What Condition Determines Eigenket of A?

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In summary, the question asks under what condition can we conclude that the sum of two eigenkets of a hermitian operator is also an eigenket. The answer is that either the eigenvalues of the two eigenkets are equal, or the inner product of the two eigenkets is zero. Both cases must be considered.
  • #1
zhaiyujia
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Homework Statement


Suppose that |[tex]\alpha[/tex]> and |[tex]\beta[/tex]> are eigenkets(eigenfunctions) of a hermitian operator A. Under what condition can we conclude that |[tex]\alpha[/tex]> + |[tex]\beta[/tex]> is also an eigenket of A?

Homework Equations


It's quite basic, I don't think any addtional equations are needed except the definations.

The Attempt at a Solution


From the question we know that A| [tex]\alpha[/tex] > =a|[tex]\alpha[/tex]> , A|[tex]\beta[/tex]> =b|[tex]\beta[/tex]>. And A is a hermitian operator:
<[tex]\alpha[/tex]|A[tex]\beta[/tex]>=<[tex]\alpha[/tex]|b[tex]\beta[/tex]>=b<[tex]\alpha[/tex]|[tex]\beta[/tex]>,
<[tex]\alpha[/tex]|A[tex]\beta[/tex]>=<A[tex]\alpha[/tex]|[tex]\beta[/tex]>=<a[tex]\alpha[/tex]|[tex]\beta[/tex]>=a<[tex]\alpha[/tex]|[tex]\beta[/tex]>,
Therefore a=b? or <[tex]\alpha[/tex]|[tex]\beta[/tex]>=0?
But it's nothing to do with |[tex]\alpha[/tex]>+|[tex]\beta[/tex]>+
It seems no addition is need to constrain on them
 
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  • #2
zhaiyujia said:
It seems no addition is need to constrain on them
Of course not; you never applied the condition that [itex]|\alpha\rangle + |\beta\rangle[/itex] is to be an eigenket!
 
  • #3
But what is the condition? if my first part is right:
A(|[tex]\alpha[/tex]>+|[tex]\beta[/tex]>)=a(|[tex]\alpha[/tex]>+|[tex]\beta[/tex]>)
is automatic right?
 
  • #4
Well, no, it's not automatic. Your first part said
"a = b" or "[itex]\langle \alpha | \beta \rangle[/itex] = 0".​

So, you have to consider both cases, not just the "a = b" case.
 

1. What is an eigenket?

An eigenket is a vector that is associated with a specific eigenvalue of a linear transformation. It is a fundamental concept in linear algebra and is used to describe the behavior of a system in quantum mechanics.

2. What condition determines the eigenket of A?

The eigenket of a linear transformation A is determined by the eigenvectors associated with the eigenvalues of A. In other words, an eigenket is a vector that when multiplied by A, gives a scalar multiple of itself.

3. How is the eigenket of A calculated?

To calculate the eigenket of A, the eigenvectors associated with the eigenvalues of A must be found. This can be done by solving the characteristic equation of A, which is det(A-λI) = 0, where λ is the eigenvalue and I is the identity matrix.

4. Can the eigenket of A change?

Yes, the eigenket of A can change if the eigenvalues and eigenvectors of A change. This can happen if the linear transformation A is altered or if the basis of the vector space is changed.

5. What is the significance of the eigenket of A?

The eigenket of A is significant because it allows us to understand the behavior of a linear transformation. It also helps us to find important properties of the transformation, such as the dominant behavior or stability of the system.

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