Ladder Operators: Commutation Relation & Beyond

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).
  • #1
Gbox
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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?
 
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  • #2
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.
 
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Related to Ladder Operators: Commutation Relation & Beyond

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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