Variable separation - Schrödinger equation

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1. Feb 5, 2016

Konte

Hello everybody,

My question is about variable separation applied in the solution of general time-independent Schrodinger equation, expressed with spherical coordinates as:

$\hat{H} \psi (r,\theta,\phi) = E \psi (r,\theta,\phi)$

Is it always possible (theoretically) to seek a solution such as:

$\psi (r,\theta,\phi) = R(r) . \Theta(\theta).\Phi(\phi)$

Thank you everybody.

Konte

2. Feb 5, 2016

Staff: Mentor

No. It only works if the Hamiltonian is separable in spherical coordinates. It wouldn't work for example for a 3D anisotropic harmonic oscillator.

3. Feb 5, 2016

Konte

Is there a way to find or to construct a system of coordinate so that the split of Ψ is possible?

Konte

4. Feb 5, 2016

blue_leaf77

There is no general rule to apply what kind coordinate transform on the Cartesian Schroedinger equation in order to be separable in the new coordinate system. The coordinate transform must be sought for each form of Schroedinger equation, for example the Hamiltonian of hydrogen atom in the presence of DC electric field (Stark effect) can be transformed into parabolic coordinate to make the differential equation separable. Other form of Schroedinger equation will almost always require different transform.

5. Feb 5, 2016

Jilang

Only if the Hamiltonian is spherically symmetric. (This generally means that the Potential is also spherically symmetric).

6. Feb 5, 2016

Staff: Mentor

That's not true: symmetry is not necessary, only separability. If $V(r,\theta,\phi) = \cos^2\theta$, one can still write eigenstates as $\psi (r,\theta,\phi) = R(r) \Theta(\theta) \Phi(\phi)$.

7. Feb 5, 2016

Jilang

Are there many real situations like that?

8. Feb 5, 2016

Staff: Mentor

Yes. The example I gave comes from the interaction of a linear molecule with a linearly-polarized laser field. You get similar potentials with (separable) angular dependence for uniform electric or magnetic fields.

9. Feb 5, 2016

Jilang

Accepted. So generally not occurring naturally and more generally.