When does the separation of variables work

In summary, the separation of variables technique works for solving partial differential equations (PDEs) when the equation can be expressed as a product of functions, each depending only on a single variable. This method is effective under conditions such as linearity, homogeneity, and when boundary conditions allow for separable solutions. It is commonly applied in systems with distinct spatial and temporal variables, facilitating the reduction of complex PDEs into simpler ordinary differential equations (ODEs) that can be solved individually.
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When studying the hydrogen atom, given that the potential depends only on the distance and not an any angle, we can do a separation of variables of the wavefunction as the product between a function depending only on the distance between particles (protons and electrons) and a spherical harmonic. However I saw this done even when the potential does depend on angle, for example when having the interaction between 2 electric dipoles. What is the criterion based on which I know if I can just a factorizable wavefunction or not? Thank you!
 
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Two answers:

(1) You just have to try it. Sometimes it helps and sometimes it just makes a mess.

(2) There is a deep relation between separation of variables and symmetries. The hydrogen atom can be solved by separation of variables in two different coordinate systems, spherical and parabolic. This is why you have the degeneracy of radial and angular energy levels.
 
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Separation variables works when it works. If it doesn't work, it can often lead to a series solution like the Legendre polynomial expansion.
 

FAQ: When does the separation of variables work

What is the separation of variables technique?

Separation of variables is a mathematical method used to solve differential equations by separating the variables involved so that each side of the equation depends on only one variable. This technique simplifies the equation into two or more simpler equations that can be solved individually.

When can separation of variables be applied?

Separation of variables can be applied when a differential equation can be written in the form where each side of the equation is a function of a single variable. This typically works for linear partial differential equations and some nonlinear equations where the variables can be isolated on different sides of the equation.

What are the limitations of the separation of variables method?

The primary limitation of the separation of variables method is that it can only be applied to differential equations that are separable. If the equation cannot be manipulated into a form where the variables can be completely separated, this method will not work. Additionally, initial and boundary conditions must be compatible with the separated form of the solution.

How do you determine if a differential equation is separable?

To determine if a differential equation is separable, try to rewrite it such that all terms involving one variable and its derivatives are on one side of the equation, and all terms involving the other variable are on the other side. If this is possible, the equation is separable. In mathematical terms, you should be able to express the equation as \( g(y) dy = f(x) dx \).

Can separation of variables be used for systems of differential equations?

Separation of variables can sometimes be extended to systems of differential equations, but this is generally more complex and less common. It requires each equation in the system to be independently separable, and the solutions must be consistent with each other. Often, other methods such as matrix techniques or numerical methods are used for systems of differential equations.

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