Completeness Relation in Quantum Mechanics Explained

[nolanp2]

can someone please give me a quick description/definition of a completeness relation in QM?

 

[malawi_glenn]

Here you can get some info:

http://phyastweb.la.asu.edu/phy576-schmidt/dirac/index.html

 

[blechman]

very roughly: if a set of functions are "complete", then you can always expand a function with the same boundary conditions in terms of these functions:

$$\{\phi_n(x)\} {\rm complete} \Rightarrow \psi(x)=\sum_n a_n\phi_n(x)$$

In particular, if you can find a set of energy eigenfunctions (which are complete due to theorems about hermitian operators on Hilbert spaces), you can always decompose any wavefunction in terms of them. This is very useful.

Slightly more formally, a complete set of functions satisfies:

$$\sum_n \phi_n(x)\phi_n(y)=\delta(x-y)$$

The above follows (more or less) from these hypotheses.

 

[dextercioby]

If you search the internet for the "spectral decomposition theorem", I'm sure you'll get many useful results.

 

[soundscape]

This old thread keeps popping up in Google, so ...

Quoting from J. J. Sakurai - Modern Quantum Mechanics, Sec. 1.3

... we must have

$$\sum_{a'} \left| a' \right\rangle \left\langle a' \right| = 1$$ (1.3.11)

where the 1 on the right-hand side is to be understood as the identity operator. Equation (1.3.11) is known as the completeness relation or closure.

The $$\left| a' \right\rangle$$ signify orthonormal eigenkets. The above sum is a useful mathematical tool as it can be inserted wherever the identity operator could appropriately be inserted. Check out the book!