- #1

- 394

- 87

- Homework Statement
- ##\left [ L_x , L_y \right ] \neq 0##, then do they have a same eigenstate?

- Relevant Equations
- None

Certainly, ##\left [ A ,B \right ] \neq 0## does not mean that they do not have a same eigenstate.

But how to construct a same eigenstate for ##L_x## and ##L_y## if it exists?

Since ##L_x Y_l^m = \frac \hbar 2 \left ( \sqrt { l \left ( l+1 \right ) -m \left ( m+1 \right )} Y_l^{m+1} + \sqrt {l \left ( l+1 \right ) -m \left ( m-1 \right ) } Y_l^{m-1} \right )##,

and ##L_y Y_l^m = \frac \hbar {2i} \left ( \sqrt { l \left ( l+1 \right ) -m \left ( m+1 \right )} Y_l^{m+1} - \sqrt {l \left ( l+1 \right ) -m \left ( m-1 \right ) } Y_l^{m-1} \right )##, I have troubles when constructing the eigenstate with ##Y_l^m##.

At the same time, I am wondering if the following statements are true.

if ##\left [ A ,B \right ] \neq 0## but ## \left < \left [ A ,B \right ] \right > =0##, then they have same eigenstates;

if ##\left [ A ,B \right ] \neq 0## and ## \left < \left [ A ,B \right ] \right > \neq 0##, then they do not have same eigenstates.

If these statements are true, then ##L_x## and ##L_y## do not have same eigenstates since ## \left < \left [ L_x ,L_y \right ] \right > = i {\hbar}^2 m \neq 0##.

Thank you for your time.

But how to construct a same eigenstate for ##L_x## and ##L_y## if it exists?

Since ##L_x Y_l^m = \frac \hbar 2 \left ( \sqrt { l \left ( l+1 \right ) -m \left ( m+1 \right )} Y_l^{m+1} + \sqrt {l \left ( l+1 \right ) -m \left ( m-1 \right ) } Y_l^{m-1} \right )##,

and ##L_y Y_l^m = \frac \hbar {2i} \left ( \sqrt { l \left ( l+1 \right ) -m \left ( m+1 \right )} Y_l^{m+1} - \sqrt {l \left ( l+1 \right ) -m \left ( m-1 \right ) } Y_l^{m-1} \right )##, I have troubles when constructing the eigenstate with ##Y_l^m##.

At the same time, I am wondering if the following statements are true.

if ##\left [ A ,B \right ] \neq 0## but ## \left < \left [ A ,B \right ] \right > =0##, then they have same eigenstates;

if ##\left [ A ,B \right ] \neq 0## and ## \left < \left [ A ,B \right ] \right > \neq 0##, then they do not have same eigenstates.

If these statements are true, then ##L_x## and ##L_y## do not have same eigenstates since ## \left < \left [ L_x ,L_y \right ] \right > = i {\hbar}^2 m \neq 0##.

Thank you for your time.