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).(adsbygoogle = window.adsbygoogle || []).push({});

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,

[tex]\psi = \sum_{n} a_{n}\psi_{n}[/tex]

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

[tex]\widehat{f}\psi_{n} = f_{n}\psi_{n}[/tex]

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

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

Given, is

[tex]\int |\psi_{n}(q)|^{2}dq = 1[/tex]

and

[tex]\int |\psi(q)|^{2}dq = 1[/tex]

How does it follow that [tex]|a_{n}|^{2}[/tex] is the probability of the physical quantity [tex]f[/tex] having the value [tex]f_{n}[/tex]? 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,

[tex]\sum |a_{n}|^{2} = 1[/tex]

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:

[tex]\psi = a_{0}\psi_{0} + a_{1}\psi_{1} + ... + a_{n}\psi_{n}[/tex]

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

[tex]\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[/tex]

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

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

[tex]|(a+bi)(c+di)|^{2} = |a+bi|^{2}|c+di|^{2} IDENTITY ONE[/tex]

According to my calculations this is true.

[tex]|(a+c) + (b+d)i|^{2} = |a+bi|^{2} + |c+di|^{2} IDENTITY TWO[/tex]

According to my calculations this is false, unless there is a constraint [tex]2ac = -2bd[/tex].

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

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