Separability of variables

[Titan97]

In a hydrogen atom, the wave function is written as R(r).Θ(θ).Φ(φ). But how is it separable when the electron is interacting with the nucleus?

 

[Nugatory]

In a hydrogen atom, the wave function is written as R(r).Θ(θ).Φ(φ). But how is it separable when the electron is interacting with the nucleus?

The "separability" refers to the form of the partial differential equation (Schrodinger's equation for the $1/r$ potential) we're trying to solve. You can show by substitution that a function of the form $\psi=R(r)Y(\theta,\phi)$ will be a solution. When you make this substitution two separate and simpler differential equations, one for $R(r)$ and the other for $Y(\theta,\phi)$, will emerge.

 

[Vanadium 50]

The interaction only depends on r.

 

[Titan97]

@Nugatory I am asking how can one be sure that a function f(x,y,z)=f(x)f(y)f(z)?

 

[Titan97]

For example, why isn't F(z,y,z)=f(x)+g(y)+h(z) and only f(x)g(y)h(z)?

 

[bhobba]

@Nugatory I am asking how can one be sure that a function f(x,y,z)=f(x)f(y)f(z)?

There are theorems from the theory of partial differential equations that many solutions have that form - the method is called separation of variables and often works:
http://math.stackexchange.com/questions/575205/why-separation-of-variables-works-in-pdes

Generally in physics we don't worry about theorems like that, but if you did a degree in math as well you might - or might not depending on the inclination of your lecturer. I did a subject partial differential equations as part of my degree many many moons ago:
http://pdf.courses.qut.edu.au/coursepdf/qut_MS01_31556_dom_cms_unit.pdf

It was a bit different in those days - now its combined with complex analysis - back then it was two separate subjects. The teacher sounded out at the beginning of class if we would like to see some of the proofs of some of this stuff. I like that sort of thing and said yes - but most couldn't care less so it wasn't covered.

If you are interested there are books that do it eg:
https://www.amazon.com/dp/052129746X/?tag=pfamazon01-20

But they use advanced methods of functional analysis (such as distribution theory that makes sense of that damnable Dirac Delta function - you should study it anyway - but that is a whole new thread) which fortuneately I did study - but it's not for the beginning student.

I remember looking it up, but have to say it was basically simply for curiosity - it played no role in any future studies I did at all. Still if you are interested after learning how to use it you can spend a bit of time on the theory - it won't play a role in your future physics but those of a particular mind set like me don't like loose ends.

Thanks
Bill

 

[PeroK]

@Nugatory I am asking how can one be sure that a function f(x,y,z)=f(x)f(y)f(z)?

You can't be sure. It's an educated guess. If you find a solution of the form $f(x)g(y)h(z)$, then all well and good. Otherwise, you'll have to try something else.

You could look for a solution of the form $f(x) + g(y) + h(z)$ but you may not be successful very often.

 

[Vanadium 50]

I am asking how can one be sure that a function f(x,y,z)=f(x)f(y)f(z)?

You can't. In general, it's not. If you mean Cartesian x,y, z, it's not true for the hydrogen atom (but is true for the harmonic oscillator). If you mean r, theta, phi, it is true provided the interaction term only depends on r.

 

[dextercioby]

In a hydrogen atom, the wave function is written as R(r).Θ(θ).Φ(φ). But how is it separable when the electron is interacting with the nucleus?

The separation of spherical variables occurs after the separation of the general 2 body-motion into the CoM motion (i.e. the motion of a „virtual” particle of mass = m_p + m_el) and the motion of a „virtual” massive (with mass = reduced mass = (m_p x m_el)/(m_p + m_el)) particle relative to the CoM and this separation of spherical variable applies only the to the wave function of the „virtual” particle of reduced mass.This separation of motion in two „virtual” motions is glossed over in introductory QM texts or at least not properly emphasized. There one usually jumps to separation of variables for the „virtual” particle of reduced mass and the reader is left to wonder: what is the small „r” in the potential function exactly and where does it come from?

 

[vanhees71]

A very illuminating article on this topic can be found here:

http://scitation.aip.org/content/aapt/journal/ajp/66/10/10.1119/1.18977
http://arxiv.org/pdf/quant-ph/9709052