Hermitian conjugate of the annihilation operator

[dyn]

Hi
I have been looking at the solutions to a past exam question. The question gives the annihilation operator for the harmonic oscillator as a= x + ip ( I have left out the constants ). The question then asks to calculate the Hermitian conjugate a(dagger).
I thought to find the Hermitian conjugate I just change the sign of the imaginary component and this does give the right answer and most books just state this as the conjugate but the exam solution goes on to use inner products and integration by parts. I'm confused !

 

[strangerep]

I have been looking at the solutions to a past exam question. The question gives the annihilation operator for the harmonic oscillator as a= x + ip ( I have left out the constants ). The question then asks to calculate the Hermitian conjugate a(dagger).
I thought to find the Hermitian conjugate I just change the sign of the imaginary component and this does give the right answer and most books just state this as the conjugate but the exam solution goes on to use inner products and integration by parts.

I'd guess that you're also required to show that x and p are self-adjoint operators. (If they're not, then the simple recipe you described is not applicable.)

 

[dyn]

I'd guess that you're also required to show that x and p are self-adjoint operators. (If they're not, then the simple recipe you described is not applicable.)

No ,that wasn't part of the question. It just wanted the inner product integral. I have looked in several books and they just state the 2 operators as a and a(dagger) with the minus sign as though its obvious. Is it obvious or does it need to be calculated ?

 

[strangerep]

No ,that wasn't part of the question. It just wanted the inner product integral. I have looked in several books and they just state the 2 operators as a and a(dagger) with the minus sign as though its obvious. Is it obvious or does it need to be calculated ?

It depends on exactly what was being asked. You're still being a bit vague, so I don't know what level of answer is appropriate here.

Maybe you should post the exact question in the "Advanced Physics" homework forum? (If you want to work through it, that is.)

 

[dyn]

The question just gave the annihilation operator and asked to calculate the Hermitian conjugate. I have the solution which involves the inner product integral and I understand the solution. I just wanted to see why it needed to be calculated when it could just be stated. My understanding is as follows - to form the Hermitian conjugate of a sum : apply the following to each term ; complex conjugate any complex constants and take the Hermitian conjugate of any operators. If the operators are Hermitian they remain unchanged.
On a sidenote I previously asked about the 1-D momentum operator : -i(h bar)d/dx . In this case to form the conjugate I change -i to +i and take the conjugate of d/dx which is -d/dx thus showing that the operator is Hermitian.
Have i got this all right ?

 

[dextercioby]

You don't need a particular realization of the Hilbert space (such as L^2(R)), you can work directly from the abstract theorems of functional analysis. If [x,p] = 1 on their common domain of x and p, then

$$a^{\dagger} = (x+ip)^{\dagger}\supset x^{\dagger} + (ip)^{\dagger}\supset x - ip$$

 

[dyn]

Hi ,
Thanks. I think you are telling me that I am right but I don't know functional analysis