In Griffith's intro to QM it says on page 95 (in footnote 6) :(adsbygoogle = window.adsbygoogle || []).push({});

"In Hilbert space two functions that have the same square integral are considered equivalent. Technically, vectors in Hilbert space represent equivalence classes of functions."

But that means that if we take for example

f(x) = 1 | 0 < x < 1

and

g(x) = [itex]\sqrt{1/10}[/itex] | 0 < x < 10

both f(x) and g(x) have the same square integral ∫|f(x)|^2 dx = ∫|g(x)|^2 dx = 1

But how can they be considered "equivalent" - if they are wavefunctions for a particle, for example, then they represent completely different probability predictions! f(x) says you can find a particle with equal probability in [0,1], while for g(x) it's [0,10]

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Equivalent vectors in a Hilbert space

**Physics Forums | Science Articles, Homework Help, Discussion**