Finding the Eigenstate for Harmonic Oscillator?

[unscientific]

Homework Statement

Given the a|n> = α|n-1>, show that α = √n :

Homework Equations

The Attempt at a Solution

<n|a^{+}\hat {a}|n> = \alpha <n|a^{+}|n-1> = | \hat a|n>|^2

\alpha = \frac{<n|a^{+}\hat {a}|n>}{<n|a^{+}|n-1>}

Taking the complex conjugate of both sides:

\alpha* = \frac{<n|a^{+}\hat {a}|n>}{\alpha<n-1|n-1>}
|\alpha^2| = \frac{<n|a^{+}\hat {a}|n>}{<n-1|n-1>}

Where $\hat {a} a^{+} = \frac{1}{2m\omega} \left( (m\omega x)^2 + p^2 + im\omega [\hat{x},\hat{p}] \right)$

 

[BruceW]

nice work. I think you're getting there. although remember that $\alpha$ isn't necessarily real, so when you took the complex conjugate of both sides, you should also take complex conjugate of $\alpha$. For the final equation you wrote down, I think that is the right idea. But it's definitely missing some $\hbar$ and the square root should not still be there. it's just a minor mistake.

 

[unscientific]

nice work. I think you're getting there. although remember that $\alpha$ isn't necessarily real, so when you took the complex conjugate of both sides, you should also take complex conjugate of $\alpha$. For the final equation you wrote down, I think that is the right idea. But it's definitely missing some $\hbar$ and the square root should not still be there. it's just a minor mistake.

The problem is, I have no idea how to proceed now..

 

[BruceW]

once you've fixed the minor mistake, you have the operator $aa^\dagger$. what other operator does this operator look like? hint: the physical model of this system is simple harmonic motion.

 

[unscientific]

once you've fixed the minor mistake, you have the operator $aa^\dagger$. what other operator does this operator look like? hint: the physical model of this system is simple harmonic motion.

Okay, The square root has gone away, due to it appearing twice from a and a⁺ and I've fixed the square term in the bracket.

 

[unscientific]

once you've fixed the minor mistake, you have the operator $aa^\dagger$. what other operator does this operator look like? hint: the physical model of this system is simple harmonic motion.

USing $[\hat x,\hat p] = i\hbar$

\hat a a^{+} = \frac{\hat H}{\hbar \omega} - \frac{1}{2}

|\alpha^2| = \frac{<n| \frac{\hat H}{\hbar \omega} - \frac{1}{2}|n>}{<n-1|n-1>}

How do I proceed from here?

 

[BruceW]

hmm. wait hold on, since they ask you to consider $|\hat{a}|n\rangle |^2$ I think you should use the operator that you were originally working with, $\hat{a}^\dagger \hat{a}$

edit: sorry about that, the rest of your working looks good. you've got the right idea.

 

[unscientific]

hmm. wait hold on, since they ask you to consider $|\hat{a}|n\rangle |^2$ I think you should use the operator that you were originally working with, $\hat{a}^\dagger \hat{a}$

edit: sorry about that, the rest of your working looks good. you've got the right idea.

How do I evaluate the inner product:

|\alpha^2| = \frac{<n| \frac{\hat H}{\hbar \omega} - \frac{1}{2}|n>}{<n-1|n-1>} = \frac{<n|a^{+}\hat {a}|n>}{<n-1|n-1>}It should give $n$ on the RHS.

 

[BruceW]

you're almost there now. In the question, they tell you $E_n = (n+1/2)\hbar \omega$ they mean this is the energy of the $|n\rangle$ eigenstate. so this should be enough to evaluate that inner product.

 

[unscientific]

you're almost there now. In the question, they tell you $E_n = (n+1/2)\hbar \omega$ they mean this is the energy of the $|n\rangle$ eigenstate. so this should be enough to evaluate that inner product.

How do I evaluate the numerator?

<n| \frac{\hat H}{\hbar \omega} - \frac{1}{2}|n>
= \frac{1}{\hbar \omega} <n|\hat H|n> - \frac{1}{2}<n|n>
= \frac{1}{\hbar \omega} <n|\hat H|n> - \frac{1}{2}<n|n>
= \frac{E_n}{\hbar \omega}<n|n> - \frac{1}{2}<n|n>
= \left (\frac{E_n}{\hbar \omega} - \frac{1}{2} \right) <n|n>

Thus,
|\alpha|^2 = \left (\frac{E_n}{\hbar \omega} - \frac{1}{2} \right)\frac{ <n|n>}{<n-1|n-1>}
|\alpha|^2 = n \frac{ <n|n>}{<n-1|n-1>}

 

[BruceW]

yep. that is good so far for the numerator. But the denominator is not (n-1)². each of the $|n\rangle$ are states, right? so what (conventionally), is the inner product of the state with itself?

edit: and now, there is a similar question for what is left over in the numerator.

 

[unscientific]

yep. that is good so far for the numerator. But the denominator is not (n-1)². each of the $|n\rangle$ are states, right? so what (conventionally), is the inner product of the state with itself?

edit: and now, there is a similar question for what is left over in the numerator.

Can I assume they are linearly independent, and have norm = 1?

 

[BruceW]

yeah! I guess it is not obvious really. But I think in these kinds of problems, you can assume that if they give you a set of states, those states will be orthonormal. (unless they say otherwise).