Sympy 量子谐波振荡器的单位

Question

I am using the Sympy function for the generating the 3D Quantum Harmonic Oscillator radial wavefunction in order to use the Harmonic Oscillator Basis to solve for the energies of the potential well.

The documentation for the radial wavefunction is not very clear on what units are being used and I am very confused. What does "in atomic units nu == omega/2 " mean? Should I assume that r is in femtometers or nanometers or something else?

What about the units for mass, omega, nu, and energy?

Answer 1

The documentation says that the energy is returned in the units of hw (that is h.omega), so you use whichever units you want (they are just rescalings). That is, hw is J if you take h(bar) = 1.055e-34 Js and w in s-1 and use SI units for everything else as well (ie measure lengths in m).

Alternatively, you can work in atomic units by taking h(bar)=1, lengths in units of the Bohr radius, a0 (5.292e-11 m in SI), etc.

The statement "in atomic units nu == omega/2" would seem only to be true if the mass in question is that of the electron, m_e, which is taken as unity in atomic units. More generally, nu = m.omega/(2hbar)