Checking that a coherent state is an eigenfunction of an operator

[Dixanadu]

Homework Statement

Hey guys, I'll type this thing up in Word.

http://imageshack.com/a/img716/8219/wycz.jpg

 

[hilbert2]

What is the physical system in question, is it a harmonic oscillator? If it is, then the lowering operator acts on the energy eigenstates as $\hat{a}\psi_{n}(x)=\sqrt{n}\psi_{n-1}(x)$. This is the only info you need in order to solve the problem.

 

[Dixanadu]

The question doesn't say that it is a harmonic oscillator, but it does say that "these states closely resemble classical particles" so I think you're right. If I do as you say, I end up with this:
http://imageshack.com/a/img69/8361/loix.jpg

 

[hilbert2]

The lowering operator annihilates the ground state: $\hat{a}\psi_{0}(x)=0$. Also, you can change the variable over which the summation is, e.g. $k=n-1$. That way you should be able to show that acting on the coherent state with the lowering operator is equivalent to multiplying with a constant.

 

[Dixanadu]

Okay I'm lost...T_T Is this what you mean?
http://imageshack.com/a/img703/889/xvhg.jpg

I feel kinda stupid now ¬_¬

 

[hilbert2]

After you change the index, the summation should be from $k=0$ to $k=\infty$ and the exponent of $\lambda$ becomes $k+1$...

 

[Dixanadu]

So you mean this?
http://imageshack.com/a/img842/8932/5a4q.jpg

which means that the eigenvalue is just...$λ$?

 

[hilbert2]

Yes, that's the correct answer.

 

[Dixanadu]

Wow thank you :d !

 

[Dixanadu]

that was supposed to be all caps but i guess it got filtered :(