Griffiths QM Ground State of Harmonic Oscillator

[bugatti79]

Homework Statement

Folks, I am looking at a past exam question regarding the Harmonic Oscillator. The question ask
'Justify that the ground state of a harmonic oscillator

$$a_\psi_0=0$$ equation 2.58 on page 45 of griffiths.

THis was not covered in my notes. Any ideas how to justify this expression?

Thanks

Eddie

PS, how come [tex][/tex] isn't working for me?

 

[ideasrule]

If there wasn't a state for which a_*psi=0, you could keep on applying a_ and eventually get a state with negative energy. That's obviously not possible, since V(x) never dips below 0.

 

[dextercioby]

There must be an algebraic way to do it, too. N|0> = 0 |0> = 0_vector. And now use how a_+ and a_- are related to N.

 

[vela]

how come [tеx] [/tеx] isn't working for me?

You have a double subscript in your expression, which isn't allowed.

I'm guessing you wanted $a\psi_0 = 0$, so you don't want that first underline to be there.

 

[bugatti79]

There must be an algebraic way to do it, too. N|0> = 0 |0> = 0_vector. And now use how a_+ and a_- are related to N.

Hmmmm...we are not introduce to that linear algebra/Dirac notation at this stage yet...any other suggestions? :-0
DO I put it into Schrödinger equation and see does it go to 0?

 

[vela]

Dextercloby is simply suggesting you use the fact that $\hat{N}\psi_0 = 0$ where $\hat{N}$ is the number operator.

 

[bugatti79]

Dextercloby is simply suggesting you use the fact that $\hat{N}\psi_0 = 0$ where $\hat{N}$ is the number operator.

But how do we show this is true? I wikied it and found that

$$\displaystyle \hat a=\sqrt{\frac{m \omega}{2 \hbar}}(\hat x +\frac{i}{m \omega} \hat p )$$ and $$\displaystyle \hat a^{ \dagger}=\sqrt{\frac{m \omega}{2 \hbar}}(\hat x -\frac{i}{m \omega} \hat p )$$

and $$\hat N = \hat a \hat a^{\dagger}$$

and ye say $$\hat N \psi_0=0$$ but how, is there more to it? You see the past exam paper states the following for 10 marks,

Justify the annihilation of a certain state by a_ and hence use the resulting differential to find this state? (I can find the state which is short enough probably only worth 5 marks so there must be more to the first part :-))

Thanks

 

[ideasrule]

Justify the annihilation of a certain state by a_ and hence use the resulting differential to find this state? (I can find the state which is short enough probably only worth 5 marks so there must be more to the first part :-))

Griffith's book gives the full solution to this question, including the justification. It's on page 45 and goes on until the middle of page 46.

 

[bugatti79]

Ok, thanks. I am aware of that. I believe the paragraph on pg 45 is only a descriptive solution to the justification. I think the question is asking for an algebraic solution... :-)

 

[ideasrule]

For an algebraic solution, applying this operator:

$$\displaystyle \hat a=\sqrt{\frac{m \omega}{2 \hbar}}(\hat x +\frac{i}{m \omega} \hat p )$$

to the ground state wavefunction. You should get 0.

 

[bugatti79]

yea but we don't know what the ground state is as yet, that is the next part of the question. :-0

 

[vela]

So what exactly do you know about the harmonic oscillator? It's kind of hard to advise you when we have to guess what you can assume is true.

By the way, it's $\hat{N}=a^\dagger a$. You got the order of the annihilation and creation operators backwards.

 

[bugatti79]

Guys,

I have just contacted my teacher. The description for the justification in Griffiths will suffice. Sorry for bothering ye! Thanks for you help.

PS - I have just noticed there seem to be some cross posting and some post requesting help for exams. I thought this stuff was forbidden? Just wondering

Cheers

 

[vela]

If you know people are asking for help when they shouldn't be, please use the report button in the thread to notify a moderator about the rule violation.