How Does Quantum Superposition Explain Probability in Wavefunctions?

[Nick R]

Hello, I am brand new to this stuff and am trying to get my head around it all. I've spent considerable time trying to understand this from Landau's book on the subject (chapter 1 of course).

I bet I'd get more answers by being more brief but I always find that asking the problem carefully sometimes helps me understand the problem better.

A wavefunction, which completely describes the states of a quantum object, can be decomposed in terms of its eigenfunctions,

\psi = \sum_{n} a_{n}\psi_{n}

Eigenvalues (maybe a physical quantity) correspond to the eigenfunctions by

\widehat{f}\psi_{n} = f_{n}\psi_{n}

Where \widehat{f} is the operator that corresponds to the quantity in question.

From this, we see that the value of a_{n} for a given eigenfunction in the decomposition is (somehow) related to the "probability" that the physical quantity f has the value f_{n}.

Given, is

\int |\psi_{n}(q)|^{2}dq = 1

and

\int |\psi(q)|^{2}dq = 1

How does it follow that |a_{n}|^{2} is the probability of the physical quantity f having the value f_{n}? The reasoning presented in the book is not clear to me - it is a sort of deductive reasoning that seems like guesswork.

Of course if this is a probability then,

\sum |a_{n}|^{2} = 1

I don't understand how this follows from the other things.

Here is why I am having a problem with this:

I can see it all works if the following is true:

\psi = a_{0}\psi_{0} + a_{1}\psi_{1} + ... + a_{n}\psi_{n}

|\psi| = \sqrt{|a_{0}\psi_{0}|^{2} + |a_{1}\psi_{1}|^{2} + ... + |a_{n}\psi_{n}|^{2}}

\int |\psi|^{2}dq = \int |a_{0}\psi_{0}|^{2}dq + \int |a_{1}\psi_{1}|^{2}dq + ... + \int |a_{n}\psi_{n}|^{2}dq

= |a_{0}|^{2}\int |\psi_{0}|^{2}dq + |a_{1}|^{2}\int |\psi_{1}|^{2}dq + ... + |a_{n}|^{2}\int |\psi_{n}|^{2}dq

Truth of this rests on the truth of two identities for complex numbers.

|(a+bi)(c+di)|^{2} = |a+bi|^{2}|c+di|^{2} IDENTITY ONE
According to my calculations this is true.

|(a+c) + (b+d)i|^{2} = |a+bi|^{2} + |c+di|^{2} IDENTITY TWO
According to my calculations this is false, unless there is a constraint 2ac = -2bd.

What is going on here? Is there some sort of constraint?

 

[jtbell]

I think your missing piece of information is that the \psi_n are orthogonal, that is,

\int \psi^*_m(q) \psi_n(q) dq = 0

for m \ne n.

This is what let's you go from

<br /> \psi = a_{0}\psi_{0} + a_{1}\psi_{1} + ... + a_{n}\psi_{n}<br />

to

<br /> \int |\psi|^{2}dq = \int |a_{0}\psi_{0}|^{2}dq + \int |a_{1}\psi_{1}|^{2}dq + ... + \int |a_{n}\psi_{n}|^{2}dq<br />

 

[Nick R]

Thanks I think that makes a lot of sense.

Basically "cross terms" looking similar to

\psi^*_m(q) \psi_n(q)

arise in the expression for |\psi|^{2}, and are eliminated when they are integrated, leaving only the terms looking like

\psi^*_n(q) \psi_n(q)

Thanks a bunch.