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,
<br /> |\chi_3\rangle=|a+ib\rangle ~~\mbox{and we want:}~~ \langle\chi_3|H|\chi_3\rangle<br />

is it correct to say:
<br /> \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 ??<br />

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

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

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.