What is the final expression for the energy in separation of variables?

In summary, when using separation of variables to solve a physical system with 2 degrees of freedom, the energy is not dependent on which equation it is placed in. The Hamiltonian acts on each piece separately and the energy is the sum of the two constants from each equation.
  • #1
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Suppose you have some partiel DE describing a physical system with 2 degrees of freedom (e.g. the SE). If you try separation of variables you get something like:

Hg(x)h(y) = Eg(x)h(y)

now you can separate this to two equations, but the energy has to go in one of them. Is the final expression for the energy dependent on which one you choose to put it in?
 
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  • #2
E typically doesn't go with one or the other. When separation works, what happens is you get
$$\hat{H}[g(x)h(y)] = G(x)h(y) + g(x)H(y).$$ The Hamiltonian acts on each piece separately. Then you can divide both sides by g(x)h(y) to get
$$\frac{G(x)}{g(x)} + \frac{H(y)}{h(y)} = E.$$ The only way this can be satisfied for all x and y is if the two terms on the left are each constants. The energy is the sum of those constants.
 

FAQ: What is the final expression for the energy in separation of variables?

What is separation of variables?

Separation of variables is a mathematical technique used to solve differential equations by separating the variables into individual equations and solving each one separately.

When is separation of variables used?

Separation of variables is typically used when solving partial differential equations, where the variables are separated into different equations in order to simplify the problem and make it easier to solve.

How is separation of variables performed?

To perform separation of variables, the equation is rearranged so that all terms involving one variable are on one side of the equation and all terms involving the other variable are on the other side. The resulting equations are then solved separately, and the solutions are combined to obtain the final solution.

What are some applications of separation of variables?

Separation of variables is commonly used in physics, engineering, and other fields to solve problems involving heat transfer, wave motion, and other physical phenomena. It is also used in financial mathematics to solve problems involving interest rates and investment strategies.

Are there any limitations to separation of variables?

Yes, there are some limitations to separation of variables. It can only be used for certain types of equations, and it may not always lead to a solution. In some cases, the resulting equations may be difficult or impossible to solve, or the solutions may not be applicable to the original problem.

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