Expand ψ(o) in terms of eignestates

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In summary, expanding ψ(o) in terms of eigenvectors allows us to express a given vector in a different basis, which can be useful in solving mathematical problems or understanding system behavior. This is calculated by finding the eigenvalues and eigenvectors, then determining the coefficients using inner product or projection. However, ψ(o) can only be expanded if it is a linear combination of the eigenvectors and the eigenvectors must form a complete basis. The expansion can provide information about the relative importance of each eigenvector and their behavior, but limitations include the need for a complete basis and potential degeneracy of the eigenvectors.
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M. next
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if ψ(o)=(1 0)[itex]^{T}[/itex] at time t=0.

According to some Hamiltonian, it was found that the corresponding eigenstates are |ø[itex]_{1}[/itex]> = 1/√2(1 i)[itex]^{T}[/itex] and |ø[itex]_{2}[/itex]> = 1/√2(1 -i)[itex]^{T}

so then we wanted to expand ψ(0) in terms of |ø[itex]_{1}[/itex]> and |ø[itex]_{2}[/itex]>:

the author got: 1/√2|ø[itex]_{1}[/itex]> + 1/√2 |ø[itex]_{2}[/itex]>

My question is that where did he get the coefficients of |ø[itex]_{1}[/itex]> and |ø[itex]_{2}[/itex]>?? Is there a certain rule to this?

Note: this is an easy example, I can give a more detailed one if needed.
 
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  • #2
Just solved this:
[itex] \psi(0)= a |1> + b |2> [/itex]
for [itex]a,b[/itex]
 

Related to Expand ψ(o) in terms of eignestates

1. What is the purpose of expanding ψ(o) in terms of eigenvectors?

The expansion of ψ(o) in terms of eigenvectors allows us to express a given vector in a different basis. This can be useful in solving certain mathematical problems or understanding the behavior of a system.

2. How is the expansion of ψ(o) in terms of eigenvectors calculated?

The expansion is calculated by first finding the eigenvalues and eigenvectors of the given vector. Then, the coefficients in the expansion are determined using the inner product or projection of the vector onto each eigenvector.

3. Can ψ(o) be expanded in terms of any set of vectors?

No, ψ(o) can only be expanded in terms of eigenvectors if the vector is a linear combination of those eigenvectors. If the vector is not a linear combination, then the expansion is not possible.

4. What information can be gained from expanding ψ(o) in terms of eigenvectors?

By expanding ψ(o) in terms of eigenvectors, we can determine the relative importance of each eigenvector in the overall vector. This information can be used to analyze the behavior of the vector or system.

5. Are there any limitations to expanding ψ(o) in terms of eigenvectors?

One limitation is that the eigenvectors must form a complete basis for the vector space. Additionally, the expansion may not be possible if the eigenvectors are degenerate, meaning they have the same eigenvalue.

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