- #1

barefeet

- 59

- 2

## Homework Statement

Using

[tex]

J^2 \mid j,m_z \rangle = h^2 j(j+1) \mid j,m_z \rangle

[/tex]

[tex]

J_z \mid j,m_z \rangle = hm_z \mid j,m_z \rangle

[/tex]

Derive that :

[tex]

\langle j,m_z \mid J_-J_+ \mid j,m_z \rangle = h^2[ j(j+1) - m_z(m_z+1)]

[/tex]

## Homework Equations

[tex]

J_- = J_x - iJ_y

[/tex]

[tex]

J_+ = J_x + iJ_y

[/tex]

## The Attempt at a Solution

[tex]

J_-J_+ = (J_x- iJ_y)(J_x + iJ_y) = J_x^2 + J_y^2 = J^2 - J_z^2

[/tex]

[tex]

\langle j,m_z \mid J_-J_+ \mid j,m_z \rangle = \langle j,m_z \mid J^2 - J_z^2 \mid j,m_z \rangle = h^2[ j(j+1) - m_z^2]

[/tex]

Apparently I am missing a term here but I don't know where it should come from. I thought this should be true

[tex]

J_z^2 \mid j,m_z \rangle = J_zJ_z \mid j,m_z \rangle = h^2m_z^2 \mid j,m_z \rangle

[/tex]

(Note: h is hbar everywhere )