## Orthogonal Eigenfunctions (Landau Lifshitz)

I've been reading QM by Landau Lifshitz, in which I've come across a statement I can't seem to get my head around.

It states (just before equation 3.6):

a_n = SUM a_m. INTEGRAL f_m. f_n. dq

( a_n is the nth coefficient, f_m is the mth eigenfunction of an operator, dq is the differential element.

It follows from the equation for determining the coefficients in a wavefunction composed of a linear sum of eigenfunctions of an operator:

a_n = INTEGRAL f_n. f. dq )

It then states that it is evident from this that the eigenfunctions must be orthogonal. I don't see how this is? I would like to understand this, as it would imply that orthogonality of eigenfunctions would fall directly out of the mathematics of QM!

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 Is it assumed in QM that all eigenvalues f_n of a given operator ARE distinct? Why?

## Orthogonal Eigenfunctions (Landau Lifshitz)

Is it assumed in QM that all eigenvalues f_n of a given operator ARE distinct? Why?

 Hi, Master J. Operators corresponds to physical variables, e.g. energy, position, and eigenvalues of operator corresponds to values of physical variables e.g. 0.12 Joule, 3.45 meter. Observed value should be distinctive. Regards.

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 Quote by Master J I've been reading QM by Landau Lifshitz, in which I've come across a statement I can't seem to get my head around. It states (just before equation 3.6): a_n = SUM a_m. INTEGRAL f_m. f_n. dq ( a_n is the nth coefficient, f_m is the mth eigenfunction of an operator, dq is the differential element. It follows from the equation for determining the coefficients in a wavefunction composed of a linear sum of eigenfunctions of an operator: a_n = INTEGRAL f_n. f. dq ) It then states that it is evident from this that the eigenfunctions must be orthogonal. I don't see how this is? I would like to understand this, as it would imply that orthogonality of eigenfunctions would fall directly out of the mathematics of QM!
The orthogonality "falls out" from that definition of the coefficients.

In particular, if you look for the coefficient, an, in the expansion of fm, you get
$$an= \int fn fm dq$$
But, fm is itself a basis function, so fm is just 1 times fm plus 0 times all the other basis functions. That is
$$am= \int fm^2 dq= 1$$
[tex]an= \int fn fm dqa= 0[tex]
for n not equal to m.

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