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Quantum Matrix Calculation

  • Thread starter quixi
  • Start date
  • #1
3
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Homework Statement



From a given [itex] \left| \psi \right> [/itex] I have calculated an expression for [itex]\eta[/itex] which results in a 4x4 matrix (as required) however I now need to find [itex]\eta^{2}[/itex] but don't know how to do this with still keeping a resulting 4x4 matrix.

Homework Equations



[tex]\eta = \alpha \left| 0000 \right> + \beta \left| 0001 \right> + ... + \theta \left| 1111 \right> [/tex] etc.

The Attempt at a Solution



I'm pretty sure I can't just do:

[tex]\eta^{2} = \alpha^{2} \left| 0000 \right> + \beta^{2} \left| 0001 \right> + ... + \theta^{2} \left| 1111 \right> [/tex] etc.

But if I do [itex]\eta^{2}[/itex] I'll end up with:

[tex]\eta^{2} = \alpha^{2} \left| 00000000 \right> + \beta^{2} \left| 00010001 \right> + ... + \theta^{2} \left| 11111111 \right> [/tex] etc.

which isn't any good, I need a 4x4 matrix not a 8x8 matrix.

Perhaps:

[tex]\eta^{2} = \eta \eta*[/tex]

but I still don't think that would do.

Hmm. :frown:
 

Answers and Replies

  • #2
318
0
The eta you gave is no 4x4 matrix. It is vector in a 2^4 dimensional space.
If you are give [tex]\psi[/tex] then [tex]\eta[/tex] is defined as [tex]\eta = |\psi><\psi|[/tex]. Then you can square the matrix which will result in having the scalar product between your basis vectors.
 
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