# The Eigenstate Existence Problem in Dirac's book 'Principles of Quantum Mechanics'

In Chapter II of Dirac's book Principles of Quantum Mechanics, Dirac explains that in general it is very difficult to know whether, for a given real linear operator, that any eigenvalues/eigenvectors exist and (if they do) how to find them.

He then goes on to state that a special tractable case can be found in the event that a real linear operator, $\xi$ can be expressed as the algebraic expression:

$\phi$($\xi$) $\equiv$ $\xi$n + a1 $\xi$n-1 + a2 $\xi$n-2 + ... + an = 0

Where the ai are all numbers. He then factorises this as:

$\phi$($\xi$) $\equiv$ ($\xi$- c1)($\xi$- c2)($\xi$- c3)...($\xi$- cn).

Where the ci are also numbers.

I'm unsure how this factorisation has been achieved. Since Dirac didn't explicitly state the difference between $\phi$($\xi$) and plain $\xi$, then I imagine that $\xi$ represents a given dynamical variable and $\phi$($\xi$) the corresponding operator. This may be incorrect however. This is the only step in the logic that I can't work out, I understand his subsequent arguments but this step is eluding me.

• He factorizes suposing that, in the case that you act over a proper eigenvector, the operator behaves as a normal number, and since you have a polinomial equation, with real values, one can, thus, use the fundamental theorem of algebra do factorize it. – Hydro Guy Jul 23 '14 at 14:10

From the equations, $\phi$ is the operator acting on the variable/state $\xi$. It is important to notice also that in the factorization the $a_i$ numbers are real and the $c_i$ are complex. This factorization comes just from the mathematical fact that for a polynomial equation of degree $n$, such as the first equation, there are $n$ complex roots.