- #1
Nick R
- 69
- 0
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?
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?