Similarity Transformation Involving Operators

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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]

λ is an arbitrary complex number. There are six [itex]Q\left(\lambda\right)[/itex]s to compute.

(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!

 

[blue_leaf77]

There is one relation which can help you with the similarity transform if the transform is also unitary like the one you are dealing with, this relation reads
$$
e^ABe^{-A} = B + [A,B] + \frac{1}{2!}[A,[A,B]] + \frac{1}{3!}[A,[A,[A,B]]] + \ldots
$$

 

[loginorsinup]

Hey, thanks for that. I knew I was on the right track with that. So I just have to compute a ton of commutators, basically for part A. At what point do I get to truncate this power series relation you gave me (which is very similar to the one I had presented earlier)?

 

[blue_leaf77]

Hey, thanks for that. I knew I was on the right track with that. So I just have to compute a ton of commutators, basically for part A. At what point do I get to truncate this power series relation you gave me (which is very similar to the one I had presented earlier)?

For now, let's see what do you get after calculating ## e^{\lambda a}ae^{-\lambda a}## and ## e^{\lambda a}a^\dagger e^{-\lambda a}## using the above formula? Find out the first two terms of the expansion, ##B## and ##[A,B]##.

 

[loginorsinup]

For now, let's see what do you get after calculating ## e^{\lambda a}ae^{-\lambda a}## and ## e^{\lambda a}a^\dagger e^{-\lambda a}## using the above formula? Find out the first two terms of the expansion, ##B## and ##[A,B]##.

So, ##e^{\lambda a}ae^{-\lambda a}## would require me to compute a few commutation relations.

• ##\left[\lambda a,a\right]## \begin{align*}\left[\lambda a,a\right] &= \lambda aa - a\lambda a\\ &= \lambda aa - \lambda aa\\&= 0\end{align*}
  • Any commutation relation that depends on this commutation relation, would also be 0 e.g. ##\left[\lambda a, \left[\lambda a, a\right]\right] = 0##
• This means that the expansion would simply be the first term. [tex]e^{\lambda a}ae^{-\lambda a} = a[/tex] since all the other terms would die off.

And, ##e^{\lambda a}a^{\dagger}e^{-\lambda a}## would lead me to the observations below.

• ##\left[\lambda a,a^{\dagger}\right]## \begin{align*}\left[\lambda a,a^{\dagger}\right] &= \lambda aa^{\dagger} - a^{\dagger}\lambda a\\ &= \lambda aa^{\dagger} - \lambda a^{\dagger}a\\&= \lambda\left(aa^{\dagger} - a^{\dagger}a\right)\\&= \lambda\left[a,a^{\dagger}\right]\\&= \lambda\left(1\right)\\&= \lambda\end{align*}
• I would then have to compute ##\left[\lambda a,\left[\lambda a,a^{\dagger}\right]\right]## \begin{align*}\left[\lambda a,\left[\lambda a,a^{\dagger}\right]\right] &= \left[\lambda a,\lambda\right]\\&= \lambda a\lambda - \lambda\lambda a\\&= \lambda^2 a - \lambda^2 a\\&= 0\end{align*}
  • Any commutation relation that depends on this commutation relation, would also be 0 e.g. ##\left[\lambda a,\left[\lambda a, \left[\lambda a, a\right]\right]\right] = 0##
• This means that the expansion would simply be the first and second terms only. [tex]e^{\lambda a}ae^{-\lambda a} = a + \lambda[/tex] since all the other terms would die off.

I think I did all the math correctly, but I honestly have no clue what this result actually means or implies. It does answer (A) (i) in part at least.

 

[blue_leaf77]

This means that the expansion would simply be the first term.

eλaae−λa=aeλaae−λa=a​

e^{\lambda a}ae^{-\lambda a} = a since all the other terms would die off.

Yes.

This means that the expansion would simply be the first and second terms only.

eλaae−λa=a+λeλaae−λa=a+λ​

e^{\lambda a}ae^{-\lambda a} = a + \lambda since all the other terms would die off.

Yes.

EDIT: In post #2, I shouldn't have said "if the transform is also unitary" because that relation can also hold if the transform is similar but not unitary, as is the case in the current problem.

 

[loginorsinup]

Hey, thanks for that. You helped me understand the mechanics of (A), but I am very much stuck on (B). I don't even know where to begin other than to compute ##e^{\lambda W}Qe^{-\lambda W}## for ##W = a^{\dagger}a^{\dagger} - aa##, but for what ##Q##? What does any of this stuff even physically mean? What is the value in computing all of this? I'm not trying to be flippant. I just genuinely want to understand the value of such computations, and it's possible that my linear algebra intuition is not as good as it needs to be to understand this without this, sadly, considerable amount of hand-holding. I really am trying!

 

[blue_leaf77]

I don't even know where to begin other than to compute eλWQe−λWeλWQe−λWe^{\lambda W}Qe^{-\lambda W} for W=a†a†−aaW=a†a†−aaW = a^{\dagger}a^{\dagger} - aa, but for what QQQ?

Are not ##a## and ##a^\dagger##? (B) asks you to calculate ##a(\lambda)## and ##a^\dagger(\lambda)##, compare them with how ##Q(\lambda)## is defined.

What does any of this stuff even physically mean? What is the value in computing all of this?

In an exponential operator ##e^{i \lambda \hat{G}}## where ##\lambda## is a real parameter and ##\hat{G}## is Hermitian, it represents a symmetry transformation. To give a real example, the operator ##e^{i \mathbf{a} \cdot \mathbf{p}/\hbar}## corresponds to a translation of the system along the direction specified by the vector ##\mathbf{a}##, another symmetry transformation includes ##e^{i \phi L_x/\hbar}## which physically means rotating the system around the ##x## axis by an angle ##\phi##.
In your problem however, ##\lambda## is unknown whether real or complex and all the ##W## considered are not Hermitian (except the one in (B) which is anti-Hermitian though). Hence, it cannot be interpreted as a symmetry transformation corresponding to a physically meaningful transformation in space and time. However, there is the so-called coherent state and squeezed state, typically relevant in the discussion of harmonic oscillator. In particular, the so-called squeezed state is a state resulting from the action of the squeezing operator ##S(\xi) = \exp( \frac{1}{2}(\xi^*a^2 - \xi (a^\dagger)^2 ) )## on the ground state ##|0\rangle##. It has been found that in order to calculate the expectation value of either ##x## or ##p## operators, it can be helpful to simplify ##S^\dagger (\xi) a S(\xi)## and ##S^\dagger (\xi) a^\dagger S(\xi)##.

 

[loginorsinup]

Are not ##a## and ##a^\dagger##? (B) asks you to calculate ##a(\lambda)## and ##a^\dagger(\lambda)##, compare them with how ##Q(\lambda)## is defined.

In an exponential operator ##e^{i \lambda \hat{G}}## where ##\lambda## is a real parameter and ##\hat{G}## is Hermitian, it represents a symmetry transformation. To give a real example, the operator ##e^{i \mathbf{a} \cdot \mathbf{p}/\hbar}## corresponds to a translation of the system along the direction specified by the vector ##\mathbf{a}##, another symmetry transformation includes ##e^{i \phi L_x/\hbar}## which physically means rotating the system around the ##x## axis by an angle ##\phi##.
In your problem however, ##\lambda## is unknown whether real or complex and all the ##W## considered are not Hermitian (except the one in (B) which is anti-Hermitian though). Hence, it cannot be interpreted as a symmetry transformation corresponding to a physically meaningful transformation in space and time. However, there is the so-called coherent state and squeezed state, typically relevant in the discussion of harmonic oscillator. In particular, the so-called squeezed state is a state resulting from the action of the squeezing operator ##S(\xi) = \exp( \frac{1}{2}(\xi^*a^2 - \xi (a^\dagger)^2 ) )## on the ground state ##|0\rangle##. It has been found that in order to calculate the expectation value of either ##x## or ##p## operators, it can be helpful to simplify ##S^\dagger (\xi) a S(\xi)## and ##S^\dagger (\xi) a^\dagger S(\xi)##.

Hi, thanks for the response. I'm not sure I follow all of it. I should mention that I come from an engineering background and never took formal physics courses aside from the introductory series before taking this quantum course. I am not really sure what a symmetry transformation is, but from what I can find online, it can mean simply translating or rotating. I don't quite understand why an exponential version of the position operator is the translation operator, but I think I'll leave such things for a different thread. Sorry for the diversion. I'm stuck on (B).

I feel like I'm just grinding through math and not really understanding why I am doing what I am doing. What is the point of (B)? If I take ##Q## to be either ##a## or ##a^{\dagger}##, where on Earth do I get a hyperbolic cosine term? So far, everything I've been doing in this problem (finished (A), now on (B)) has involved merely operators. How do the ladder operators end up depending on this parameter ##\lambda##?

 

[loginorsinup]

I can't edit, but I meant exponential version of the momentum operator.

 

[blue_leaf77]

EDIT to post #8:
1) There is one ##W## which is Hermitian, ##W=a^\dagger a##.
2) Judging from the right hand sides of the relations given in B), ##\lambda## seems to be real (you can prove this parallelly when doing B)).
3) ##S(\xi)## is a unitary operator since ##(S(\xi))^\dagger = (S(\xi))^{-1}##. Thus, along with 2) I think it can be associated to some symmetry transformation. To know its kinematical effect, you can try calculating how it transforms the position and momentum operators: ##S^\dagger (\xi) x S(\xi)## and ##S^\dagger (\xi) p S(\xi)##. These can be calculated from the relations given in B).

I feel like I'm just grinding through math and not really understanding why I am doing what I am doing. What is the point of (B)? If I take QQQ to be either aaa or a†a†a^{\dagger}, where on Earth do I get a hyperbolic cosine term?

The hyperbolic functions result from its power expansion which results from the commutation expansion in post #2. For a starter, calculate ##[W,a]## and ##[W,a^\dagger]## where ##W = (a^\dagger)^2-a^2##. As for the commutation expansion, it's actually not too difficult to calculate if you had recognized a certain pattern resulting from the chain of commutation ##[W,[W,[...,[W[W,a]]...]]]##.

 

[loginorsinup]

Hi, I really tried to see a pattern that led to the inclusion of hyperbolic trigonometric functions, but I am stuck.

I calculated ##[\lambda W,a]## and ##[\lambda W,a^{\dagger}]## for ##W = a^{\dagger}a^{\dagger} - aa## and got the following.

For ##[\lambda W,a]##
[tex]\tilde{a}+\sum_{k=1}^{\infty}\left(-1\right)^{k}\left(2\right)^{k}\lambda^{k}\tilde{a}^{\dagger^{k}}[/tex]

For ##[\lambda W,a^{\dagger}]##
[tex]\tilde{a^{\dagger}}+\sum_{k=1}^{\infty}\left(-1\right)^{k-1}\left(2\right)^{k}\lambda^{k}\tilde{a}^{\dagger^{k-1}}
[/tex]

The Taylor series expansion of the cosh and sinh functions don't look like that, as far as I can tell.

http://planetmath.org/taylorseriesofhyperbolicfunctions

What am I missing? :(

 

[blue_leaf77]

Hi, I really tried to see a pattern that led to the inclusion of hyperbolic trigonometric functions, but I am stuck.

I calculated ##[\lambda W,a]## and ##[\lambda W,a^{\dagger}]## for ##W = a^{\dagger}a^{\dagger} - aa## and got the following.

For ##[\lambda W,a]##
[tex]\tilde{a}+\sum_{k=1}^{\infty}\left(-1\right)^{k}\left(2\right)^{k}\lambda^{k}\tilde{a}^{\dagger^{k}}[/tex]

For ##[\lambda W,a^{\dagger}]##
[tex]\tilde{a^{\dagger}}+\sum_{k=1}^{\infty}\left(-1\right)^{k-1}\left(2\right)^{k}\lambda^{k}\tilde{a}^{\dagger^{k-1}}
[/tex]

The Taylor series expansion of the cosh and sinh functions don't look like that, as far as I can tell.

http://planetmath.org/taylorseriesofhyperbolicfunctions

What am I missing? :(

##\lambda## is not an operator, you can take it out of the commutator.
Ok let's try doing ##\lambda[a^\dagger a^\dagger - aa , a]##, from now I will omit ##\lambda##
$$
[a^\dagger a^\dagger - aa , a] = [a^\dagger a^\dagger , a] - [aa , a]
$$
The second term obviously vanishes. So, we can only focus on the first one
$$
[a^\dagger a^\dagger , a] = a^\dagger (a^\dagger a) - a a^\dagger a^\dagger
$$
Replace the term inside the bracket using the identity ##[a,a^\dagger]=1## to obtain
$$
a^\dagger (a a^\dagger -1) - a a^\dagger a^\dagger
$$
Let's see how you can simply the last equation.

 

[loginorsinup]

From
\begin{align*}
[a^{\dagger}a^{\dagger},a] &= a^{\dagger}(a^{\dagger}a) - aa^{\dagger}a^{\dagger}\\
&= a^{\dagger}a^{\dagger}a + 0 - aa^{\dagger}a^{\dagger}\\
&= a^{\dagger}a^{\dagger}a - a^{\dagger}aa^{\dagger} + a^{\dagger}aa^{\dagger} - aa^{\dagger}a^{\dagger}\\
&= a^{\dagger}(a^{\dagger}a-aa^{\dagger})+(a^{\dagger}a-aa^{\dagger})a^{\dagger}\\
&= a^{\dagger}[a^{\dagger},a]+[a^{\dagger},a]a^{\dagger}\\
&= a^{\dagger}(-1) + (-1)a^{\dagger}\\
&= -2a^{\dagger}\end{align*}

After that, I'm not sure what you did, sorry.

 

[blue_leaf77]

Right! Now do the same for ##[W,a^\dagger]##.

 

[loginorsinup]

Right! Now do the same for ##[W,a^\dagger]##.

\begin{align*}
[a^{\dagger}a^{\dagger}-aa,a^{\dagger}] &= [a^{\dagger}a^{\dagger},a^{\dagger}]-[aa,a^{\dagger}]\\
&= (a^{\dagger}a^{\dagger}a^{\dagger}-a^{\dagger}a^{\dagger}a^{\dagger})-(aaa^{\dagger}-a^{\dagger}aa)\\
&= -aaa^{\dagger}+a^{\dagger}aa\\
&= a^{\dagger}aa-aaa^{\dagger}\\
&= a^{\dagger}aa+0-aaa^{\dagger}\\
&= a^{\dagger}aa-aa^{\dagger}a+aa^{\dagger}a-aaa^{\dagger}\\
&= (a^{\dagger}a-aa^{\dagger})a+a(a^{\dagger}a-aa^{\dagger})\\
&= [a^{\dagger},a]a+a[a^{\dagger},a]\\
&= (-1)a+a(-1)\\
&= -2a
\end{align*}

After trying to see how this can be "expanded," I've largely given up. I just don't see how this leads me to hyperbolic trigonometric functions. Any tips? Sorry for my slow progress. I haven't had enough time lately. Thanks again.

 

[blue_leaf77]

Now that you have ##[W,a] = -2a^\dagger## and ##[W,a^\dagger] = -2a##, you should be able to calculate all five terms shown in the series below
$$
e^{\lambda W} a e^{-\lambda W} = a + \lambda[W,a] + \frac{\lambda^2}{2!}[W,[W,a]] + \frac{\lambda^3}{3!}[W,[W,[W,a]]] + \frac{\lambda^4}{4!}[W,[W,[W,[W,a]]]] + \ldots
$$
Do it.

 

[loginorsinup]

Now that you have ##[W,a] = -2a^\dagger## and ##[W,a^\dagger] = -2a##, you should be able to calculate all five terms shown in the series below
$$
e^{\lambda W} a e^{-\lambda W} = a + \lambda[W,a] + \frac{\lambda^2}{2!}[W,[W,a]] + \frac{\lambda^3}{3!}[W,[W,[W,a]]] + \frac{\lambda^4}{4!}[W,[W,[W,[W,a]]]] + \ldots
$$
Do it.

\begin{align*}
e^{\lambda W} a e^{-\lambda W} &= a + \lambda[W,a] + \frac{\lambda^2}{2!}[W,[W,a]] + \frac{\lambda^3}{3!}[W,[W,[W,a]]] + \frac{\lambda^4}{4!}[W,[W,[W,[W,a]]]] + \ldots\\
[W,a] &= -2a^{\dagger}\\
[W,a^{\dagger}] &= -2a\\
[W,[W,a]] &= [W,-2a^{\dagger}]\\
&= W(-2a^{\dagger}) - (-2a^{\dagger})W\\
&= -2( Wa^{\dagger} - a^{\dagger}W )\\
&= -2[W,a^{\dagger}]\\
&= -2(-2a)\\
&= 4a\\
[W,[W,[W,a]]] &= [W,4a]\\
&= W(4a) - (4a)W\\
&= 4( Wa - aW )\\
&= 4[W,a]\\
&= 4(-2a^{\dagger})\\
&= -8a^{\dagger}
\end{align*}

This matches what I had done on my own, too. The pattern is that the sign flips, the magnitude doubles, and an additional dagger is incurred for each nested commutator computation (and note that the dagger of a dagger returns itself).

So, I can say

\begin{align*}
e^{\lambda W} a e^{-\lambda W} &= a + \lambda(-2a^{\dagger}) + \frac{\lambda^2}{2!}(4a) + \frac{\lambda^3}{3!}(-8a^{\dagger}) + \ldots\\
e^{\lambda W} a e^{-\lambda W} &= a - 2\lambda a^{\dagger} + 2\lambda^2 a - \frac{4\lambda^3}{3}a^{\dagger} + \ldots\\
\end{align*}

But, it's at this stage that I get stumped. Any pointers on how to link this up to cosh? Thanks again!

 

[blue_leaf77]

$$
\begin{align*}
e^{\lambda W} a e^{-\lambda W} &= a + \lambda(-2a^{\dagger}) + \frac{\lambda^2}{2!}(4a) + \frac{\lambda^3}{3!}(-8a^{\dagger}) + \ldots\\
e^{\lambda W} a e^{-\lambda W} &= a - 2\lambda a^{\dagger} + 2\lambda^2 a - \frac{4\lambda^3}{3}a^{\dagger} + \ldots\\
\end{align*}
$$

Just stay with the first form. Let the numbers to be in power form, namely
$$
e^{\lambda W} a e^{-\lambda W} = a -2\lambda a^{\dagger} + \frac{(2\lambda)^2}{2!}a - \frac{(2\lambda)^3}{3!}a^{\dagger} + \frac{(2\lambda)^4}{4!}a+\ldots\\
$$
At this point, it should be very easy to predict how the pattern will determine the higher order terms. Next separate the terms containing ##a## and those containing ##a^\dagger## and compare with the Taylor series of sinh and cosh. If you don't know their expansion, look it up online.

 

[loginorsinup]

Just stay with the first form. Let the numbers to be in power form, namely
$$
e^{\lambda W} a e^{-\lambda W} = a -2\lambda a^{\dagger} + \frac{(2\lambda)^2}{2!}a - \frac{(2\lambda)^3}{3!}a^{\dagger} + \frac{(2\lambda)^4}{4!}a+\ldots\\
$$
At this point, it should be very easy to predict how the pattern will determine the higher order terms. Next separate the terms containing ##a## and those containing ##a^\dagger## and compare with the Taylor series of sinh and cosh. If you don't know their expansion, look it up online.

I am pretty sure that is enough to solve what I have been missing. Thank you very much. I wish I could have seen that as I was solving it. I felt like I was in the dark when I first did it alone, prior to asking here. How did you know how to solve these? Does it come from experience?

 

[blue_leaf77]

I had once solved the same problem in my previous class.

 

[loginorsinup]

I had once solved the same problem in my previous class.

Hi. Thanks. Interesting that you also had the same problem!

I was able to go through the calculus/algebra to show (and derive) how to work the cosh and sinh expansions, but, I ended up with

$$a(\lambda) = a\cosh(2\lambda) -a^{\dagger}\sinh(2\lambda)$$

Just a quick look at the last post where we agreed that

$$e^{λW}ae^{-λW}=a−2λa^{\dagger}+\frac{(2\lambda)^2}{2!}−\frac{(2\lambda)^3}{3!}a^{\dagger}+\frac{(2\lambda)^4}{4!}a-\frac{(2\lambda)^5}{5!}a^{\dagger}+\cdots$$

All the odd powers have a leading negative sign and the sinh Taylor series expansion contains only the odd powers. So, it seems to me that the problem statement is incorrect (off by just the one negative sign on the sinh) unless I'm missing something?

 

[blue_leaf77]

it seems to me that the problem statement is incorrect (off by just the one negative sign on the sinh) unless I'm missing something?

I think the problem statement is incorrect. I have done a cross check with old note and it should be like you did.