# What is the significance of a complex commutator?

• Darkmisc
In summary, the result of exp(A+B) = exp(A)exp(B)exp([A,B]/2) is always positive, regardless of the order of the parameters. The imaginaryness of λ and μ is important for a formal reason relating to the infinite power series of exponents. Exponentiating an operator means nothing more than the infinite power series of the exponent (Taylor series).
Darkmisc

## Homework Statement

If A and B are two operators such that
[A,B] = λ , where l is a complex number, and
if μis a second complex number, show that:
exp[μ(A + B)] = exp(μA)exp(μB)exp(- λμ^2/ 2).

## The Attempt at a Solution

I'm stuck on where to begin. I know that the result [x, p_x] = ih_bar indicates that x and p_x relate by the uncertainty principle. I'm not sure if this conclusion applies to complex commutators in general.

Also, I don't understand the significance of the exponential function. Do I lose anything by expressing that information as

μ(A+B)=(-λμ^2)/2 ?

Any ideas?

Thanks

I can't figure out why the imaginaryness of λ and μ is important (at least formally). Exponentiating an operator means nothing more than the infinite power series of the exponent (Taylor series). I think that it's actually technically defined this way. So, for example:

exp(μA) = 1 + μA + (1/2)μ2A2 + (1/6)μ3A3 ...

So, just expand each exponential, multiply the r.h.s. out, and then collect terms of the same order in μ. If you can stand it, you might work all the way up to order μ4, but you will begin to see the point even at order μ2.

CAUTION: WHEN YOU EXPAND OPERATOR EXPRESSIONS, THE COMMUTATIVE LAW FOR MULTIPLICATION MAY NOT APPLY ...

Darkmisc said:
exp[μ(A + B)] = exp(μA)exp(μB)exp(- λμ^2/ 2).

Do I lose anything by expressing that information as

μ(A+B)=(-λμ^2)/2 ?
Where did you get that?

Last edited:
oops, I messed up that expression. I meant to equate the indices.

Darkmisc said:
I meant to equate the indices.
There are operators A and B, and then there are parameters λ and μ; what indices?

Could this be done easier using BCH formula due to non-commuting? Expansion doesn't seem to get anywhere.

edit: I guess I don't mean putting it through the entire thing, just the result for exp(A +B) = exp(A)exp(B)exp([A,B]/2), but then again I'm also clueless.

sgnl03 said:
Could this be done easier using BCH formula ...
Sure, it would be MUCH easier using the BCH formula (almost trivial). I was assuming that the problem is an excercise to demonstrate a nontrivial exponentiation of operators, and if the point was to use the BCH formula, then they would have at least made the commutator more complicated (like proportional to A or B). Of course, this is just an assumption, and since the problem statement doesn't disallow it, BCH formula is probably the way to show something like this in practice.

sgnl03 said:
Expansion doesn't seem to get anywhere.
I don't know what you mean. Expansion must get somewhere, and it is a good excercise to formally match terms according to an order parameter (μ in this case.) Of course, then you must also make use of
BA = AB - [A,B]
etc.

sgnl03 said:
edit: I guess I don't mean putting it through the entire thing, just the result for exp(A +B) = exp(A)exp(B)exp([A,B]/2), but then again I'm also clueless.
This is correct. You know that you can stop there because the commutator is a scalar. In effect, that IS "the entire thing", and anyway "the entire thing" apparently doesn't have (what a would call) a closed form (at least it doesn't look very elegant in general).

Cheers, Thanks.

Last edited:

## 1. What is a complex commutator?

A complex commutator is a mathematical term used to describe the relationship between two operators in quantum mechanics. It is a measure of how much two operators do not commute, or how much their order of operations affects the final result.

## 2. How is a complex commutator calculated?

The complex commutator is calculated by taking the difference between the product of the two operators and the product of the same operators in reverse order. It is represented by the symbol [A,B] and is written as [A,B] = AB - BA.

## 3. What is the significance of a complex commutator in quantum mechanics?

In quantum mechanics, the complex commutator is a fundamental concept that helps determine the uncertainty principle, which states that certain physical quantities cannot be precisely measured at the same time. It also plays a crucial role in defining the Heisenberg uncertainty principle.

## 4. How does a complex commutator relate to observables in quantum mechanics?

In quantum mechanics, physical quantities such as position, momentum, and energy are represented by operators. The complex commutator measures the non-commutativity between these operators, which affects the precision with which these quantities can be measured.

## 5. What are some real-world applications of a complex commutator?

The complex commutator has various real-world applications, including in quantum computing, where it is used to understand the relationships between quantum gates. It is also essential in nuclear magnetic resonance spectroscopy, which is used to study the structure and dynamics of molecules. Additionally, it plays a significant role in the mathematical formulation of quantum field theory.

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