The concept of separation of variables in QM

The concept of separation of variables is a powerful technique used in solving certain types of mathematical equations, particularly partial differential equations (PDEs). It allows us to simplify the process of solving complex equations by breaking them down into simpler equations that can be solved individually.

The idea behind the separation of variables is based on the assumption that a solution to a multi-variable equation can be expressed as a product of functions, each depending on only one variable. By making this assumption and substituting this product form into the original equation, we can often transform a complicated equation into a set of simpler ordinary differential equations (ODEs) or algebraic equations.

The process of separation of variables typically involves the following steps:

1. Assume a separable solution:

Assume that the solution to the equation can be expressed as a product of functions, each depending on a different variable. For example, if we have a function f(x, y), we assume that it can be written as f(x, y) = X(x)Y(y).

2. Substitute the assumed solution:

Substitute the assumed separable solution into the original equation.

3. Separate the variables:

Collect terms that depend on different variables on opposite sides of the equation. This typically involves grouping terms with the same variable together.

4. Equate each side to a constant:

Since the separated terms depend on different variables, they must be equal to a constant (known as a separation constant). This step effectively converts the original PDE into a set of ODEs or algebraic equations.

5. Solve the resulting equations:

Solve the simplified equations obtained in the previous step, which are typically ordinary differential equations or algebraic equations.

6. Combine the solutions:

Once the separate solutions for each variable are obtained, combine them using the assumed separable form to obtain the general solution to the original equation.

The key benefit of the separation of variables is that it simplifies the process of solving complex equations by breaking them down into simpler equations that can be solved individually. It allows us to solve problems that would be otherwise difficult or impossible to solve directly.

This technique is widely used in various fields, including physics (such as in solving the wave equation, heat equation, and Schrödinger equation), engineering, and applied mathematics, where equations with separable variables arise frequently.