Nondegenerate Eignefunctions as Linear Combinations

[zachzach]

It is easily shown that two eigenfunctions with the same eigenvalues can be combined in a linear combination so that the linear combination is itself an eigenfunction. But what if the two eigenvalues are not the same? Can you still find a linear combination of the two functions that is an eigenfunction?

<br /> <br /> aE_1 \psi_1+ b E_2 \psi_2 = E(\psi_1 + \psi_2)<br /> <br />

 

[RedX]

The equation you have written down you can set aE_1=bE_2 and it will satisfy the equation with E=aE1.

But what you're really looking for is:

<br /> <br /> <br /> aE_1 \psi_1+ b E_2 \psi_2 = E(a\psi_1 + b\psi_2)<br /> <br /> <br />

which can only be satisfied if E_1=E_2=E

 

[zachzach]

Yes, I meant to write the equation that you did. So the answer is no then. You can never have a linear combination of non degenerate eigenfunctions that is itself an eigenfunction.

 

[RedX]

Yes, I meant to write the equation that you did. So the answer is no then. You can never have a linear combination of non degenerate eigenfunctions that is itself an eigenfunction.

The idea is that the set of nondegenerate eigenfunctions is linearly independent. One eigenfunction cannot be the sum of two others, or else the set would not be linearly independent.