Variable separation - Schrödinger equation

[Konte]

Hello everybody,

My question is about variable separation applied in the solution of general time-independent Schrödinger 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

 

[DrClaude]

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

Ψ(r,θ,φ) = R(r).Θ(θ).Φ(φ)

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

 

[Konte]

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

Thanks for your answer.

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

Konte

 

[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.

 

[Jilang]

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

 

[DrClaude]

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

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

 

[Jilang]

Are there many real situations like that?

 

[DrClaude]

Are there many real situations like that?

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.

 

[Jilang]

Accepted. So generally not occurring naturally and more generally.