Relationship between several operators and their eigenvectors.

[aop12]

Homework Statement

operators: K=LM and [L,M]=1

α is an eigenvector of K with eigenvalue λ.

Show that x=Lα and y=Mα are also eigenvectors of K and also find their eigenvalues.

Homework Equations

K=LM
[L,M]=1
Kα=λα

The Attempt at a Solution

I tried, but its not even worth putting up here.

 

[mfb]

You have to show that x eigenvector of K which means Kx=λ'x for some λ'.

Kx=KLα = LMLα
Try to modify that equation (using [L,M]=1) until you can use Kα=λα and simplify.

 

[aop12]

So, am I doing this right?
Starting with: Kx=λ'x where x=Lα

Kx=KLα=LMLα ===> using K=LM

=L(LM-1)α=LLMα-Lα ===>using [L,M]=1

=LKα-Lα
Now going back to Kx=λ'x
LKα-Lα=λ'Lα

LKα=(λ'+1)Lα

Finally,

Kα=L^-1(λ'+1)Lα

since λ'+1 is a constant,

Kα=(λ'+1)L^-1Lα
Kα=(λ'+1)α

so λ'=λ-1.

is that correct?

 

[grzz]

So, am I doing this right?
Starting with: Kx=λ'x where x=Lα

Kx=KLα=LMLα ===> using K=LM

=L(LM-1)α=LLMα-Lα ===>using [L,M]=1

=LKα-Lα
...

Good start! And correct finish!

But I would have continued like this:

= Lλα - Lα since Kα = λα.

= (λ - 1)Lα since λ commutes with the operator L.

Hence K(Lα) = (λ - 1)Lα, showing that Lα is an eigenvector of operator K with eigenvalue (λ - 1).