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Quick question about matrices & bases?

  • Thread starter jeebs
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  • #1
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Say I was given a 2x2 matrix made from a certain basis [tex]{|x\rangle, |y\rangle} [/tex] , and I split that matrix into two parts, one being the diagonal part and one being the off-diagonal part.

for example, if I had [tex] H = H_0 + W = \left(\begin{array}{cc}a&c\\b&d\end{array}\right) = \left(\begin{array}{cc}a&0\\0&d\end{array}\right) + \left(\begin{array}{cc}0&c\\b&0\end{array}\right)[/tex]

Is it true to say that H0 is still in the basis [tex]{|x\rangle, |y\rangle} [/tex], and if it is, is there a way I could determine what [tex]|x\rangle[/tex] and [tex]|y\rangle[/tex] actually look like?
 

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  • #2
fzero
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Your question is a bit ambiguous. The explicit matrix form that you've written down is in the basis

[tex] |e_1\rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix}, |e_2\rangle = \begin{pmatrix} 0 \\ 1 \end{pmatrix}.[/tex]

Then

[tex] H = a |e_1\rangle \langle e_1 | + d |e_2\rangle \langle e_2 | + b |e_2\rangle \langle e_1 | + c |e_1\rangle \langle e_2 |.[/tex]

If you want to express this in another basis, you could just rotate [tex]|e_{1,2}\rangle[/tex] into [tex]|x,y\rangle[/tex].
 
  • #3
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well to be more specific I'm attempting this:

[PLAIN][URL]http://i52.photobucket.com/albums/g33/long_john_cider/problem.jpg[/PLAIN][/URL]

I've done part one, which involves coming up with a Hamiltonian then splitting it into two parts, an initial part H0 which has the diagonal elements, and a time dependent part W(t) which has the off diagonal elements.

On the second part I need to mess around with that integral for a(t), and to do that I must need to know what [tex]|\uparrow \rangle , | \downarrow \rangle [/tex] are.

So, I'm wondering how I can find them. I have "constructed the Hamiltonian in the basis [tex]|\uparrow \rangle , | \downarrow \rangle [/tex] " in part one, so now I'm wondering how to find out what this basis actually is...
 
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  • #4
fzero
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Well [tex]|\uparrow\rangle = | e_1\rangle, |\downarrow\rangle = | e_2\rangle[/tex], but that doesn't quite answer your question. If you wanted to write an expression for a matrix in another basis, you would need a formula that relates the two bases.

(Edit: Maybe it does answer your question)
 
  • #5
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I'm not sure I understand how you've decided that [tex] |\uparrow\rangle = \left(\begin{array}{c}1&0\end{array}\right) [/tex] and [tex] |\downarrow\rangle = \left(\begin{array}{c}0&1\end{array}\right) [/tex]
 
  • #6
fzero
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I'm not sure I understand how you've decided that [tex] |\uparrow\rangle = \left(\begin{array}{c}1&0\end{array}\right) [/tex] and [tex] |\downarrow\rangle = \left(\begin{array}{c}0&1\end{array}\right) [/tex]
They're the eigenvectors of [tex]\sigma_z[/tex].
 
  • #7
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ah, so they are, and I have [tex] H_0 = B_0\sigma_z [/tex].
I might be getting somewhere.
thanks.
 

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