# Homework Help: Find annihilation operator

1. Nov 10, 2013

### wileecoyote

1. The problem statement, all variables and given/known data
Given the Hamiltonian $H(t) = \frac{P^2}{2m} + \frac{1}{2}mw^2X^2 + b(XP+PX)$ from some $b>0$. Find an annihilation operator $a_b$ s.t. $[a_b,a_b^{\dagger}]=1$ and $H = \hbar k (a_b^{\dagger}a_b+\frac{1}{2})$ for some constant $k$. Hint: $[P + aX,X]=[P,X], \forall a$.

2. Relevant equations
none

3. The attempt at a solution
I am not sure how to go about this problem. I played around with the commutators but can't seem to get it. Any help is appreciated thanks.

2. Nov 10, 2013

### Stealth95

I recently started self-studying Quantum Mechanics so I am not really sure for my answer, but I can find an operator that works for some cases.
I assumed that $\displaystyle{a_b=AX+BP}$ for some constants $\displaystyle{A,B \in \mathbb{C}^*}$. Then $\displaystyle{a_{b}^{\dagger}=A^{*}X+B^{*}P}$, because $\displaystyle{X}$ and $\displaystyle{P}$ are Hermitian operators.

Now we can use the fact that $\displaystyle{a_b}$ is annihilation operator and get some equations for $\displaystyle{A,B}$. I did that the obvious way and I found an operator which works, but without knowing if it is unique. Also, this operator works only if $\displaystyle{k^2+4b^2=w^2}$ (this comes for the equations). But this limits the values of $\displaystyle{b}$, because $\displaystyle{k \in \mathbb{R}\Rightarrow b\leq \frac{w}{2}}$.

Maybe another person can help us more.

3. Nov 10, 2013

### wileecoyote

What do you mean by the obvious way, I am not really sure what the obvious way to start this is.

4. Nov 10, 2013

### wileecoyote

Please someone, help. I am so stuck.

5. Nov 11, 2013

### Stealth95

If you use $\displaystyle{[a_b,a_{b}^{\dagger}]=1}$ you get an equation for $\displaystyle{A,B}$ (and their conjugates). Then you use $\displaystyle{a_{b}^{\dagger}a_b=\frac{H}{\hbar k}-\frac{1}{2}}$.

But now it's not so trivial to find the equations for $\displaystyle{A,B}$, because there are also $\displaystyle{X}$ and $\displaystyle{P}$ in the equation. By the obvious way I meant that you simply equate coefficients of the same variables. For example, if you had:
$$\displaystyle{CX^2+DP=EX^2+FP\Rightarrow \begin{Bmatrix} C=E\\ D=F \end{Bmatrix}}$$
(I am not sure if I am losing some solutions with the above method)

After doing this you have some equations for $\displaystyle{A,B}$. If you find a solution that verifies all of them then the coresponding operator certainly works. But from these equations you get also the restriction I mentioned in the previous post. So I am not sure if my method is a general way to solve the problem. That's why I asked for help from someone else.

Last edited: Nov 11, 2013
6. Nov 11, 2013

### hilbert2

I tried to solve this in the same way as Stealth95, using trial $a=AP+BX$, with $A$ and $B$ complex numbers. I got a nonlinear system of equations for the real and imaginary parts of $A$ and $B$ and I had to solve it with Mathematica. Maybe there's some other way that is easier to do by hand.

7. Nov 11, 2013

### Stealth95

Yes, the system is nonlinear. But based on the fact that the problem asks to find an annihilation operator I wasn't very strict with Maths. So I assumed that:
$$\displaystyle{B=\frac{i}{\sqrt{2mk\hbar }}}$$
and then I used one of the equations to find $\displaystyle{A}$. Note that the value I chose for $\displaystyle{B}$ is "stolen" from the annihilation operator for the harmonic oscillator potential.
Although guessing solutions is not a good way to solve systems, luckily in our case the solution I get verifies all the equations of the system so it gives an operator (only when $\displaystyle{k=\sqrt{w^2-4b^2}}$ ofcourse, as I mentioned above).