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## Homework Statement

Virtually all quantum mechanical calculations involving the harmonic oscillator can be done in terms of the creation and destruction operators and by satisfying the commutation relation [itex]\left[a,a^{\dagger}\right] = 1[/itex]

(A) Compute the similarity transformation [itex]Q\left(\lambda\right) = e^{\lambda W}Qe^{-\lambda W}[/itex]

for the operators [itex]Q = a[/itex] and [itex]Q = a^{\dagger}[/itex] and

- [itex]W = a[/itex]
- [itex]W = a^{\dagger}a[/itex]

- [itex]W = a^{\dagger}a^{\dagger}[/itex]

(B) For a transformation generated by [itex]W = a^{\dagger}a^{\dagger} - aa[/itex],

show that the transformed variables are [tex]a\left(\lambda\right) = a\cosh\left(2\lambda\right) + a^{\dagger}\sinh\left(2\lambda\right)\\

a^{\dagger}\left(\lambda\right) = a^{\dagger}\cosh\left(2\lambda\right) + a\sinh\left(2\lambda\right)[/tex]

## Homework Equations

In addition to the ones given, I believe two equations may be relevant. First, describing what a similarity transformation is. Second, how an exponentiated operator can be expanded in terms of a Taylor series.

- Similarity Transformation: Given a matrix [itex]M[/itex] with eigenvalue [itex]m[/itex] and eigenvector [itex]v[/itex], the following is true. [tex]Mv = mv[/tex] A similarity transformation does the same thing, but instead of focusing on the [itex]v[/itex] part, it focuses on the [itex]m[/itex] part. So given another matrix [itex]K = P^{-1}MP[/itex]. [itex]K[/itex] is consider similar to [itex]M[/itex] if an invertible matrix [itex]P[/itex] can be found. Then, [tex]Ku = P^{-1}MPu[/tex] and [itex]u[/itex] is desired to be such that [tex]v = Pu[/tex] This would mean that \begin{align*}Ku &= P^{-1}M\underbrace{Pu}_{v}\\Ku &= P^{-1}Mv\\Ku &= P^{-1}mv\\Ku &= mP^{-1}\underbrace{v}_{Pu}\\Ku &= mP^{-1}\left(Pu\right)\\Ku &= m\mathbb{I}u\\Ku &= mu\end{align*}
- So, finding a similarity transformation involves finding the invertible matrix [itex]P[/itex].

- Taylor expansion of an exponentiated operator [itex]M[/itex] [tex]e^{M} = \mathbb{I} + M + \frac{1}{2}M^2 + \frac{1}{6}M^3 + \cdots[/tex]

## The Attempt at a Solution

In addition to the attempt in the relevant equations section, I think the definitions of the ladder operators are useful, but, I want to make sure I have the bigger picture cleared up.

What is the strategy I need to use? I am to find a similarity transformation, first, and central to each of the six calculations is finding some invertible matrix [itex]P[/itex]. Maybe my linear algebra is a bit weak, but I am not sure how I am supposed to find such a matrix. It's a bit abstract and I would appreciate if someone could help me connect the dots.

I'd like to start there with (A) and then end up tackling (B) afterwards. Thanks!