Finding eigenstate for the annhilation operator

[Sandrasa]

Homework Statement

Find the eigenvector of the annhilation operator a.

Homework Equations

a|n\rangle = \sqrt{n}|{n-1}\rangle

The Attempt at a Solution

Try to show this for an arbitrary wavefunction:
|V\rangle = \sum_{n=1}^\infty c_{n}|n\rangle

a|V\rangle = a\sum_{n=1}^\infty c_{n}|n\rangle = \sum_{n=1}^\infty \sqrt{n} c_{n}|n-1\rangle

Define a new constant, k = n - 1, and put this in:

- > \sum_{k=0}^\infty \sqrt{k + 1} c_{k + 1}|k\rangle = \sum_{n=0}^\infty \sqrt{n + 1} c_{n + 1}|n\rangle

This was a trick I found in my notes from a lecture, but I do not quite understand how you can say that k = n in the last part. Is it just because k and n are an arbitrary notation for the basis set? Or is it due to the fact that since the sum goes to infity, the difference of 1 does not matter (since I really defined k to be n - 1)?

When I continue with this, I get:

a( c_{1}|1\rangle + c_{2}|2\rangle + c_{3}|3\rangle + ...) = c( \sqrt{1}c_{1}|0\rangle\ + \sqrt{2}c_{2}|1\rangle + \sqrt{3}c_{3}|2\rangle + \sqrt{4}c_{4}|3\rangle)

I am not really sure where to go next. I tried setting the coeffecient before each basisvector equal, like this:

c\sqrt{1}c_{1} = 0
c_{1} = \sqrt{2}c_{2}c
c_{2} = \sqrt{3}c_{3}c
c_{3} = \sqrt{4}c_{4}c

The problem her is that since the first part does not contain the vector 0|\rangle, either c or c_{1} has to be zero in the first equation, and then everything becomes zero, but I know that the annihilation operator has eigenstates.

 

[strangerep]

Try to show this for an arbitrary wavefunction:
|V\rangle = \sum_{n=1}^\infty c_{n}|n\rangle

Is there some reason you didn't start this sum at n=0 ?

 

[Sandrasa]

Yes, because when the annihilation operator works on the state zero, we will get zero by definition according to my book so I thought that there was no point in taking it. But I can see that that would get me on the right track again, but I don't know how I can do it?

But by taking the sum from n = 0, I end up with c_n = c_{n-1}\frac{c'}{\sqrt{n}}, where c' = \frac{1}{c}. And now I'm not really sure what to do next. Can I just put this in my expression for |V\rangle and call that an eigenstate with c' as the constant? It doesn't look like the expression I found for the eigenstate, which was |V\rangle = c_{0} \sum\limits_{n=0}^\infty \frac{c^n}{\sqrt{n!}}|n\rangle in my notation. I tried writting the expression I found out, but then I got something like:
c_0 + \frac{c'c_0}{\sqrt{1}} + \frac{c_0c'^2}{\sqrt{2}} + \frac{c_0c'^3}{\sqrt{6}}. It sort of looks like the equations I got in the start, but I'm stuck in trying to get to one place from the other. Any hints?

And thanks for the help!

 

[strangerep]

In your "relevant equations", you didn't write out the action of the creation operator $a^*$. E.g.,
$$a^*|0\rangle ~=~ ~?~$$$$(a^*)^2 |0\rangle ~=~ ~?~$$etc. Try to express each term in your sum for $|V\rangle$ in terms of $a^*$ acting on the vacuum.