How do I handle degenerate eigenvalues and eigenvectors in quantum mechanics?

In summary, when faced with a degenerate eigenvalue problem, one must choose an orthonormal basis within the eigenspace, but the specific vectors chosen do not affect the outcome. For non-degenerate eigenvalues, one can choose any vector within the eigenspace, but it is convention to choose orthonormal vectors. This also applies to finding eigenvectors for matrices with no degenerate eigenvalues.
  • #1
M. next
382
0
In Quantum, I ran across the eigenvalue problem.
They gave me a matrix, and i was asked to find eigenvalues and then eigenvectors.
But the eigenvalues, were degenerate and thus i couldn't find the exact normalized eigenvector.
What to do in this case? Shoukd i choose arbitrary values?

My other question is about another problem, they gave me a matrix and i got no degenrate eigenvakues, anyway when i wanted to find eigenvector, i tried normalizing it, so i got let's say:
y^2=4 so y=±2

What do i choose? Does it make a difference?
 
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  • #2
M. next said:
In Quantum, I ran across the eigenvalue problem.
They gave me a matrix, and i was asked to find eigenvalues and then eigenvectors.
But the eigenvalues, were degenerate and thus i couldn't find the exact normalized eigenvector.
What to do in this case? Shoukd i choose arbitrary values?

My other question is about another problem, they gave me a matrix and i got no degenrate eigenvakues, anyway when i wanted to find eigenvector, i tried normalizing it, so i got let's say:
y^2=4 so y=±2

What do i choose? Does it make a difference?

There is no unique eigenvector corresponding to degenerate eigenvalues. Instead, all the vectors in a subspace of dimension equal to the degeneracy can be its eigenvectors. Non-degenerate eigenvalue is really a special case where that dimension is 1. In case of degeneracy, you are free to choose any vectors in the eigenspace in forming a basis, it doesn't matter, but by convention, you choose orthonormal vectors with simple coordinates.

This also answers your 2nd question, if it has no degeneracy, you are choosing an orthonormal basis in 1D, but you still have freedom to choose its direction. In general, you can multiply a ket with unitary complex number without changing its physical significance.
 
  • #3
Thanks you a lot
 

FAQ: How do I handle degenerate eigenvalues and eigenvectors in quantum mechanics?

What are eigenvectors and eigenvalues?

Eigenvectors and eigenvalues are mathematical concepts that are used to study linear transformations and systems of linear equations. An eigenvector is a vector that, when multiplied by a specific matrix, results in a scalar multiple of itself. This scalar multiple is known as the eigenvalue, and it represents the amount by which the eigenvector is scaled or stretched.

What is the significance of eigenvectors and eigenvalues?

Eigenvectors and eigenvalues are important in many areas of mathematics and science, including physics, engineering, and computer science. They are used to study the behavior of linear systems, such as the movement of objects in space or the flow of electricity in a circuit. They can also be used to simplify complex calculations and equations, making them valuable tools in problem-solving and modeling.

How are eigenvectors and eigenvalues calculated?

The process of finding eigenvectors and eigenvalues involves solving a system of linear equations. The eigenvalues can be found by setting the determinant of the matrix equal to zero and solving for the scalar values. The corresponding eigenvectors can then be found by plugging in the eigenvalues into the original equation and solving for the vector components. There are also various algorithms and software programs that can be used to calculate eigenvectors and eigenvalues.

What is the relationship between eigenvectors and eigenvalues?

The eigenvalue represents the scaling factor of the eigenvector, meaning that multiplying the eigenvector by the eigenvalue will result in the same vector but scaled by that factor. Additionally, eigenvectors corresponding to different eigenvalues are orthogonal, meaning they are perpendicular to each other. This relationship is often used in applications such as principal component analysis, which is used to reduce the dimensionality of a dataset.

Can eigenvectors and eigenvalues be negative?

Yes, eigenvectors and eigenvalues can be negative. The sign of an eigenvector or eigenvalue does not change its mathematical properties or significance. However, in some applications, such as in physics or engineering, the direction or magnitude of an eigenvector may have physical meaning, so the sign may be important in those contexts.

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