Missing h-bar in showing SHO in terms of ANHIL and CREA operators is correct

[rwooduk]

this is the given:

the problem is the middle term, if the h-bar w outside the set brackets is canceled with the h-bar w of the m/2hw, then there will be a h-bar w that is left introduced from the middle term, i.e.

$$i\frac{w}{m}XP- i\frac{w}{m}PX = i\frac{w}{m}[X,P]= i\frac{w}{m}i\hbar$$

but there is no h-bar in the end result, as though h-bar w has been canceled twice??

thanks in advance for any help

 

[moriheru]

Ah so your on harmonic oscillators in susskind?! The hamiltonian is rewriten in form of a^2+b^2 with complex and imaginary numbers, yet you must add the h(bar) so that the hamiltonian stays the same.

 

[rwooduk]

Ah so your on harmonic oscillators in susskind?! The hamiltonian is rewriten in form of a^2+b^2 with complex and imaginary numbers, yet you must add the h(bar) so that the hamiltonian stays the same.

Not yet, this one was given in class, hmm not sure what you are saying, if the h-bar w outside the set bracket cancels with the one inside then how can it also be used to cancel the h-bar w that would be given from the [X,P] commutation relation?

thanks for the reply!

 

[moriheru]

What I am trying to say is that you start with the last equation in your post and then use the fact that (a-bi)(a+bi)+a^+2abi-2bi-b^2=a^2+b^2 which is of the form of 1/2mw^2x^2+p^2/2m. So you don't start woth the anhil and crea form you start with this equation and reform it. The [X,P] has nothing ,that I know of, to do woth the theme at hand.

 

[rwooduk]

ahhh i think i see what you are getting at, i'll have to run through it a few times to be sure, thanks again for your help, its appreciated!

 

[moriheru]

Wait here is the equation:
1.H=(P^2+wX^2)*1/2
We define
2.a^2+b^2=H=(P^2+wX^2)*1/2=(a+bi)(a-bi)
3.H=1/2(p+iwX)(p-iwX) (!note this is incomplete!)
4.1/2(p+iwX)(p-iwX)= 1/2*(P^2+iwXP-iwPX - i^2w^2X^2)
=1/2(P^2+iw(XP-PX)+w^2X^2
=1/2(P^2+w^2X^2)+1/2iw(XP-PX)
(XP-PX)=[X,P]=-ih(bar)

5.1/2(P^2+w^2X^2)+1/2iwih(bar)
=1/2(P^2+w^2X^2)-1/2wh(bar)

The factored expression we started with is smaller than the actuall hamiltonian so we must add wh(bar)/2. so the equation becomes

1/2(p+iwX)(p-iwX)+wh(bar)/2

and then we define the creation and anhilation operators to be: (p+iwX) and (p-iwX) with some modification.

Yet there are different versions of how to derive the anhilation operator and creation operator, for example one can define q and p and create the operators with the aid of q and p

hope this helps...

 

[rwooduk]

Wait here is the equation:
1.H=(P^2+wX^2)*1/2
We define
2.a^2+b^2=H=(P^2+wX^2)*1/2=(a+bi)(a-bi)
3.H=1/2(p+iwX)(p-iwX) (!note this is incomplete!)
4.1/2(p+iwX)(p-iwX)= 1/2*(P^2+iwXP-iwPX - i^2w^2X^2)
=1/2(P^2+iw(XP-PX)+w^2X^2
=1/2(P^2+w^2X^2)+1/2iw(XP-PX)
(XP-PX)=[X,P]=-ih(bar)

5.1/2(P^2+w^2X^2)+1/2iwih(bar)
=1/2(P^2+w^2X^2)-1/2wh(bar)

The factored expression we started with is smaller than the actuall hamiltonian so we must add wh(bar)/2. so the equation becomes

1/2(p+iwX)(p-iwX)+wh(bar)/2

and then we define the creation and anhilation operators to be: (p+iwX) and (p-iwX) with some modification.

Yet there are different versions of how to derive the anhilation operator and creation operator, for example one can define q and p and create the operators with the aid of q and p

hope this helps...

wow, thanks for your efforts! yes i see now i was working this the wrong way around, and i see now why the h-bar w must be added. Thanks once again, its appreciated!

 

[rwooduk]

The factored expression we started with is smaller than the actuall hamiltonian so we must add wh(bar)/2.

just a last question, how can you do this? it kind of comes back to my original question working the other way in that you have to remove the wh(bar)/2 term for it to be correct, why is this allowed? especially as you are trying to show one is equal to the other. thanks

 

[moriheru]

Ok, I will replie properly later but just for now: I don't know if you have done this already in class but the product of the operators crea and anhil is written as N=crea*anhil so your equation would be simpler. I know this in't an answer but I shall come back later to your question and I hope I can help! Scientists went mad defining operators!:)

 

[rwooduk]

Ok, I will replie properly later but just for now: I don't know if you have done this already in class but the product of the operators crea and anhil is written as N=crea*anhil so your equation would be simpler. I know this in't an answer but I shall come back later to your question and I hope I can help! Scientists went mad defining operators!:)

thanks any more feedback on this would be more than welcome

 

[moriheru]

I believe I understand what your problem is!? The crea and anhil operators are defined as I said(either with p and q or as factors of the hamiltonian...), yet historically the crea and anhil operators are defined differnetly which changes the complete or at least part of the hamiltonian equation

where historically they are defined with an extra factor, beeing

+-i/2wh(bar).

I believe one can leave out the factor, to make things simpler. Anyway I hope that helped a bit...

 

[moriheru]

Try this pdf. It looks at simple 1d harmonic oscillators and there operators in a different way.
http://www.cithep.caltech.edu/~fcp/physics/quantumMechanics/SHO/SHO.pdf

The last parts are not so inportant but
And here is the p-q stuff and some matrices:

This isn't the answer to your original question but I think it's a aproach.

 

[rwooduk]

Thats very helpful ! Think I have it now! Many thanks for all your help!