- #1
jumbogala
- 423
- 4
Homework Statement
Find <P>. P = i√(mhw/2)(a†-a). Note a† and a are the ladder operators. P is the momentum operator of the harmonic oscillator.
|ψ > = (1/sqrt(2))[ |1> - i |2>]
The answer should be zero, can someone check my work?
Homework Equations
a† |n> = sqrt(n+1)|n+1>
a |n> = sqrt(n)|n-1>
The Attempt at a Solution
I think my first step here might be wrong. I assumed that <ψ | = (1/sqrt(2))[ <1| + i <2|].
<P> = <ψ | P | ψ > = (1/sqrt(2))[ <1| + i <2|] i√(mhw/2)(a†-a) (1/sqrt(2))[ |1> - i |2>]
= (1/2)i√(mhw/2)[ <1| + i <2|] * (a†|1> - i*a†|2> - a|1> + i*a|2>]
= (1/2)i√(mhw/2)[ <1| + i <2|] * [sqrt(2)|2> - i*sqrt(3)|3> - sqrt(1)|0> + i*sqrt(2)|1>]
Since <x|y> = 1 when x = y but 0 otherwise, this reduces to
= (1/2)i√(mhw/2)[ <1| + i <2|] * [sqrt(2)|2> - i*sqrt(3)|3> - sqrt(1)|0> + i*sqrt(2)|1>]
=(1/2)i√(mhw/2) (isqrt(2)<1|1> + isqrt(2)<2|2>) = (1/2)i√(mhw/2) ( 2isqrt(2))
= -sqrt(mhw)
But I think the answer is supposed to be zero. What am I doing wrong here?
I already checked for sign mistakes 3 times, and I can't find any.