# QM having difficulty on proofs of operators

• kel
In summary, Kel is having trouble proving the following for arbitrary operators A,B and C: i- [A,c1B+c2C] = c1[A,B] + c2[A,C], ii-[A,BC] = [AB]C + B[A,C], iii-[A,[B,C]] + [B,[C,A]] + [C,[A,B]] = 0. Kel is able to prove the first one, but needs help with the second and third. Daniel points out that the third double commutator in the third equation should be [C,[A,B]].
kel
I know this is a simple part of Quantum Mechanics, but I seem to be having trouble with it, I'm not sure if my math is just wrong or if I'm applying it wrong.

The questions that I have are:

Prove the following for arbitrary operators A,B and C:
(hint-no tricks, just write them out in full)

i- [A,c1B+c2C] = c1[A,B] + c2[A,C]

So far I've got

A[c1B+c2C] - [c1B+c2C]A = (Ac1B+Ac2C) - (c1BA + c2CA)

giving

Ac1B+Ac2C - c1BA + c2CA = c1[AB] + c2[AC] - c1[BA] + c2[CA]

but I don't know what to do from here - something should cancel, but I think my workings may be wrong.

ii-[A,BC] = [AB]C + B[A,C]

iii-[A,[B,C]] + [B,[C,A]] + [C,[B,A]] = 0

I may be able to do this one based on number ii above, but I need some help on that one first please.

Cheers
Kel

Ac1B+Ac2C - c1BA + c2CA

Nothing cancels, just rearrange the terms. Btw, the second 'plus' should actually be a 'minus'.

ok, I've got the first one.

i- [A,c1B+c2C] = c1[A,B] + c2[A,C]
= (Ac1B+Ac2C) - (c1BA + c2CA)
= Ac1B + Ac2C - c1BA - c2CA
= c1AB - c1BA + c2AC - c2CA
= c1[AB-BA] + c2[AC-CA]
= c1[A,B] + c2[A,C]

but how, do I go about the second one??
[A,BC] = [AB]C + B[A,C]
is it?
[A,BC] = A[BC] - [BC]A

and if so (or not) where do I go from here??

Kel

$$[A,BC]=ABC-BCA=ABC-BAC+BAC-BCA = \ ... \$$

Daniel.

iii is written incorrectly. The third double commutator should be [C,[A,B]].

Daniel.

## 1. What is QM and why is it having difficulty with proofs of operators?

QM stands for quantum mechanics, which is a branch of physics that deals with the behavior of particles at a subatomic level. It is having difficulty with proofs of operators because these proofs involve complex mathematical equations and concepts that are difficult to fully comprehend and prove.

## 2. What are operators in QM and why are they important?

Operators in QM are mathematical representations of physical quantities, such as position, momentum, and energy. They are important because they allow us to make predictions and calculations about the behavior of particles in quantum systems.

## 3. What are some common difficulties that arise when proving operators in QM?

Some common difficulties include understanding and manipulating complex mathematical equations, dealing with uncertain and probabilistic nature of quantum systems, and reconciling classical and quantum theories.

## 4. Are there any strategies or techniques that can help in proving operators in QM?

Yes, there are various strategies and techniques that can help in proving operators in QM. These include using mathematical tools such as linear algebra and calculus, understanding the principles of quantum mechanics, and practicing problem-solving techniques.

## 5. How can one improve their understanding and proficiency in proving operators in QM?

Improving understanding and proficiency in proving operators in QM takes time and dedication. Some ways to improve include studying and practicing regularly, seeking help from experts or peers, and staying updated with new developments in the field.

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