Representing a wavefunction using bases

[Shaybay92]

Can someone please explain why the representation of a wavefunction as an expansion of basis eigenfunctions actually gives us something of physical meaning? For example, it can tell us the probabilities of measuring a particular eigenvalue (depending on the expansion coefficients)... I mean its just a way we are mathematically representing something, so how does it have meaning physically? Thanks

 

[alxm]

I mean its just a way we are mathematically representing something, so how does it have meaning physically?

How's that different from anything else in physics?

It's kind of an epistemological question. All I can say is that apparently Nature obeys logic and is self-consistent. So a mathematical description of nature will also be consistent as long as the empirical assumptions behind that description are true.

 

[Shaybay92]

I had a feeling that was the answer. Ok, well, how do we know that the expansion coefficients are the relative probabilities? is it because we observe this?

 

[maverick_starstrider]

I had a feeling that was the answer. Ok, well, how do we know that the expansion coefficients are the relative probabilities? is it because we observe this?

I assume you are familiar with Fourier Transforms. Is a given wave REALLY its overall function or is it REALLY an infinite sum of periodic sin and cos functions? The two descriptions are mathematically IDENTICAL thus there could never be any experiment that could tell the difference and thus the distinction could never have any manifestation in reality. This is an identical situation (or rather it's the same situation since a Fourier basis is a perfectly valid eigenbasis). Now, pragmatically an infinite sum perspective may make certain aspects of the math or approximations POSSIBLE where working with the whole wavefunction you get nowhere. But this is a perfect example of why physicists don't much care for philosophers in general. Most physicists will simply say that it's a pointless question (whether one representation is MORE TRUE) since by construction, the mathematical model that predicts the two also says they are absolutely identical representations, however I'm sure you can find a philosopher (who can't actually do math or understand what's actually going on of course) who will argue that one mathematical representation is the true one because of Descartes Meditations or Plato's Theory of Perfect Forms or some such silliness.

As for the fact that the coefficients are relative probabilities this is just a combination of Born's Rule and the mathematical fact that your eigenbasis is chosen to be orthonormal

 

[Shaybay92]

As for the fact that the coefficients are relative probabilities this is just a combination of Born's Rule and the mathematical fact that your eigenbasis is chosen to be orthonormal

Could someone please elaborate on this and show exactly how we come up with the coefficients being the relative probabilities..? Just keep it 1dimensional.Thank you!

 

[Amok]

Could someone please elaborate on this and show exactly how we come up with the coefficients being the relative probabilities..? Just keep it 1dimensional.Thank you!

$$\left <A \right > = \langle \psi \mid \hat{A} \mid \psi \rangle = (\sum_{n=1}^{\infty} c^*_{n} \langle \phi_n \mid) \hat{A} (\sum_{m=1}^{\infty} c_{m}\mid \phi_m \rangle) = \sum_{n,m=1}^{\infty} c^*_{n} c_{m} a_m \langle \phi_n \mid \phi_m \rangle = \sum_{n,m=1}^{\infty} c^*_{n} c_{m} a_m \delta_{nm} = \sum_{n=1}^{\infty} a_n \left | c_n \right |^2$$

Where the "phis" are eigenfunctions of A. Knowing this and from the definition of an expectation value:

$$\left <A \right > = \sum_{i=1}^{N} P_i a_i$$

You get your result.

 

[The_Duck]

how do we know that the expansion coefficients are the relative probabilities?

This is simply an axiom of the mathematical formulation of QM.

 

[Shaybay92]

Thanks for the help guys :)