(adsbygoogle = window.adsbygoogle || []).push({}); Operators and eigenstates (updated with new question)

Hi, I encountered the following HW problem which really confuses me. Could anyone please explain it to me? Thank you so much!

The result of applying a Hermitian operator B to a normalized vector |1> is generally of the form:

B|1> = b|1> + c|2>

where b and c are numerical coefficients and |2> is a normalized vector orthogonal to |1>.

My question is:Why B|1> must have the above form?Does it mean if |1> is an eigenstate of B, then b=!0 and c=0? But what if |1> is not an eigenstate of B?

I also need to find the expectation value of B (<1|B|1> ), but I think I got this part:

<1|B|1> = <1|b|1> + <1|c|2> = b<1|1> + c<1|2> = b

since |1> and |2> are orthogonal and they're both normalized. Does that look right?

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Operators and eigenstates

**Physics Forums | Science Articles, Homework Help, Discussion**