem is not holding
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 27, 2021 at 15:17 | vote | accept | user7292119 | ||
| Jan 27, 2021 at 14:07 | comment | added | Philip | That sounds about right, yes! | |
| Jan 27, 2021 at 14:06 | comment | added | user7292119 | Thank you very much! Then as a summary, to make sure I have understood: Virial Theorem holds for bound states, that means eigenfunctions with discrete energy. Singe eigenfunctions of the QHO have discrete energy, all eigenstates are bound. So Virial holds for any QHO eigenstate. However, my state does not have a definite enrgy, since it isn't an eigenstate, so it's not bound, so no Virial. | |
| Jan 27, 2021 at 13:54 | comment | added | Philip | Right, I'm not sure I understand the last bit of your comment, but globally, yes. "Bound" and "scattering" are adjectives used to describe states of definite energy (i.e, eigenfunctions of the Hamiltonian). Eigenfuctions that have a discrete spectrum are called "bound" and those that have a continuous spectrum are called "scattering". The harmonic oscillator only has eigenfunctions that have a discrete energy spectrum, and therefore all states of definite energy are "bound" states. The Virial theorem only holds for such bound states. | |
| Jan 27, 2021 at 13:36 | comment | added | user7292119 | That relation is satisfied by eigenfunctions right? That's why they have a definite energy E but my state is an infinte sum of the QHO eigenfunctions with different weight coefficients. So it does not satisfy $H\psi=E\psi$. Thus it is not a bound state so Virial does not hold. Therefore Virial only holds for eigenstates? And Bound states are those of the eigenfunctions? | |
| Jan 27, 2021 at 13:21 | comment | added | Philip | That is also correct. However, not all functions are bound states! A bound state for a particular Hamiltonian $H$ is a state that satisfies$$H \psi = E\psi,$$ where $E$ is a constant number, which we understand to be the energy. I urge you to plug the state that you have into this differential equation to see if it satisfies it. | |
| Jan 27, 2021 at 13:19 | comment | added | user7292119 | I read that all states in a quantum harmonic oscillator are bound: physics.stackexchange.com/questions/135456/… But maybe I am wrong and for classical energies there existes unbound states? | |
| Jan 27, 2021 at 13:05 | comment | added | Philip | Sure, except that the Virial theorem in Quantum Mechanics is only true for bound states, and the state you've provided isn't a bound state of the Harmonic Oscillator. So it's normal that it doesn't satisfy the theorem. Perhaps there is a misunderstanding as to what a bound state is? | |
| Jan 27, 2021 at 13:02 | comment | added | user7292119 | The state they give me is indeed not the ground state, I need to calculate it for the state given, with that factor of 4 | |
| Jan 27, 2021 at 12:55 | history | answered | Philip | CC BY-SA 4.0 |