Calculating <\chi_3|H|\chi_3> w/ Dirac Notation Algebra

[roro-rose]

Say we have,
|\chi_3>=|a+ib> and we want : <\chi_3|H|\chi_3>

is it correct to say:
<\chi_3|H|\chi_3>= <a|H|a>+<a|H|ib>+<-ib|H|a>+<-ib|H|ib> ??

 

[strangerep]

(Hint: learn how do latex on this forum. I've adjusted your stuff below accordingly.
Click on any of the expressions to get more info.)

Say we have,
$$|\chi_3\rangle=|a+ib\rangle ~~\mbox{and we want:}~~ \langle\chi_3|H|\chi_3\rangle$$

is it correct to say:
$$\langle\chi_3|H|\chi_3\rangle ~=~ \langle a|H|a\rangle ~+~ \langle a|H|ib\rangle<br /> ~+~ \langle -ib|H|a\rangle ~+~ \langle -ib|H|ib\rangle ??$$

That's almost right, except that bras are antilinear, hence
$$\langle ib| ~=~ -i \langle b|$$

So, e.g, your 3rd term should be
$$\langle ib|H|a\rangle ~=~ -i \langle b|H|a\rangle$$

HTH.

 

[Avodyne]

It really depends on what you mean by $|a+ib\rangle$, which is not a standard notation. Do you mean $|a\rangle+i|b\rangle$? if so, then strangerep's correction of your result is the right answer.