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I am reading Modern Quantum Mechanics by Sakuria and Napolitano. I either found a logical contradiction, or I am greatly misunderstanding something.

Background Information from the Textbook

a' and a'' are eigenvalues of A.

A is a Hermitian operator.

The symbol, * , implies complex conjugation.

|a'> is an eigenket of A.

The reality condition is proved by assuming that a' = a''. In other words, the two eigenvalues are the same.

Then the authors start proving the orthogonality property by assuming that a' is no longer equal to a''. The eignevalues are assumed to be different now.

But then the authors use the reality condition, which implies that a' = a'', as stated previously.

In summary, the authors are proving the orthogonality property while assuming that the eigenvalues are both the same (by using the reality condition) and different. It is my understanding that something cannot be the same and different!

More Background Information from the Textbook (bottom of page 17)

Consider the following Equations.

A |a'> = a'|a'> (Equation 1)

< a''|A = a'' < a''|* (Equation 2)

Multiply both sides of Equation 1 by < a''| on the left.

Multiply both sides of Equation 2 by |a'> on the right.

Subtract the Equation 2 from Equation 1.

The following expression is obtained.

(a' - a''*) < a'' | a'> = 0 (Equation 3)

After deriving Equation 3, the authors proceed to choose whether a' or a'' are the same or not.

The eigenvalues are the same when proving the reality condition.

The eigenvalues are different when proving the orthogonality property.

But the orthogonality property uses the reality condition, which assumes that the eigenvalues are the same!

The orthogonality property appears to have two contradictory assumptions regarding the equality of a' and a''.

Is it actually correct to prove the orthogonality property with the reality condition?

I am reading Modern Quantum Mechanics by Sakuria and Napolitano.

Background Information from the Textbook

a' and a'' are eigenvalues of A.

A is a Hermitian operator.

The symbol, * , implies complex conjugation.

|a'> is an eigenket of A.

The reality condition is proved by assuming that a' = a''. In other words, the two eigenvalues are the same.

Then the authors start proving the orthogonality property by assuming that a' is no longer equal to a''. The eignevalues are assumed to be different now.

More Background Information from the Textbook (bottom of page 17)

Consider the following Equations.

A |a'> = a'|a'> (Equation 1)

< a''|A = a'' < a''|* (Equation 2)

Multiply both sides of Equation 1 by < a''| on the left.

Multiply both sides of Equation 2 by |a'> on the right.

Subtract the Equation 2 from Equation 1.

The following expression is obtained.

(a' - a''*) < a'' | a'> = 0 (Equation 3)

After deriving Equation 3, the authors proceed to choose whether a' or a'' are the same or not.

The eigenvalues are the same when proving the reality condition.

The eigenvalues are different when proving the orthogonality property.

Is there a contradiction? On page 18, the authors prove the orthogonality property "because of the just-proved reality condition."

I am reading Modern Quantum Mechanics by Sakuria and Napolitano. I either found a logical contradiction, or I am greatly misunderstanding something.

Background Information from the Textbook

a' and a'' are eigenvalues of A.

A is a Hermitian operator.

The symbol, * , implies complex conjugation.

|a'> is an eigenket of A.

On the bottom of page 17, the reality condition is proved by assuming that a' = a''.

Then the authors start proving the orthogonality property by assuming that a' is no longer equal to a''.

But then the authors use the reality condition, which implies that a' = a'', as stated previously.

The authors are proving the orthogonality property, while assuming that the eigenvalues are both the same (by using the reality condition) and different. It is my understanding that something cannot be the same and different.

More Background Information from the Textbook (bottom of page 17)

Consider the following Equations.

A |a'> = a'|a'> (Equation 1)

< a''|A = a'' < a''|* (Equation 2)

Multiply both sides of Equation 1 by < a''| on the left.

Multiply both sides of Equation 2 by |a'> on the right.

Subtract the Equation 2 from Equation 1.

The following expression is obtained.

(a' - a''*) < a'' | a'> = 0 (Equation 3)

After deriving Equation 3, the authors proceed to choose whether a' or a'' are the same or not. The first part is the reality condition (the eigenvalues are the same), and the second part is the orthogonality property (the eigenvalues are different). These two parts of the same theorem have contradictory assumptions regarding the equality of a' and a'', so is it actually correct to prove the orthogonality property with the reality condition?

Side Note: The reality condition is:a' = a'*,.

I am reading Modern Quantum Mechanics by Sakuria and Napolitano. I either found a logical contradiction, or I am greatly misunderstanding something.

Background Information from the Textbook

a' and a'' are eigenvalues of A.

A is a Hermitian operator.

The symbol, * , implies complex conjugation.

|a'> is an eigenket of A.

The reality condition is proved by assuming that a' = a''. In other words, the two eigenvalues are the same.

Then the authors start proving the orthogonality property by assuming that a' is no longer equal to a''. The eignevalues are assumed to be different now.

But then the authors use the reality condition, which implies that a' = a'', as stated previously.

In summary, the authors are proving the orthogonality property while assuming that the eigenvalues are both the same (by using the reality condition) and different. It is my understanding that something cannot be the same and different!

More Background Information from the Textbook (bottom of page 17)

Consider the following Equations.

A |a'> = a'|a'> (Equation 1)

< a''|A = a'' < a''|* (Equation 2)

Multiply both sides of Equation 1 by < a''| on the left.

Multiply both sides of Equation 2 by |a'> on the right.

Subtract the Equation 2 from Equation 1.

The following expression is obtained.

(a' - a''*) < a'' | a'> = 0 (Equation 3)

After deriving Equation 3, the authors proceed to choose whether a' or a'' are the same or not. 

The eigenvalues are the same when proving the reality condition.

The eigenvalues are different when proving the orthogonality property.

But the orthogonality property uses the reality condition, which assumes that the eigenvalues are the same!

The orthogonality property appears to have two contradictory assumptions regarding the equality of a' and a''.

Is it actually correct to prove the orthogonality property with the reality condition?

I am reading Modern Quantum Mechanics by Sakuria and Napolitano. I either found a logical contradiction, or I am greatly misunderstanding something.

Background Information from the Textbook

a' and a'' are eigenvalues of A.

A is a Hermitian operator.

The symbol, * , implies complex conjugation.

|a'> is an eigenket of A.

On the bottom of page 17, the reality condition is proved by assuming that a' = a''.

Then the authors start proving the orthogonality property by assuming that a' is no longer equal to a''.

But then the authors use the reality condition, which implies that a' = a'', as stated previously.

The authors are proving the orthogonality property, while assuming that the eigenvalues are both the same (by using the reality condition) and different.

More Background Information from the Textbook

The proofs for the reality condition and the orthogonality property begin as follows.

A |a'> = a'|a'> (Equation 1)

Also,

< a''|A = a'' < a''|* (Equation 2)

Multiply both sides of Equation 1 by < a''| on the left.

Multiply both sides of Equation 2 by |a'> on the right.

Subtract the Equation 2 from Equation 1.

The following expression is obtained.

(a' - a''*) < a'' | a'> = 0 (Equation 3)

After deriving Equation 3, the authors proceed to choose whether a' or a'' are the same or not. The first part is the reality condition (the eigenvalues are the same), and the second part is the orthogonality property (the eigenvalues are different). These two parts of the same theorem have contradictory assumptions regarding the equality of a' and a'', so is it actually correct to prove the orthogonality property with the reality condition?

Side Note: The reality condition is:a' = a'*,.

I am reading Modern Quantum Mechanics by Sakuria and Napolitano. I either found a logical contradiction, or I am greatly misunderstanding something.

Background Information from the Textbook

a' and a'' are eigenvalues of A.

A is a Hermitian operator.

The symbol, * , implies complex conjugation.

|a'> is an eigenket of A.

On the bottom of page 17, the reality condition is proved by assuming that a' = a''.

Then the authors start proving the orthogonality property by assuming that a' is no longer equal to a''.

But then the authors use the reality condition, which implies that a' = a'', as stated previously.

The authors are proving the orthogonality property, while assuming that the eigenvalues are both the same (by using the reality condition) and different. It is my understanding that something cannot be the same and different.

More Background Information from the Textbook (bottom of page 17)

Consider the following Equations.

A |a'> = a'|a'> (Equation 1)

< a''|A = a'' < a''|* (Equation 2)

Multiply both sides of Equation 1 by < a''| on the left.

Multiply both sides of Equation 2 by |a'> on the right.

Subtract the Equation 2 from Equation 1.

The following expression is obtained.

(a' - a''*) < a'' | a'> = 0 (Equation 3)

After deriving Equation 3, the authors proceed to choose whether a' or a'' are the same or not. The first part is the reality condition (the eigenvalues are the same), and the second part is the orthogonality property (the eigenvalues are different). These two parts of the same theorem have contradictory assumptions regarding the equality of a' and a'', so is it actually correct to prove the orthogonality property with the reality condition?

Side Note: The reality condition is:a' = a'*,.