# Ladder Operators: Commutation Relation & Beyond

• Gbox
In summary, the conversation discusses the properties of the L+ and L- operators, which are equal to each other. It also mentions the commutator [L+, L-], which simplifies to 2(iL_xL_y+iL_yL_x+L_y^2). The conversation acknowledges that L_yL_x is not equal to L_xL_y, but notes that L_xL_y - L_yL_x is equal to 2(iL_xL_y+iL_yL_x+L_y^2).
Gbox
Homework Statement
Its is known that: ##L^2=L_z^2+L_{-}L_{+}-L_z##
##L_{+}=L_x+iL_y##
##L_{-}=L_x-iL_y##

a. what is ##L_{+}^{\dagger}##
b. what is ##[L_{+},L_{-}]##
c. what is ##||L_{+}|l,m>||^2 ##
d. assuming all coefficients are integer and positive what is ## L_{+}|l,m>##
Relevant Equations
##L^2=L_x^2+L_y^2+L_z^2##
a. ##L_{+}^{\dagger}=(L_x+iL_y)^{\dagger}=L_x-iL_y=L_{-}##

b.##[L_{+},L_{-}]=[L_x+iL_y,L_x-iL_y]=(L_x+iL_y)(L_x-iL_y)-(L_x-iL_y)(L_x+iL_y)=##
##=L_x^2-iL_xL_y+iL_yL_x+L_y^2-(L_x^2+iL_xL_y-iL_yL_x-L_y^2)##
##=L_x^2-iL_xL_y+iL_yL_x+L_y^2-L_x^2-iL_xL_y+iL_yL_x+L_y^2##
##=-iL_xL_y+iL_yL_x+L_y^2-iL_xL_y+iL_yL_x+L_y^2##
##=-2iL_xL_y+2iL_yL_x+2L_y^2=2(iL_xL_y+iL_yL_x+L_y^2)##

It is not ture that ##L_yL_x=L_xl_y## right? What can be done next?

Gbox said:
It is not ture that ##L_yL_x=L_xl_y## right?

Right. But ##L_x L_y - L_y L_x =## ?

Also, be careful with the +- signs. The ##L_y^2## terms should cancel.

Gbox

## 1. What are ladder operators?

Ladder operators are mathematical operators used in quantum mechanics to describe the behavior of particles. They are used to raise or lower the energy states of a system.

## 2. What is the commutation relation for ladder operators?

The commutation relation for ladder operators is a mathematical expression that describes how two operators interact with each other. In the case of ladder operators, the commutation relation is [A, A†] = 1, where A and A† are the lowering and raising operators, respectively.

## 3. How are ladder operators used in quantum mechanics?

Ladder operators are used in quantum mechanics to calculate the energy levels and transition probabilities of a system. They are also used to describe the creation and annihilation of particles.

## 4. What is the significance of the commutation relation for ladder operators?

The commutation relation for ladder operators is significant because it allows us to determine the energy levels of a system and the probabilities of transitions between those levels. It also helps us understand the behavior of particles in quantum systems.

## 5. What are some applications of ladder operators?

Ladder operators have many applications in quantum mechanics, including in the study of atomic and molecular spectra, the behavior of electrons in a magnetic field, and the properties of quantum systems such as the harmonic oscillator. They are also used in the development of quantum algorithms for quantum computing.

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