Components for the angular momentum operator L

In summary, the conversation discusses the wavefunction psi (subscript "nlm") for the stationary state of the hydrogen atom with quantum numbers n,l,m and the third component L3 for the orbital angular momentum operator L. The question is posed about the expectation value of L3 and L3^2 for this state, and the conversation explores the use of the Lz operator and its relation to the stationary state and quantum numbers. The hint suggests using spherical harmonics to solve for the orbital angular momentum operator.
  • #1
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Homework Statement



Consider wavefunction psi (subscript "nlm") describing the electron in the stationary state for the hydrogen atom with quantum numbers n,l,m and the third component L3 for the orbital angular momentum operator L. What is the expectation value of L3 and of L3^2 for the state described by psi?


Homework Equations





The Attempt at a Solution



L = sqrt(l(l+1)*hbar).

And I think L3 is the same as Lz
Lz = m*hbar
but I don't know what stationary state implies in terms of quantum numbers.
 
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  • #2
To add to my attempt at a solution:

<L3> = <Lz> = integral(-inf to inf of(psi* x L3operator x psi dz)

L3 operator = (-i x hbar) x (partial / partial x phi)
 
  • #3
stationary means that it does not depend on time, no time dependence.
 
  • #4
does that have implications on the orbital angular momentum operator?
 
  • #5
no, hint: write the solutions in terms of spherical harmonics and use the property of L_z operator on those.
 

1. What is the angular momentum operator L?

The angular momentum operator L is a mathematical operator used in quantum mechanics to describe the rotational motion of a particle or system of particles. It is represented by the symbol L and is defined as the cross product of the position vector and the momentum vector of the particle.

2. What are the components of the angular momentum operator L?

The components of the angular momentum operator L are Lx, Ly, and Lz. These correspond to the x, y, and z axes respectively and can be expressed as linear combinations of the position and momentum operators.

3. How do the components of L relate to the spin of a particle?

The spin of a particle is a type of intrinsic angular momentum and is represented by the spin operator S. The spin operator is related to the components of the angular momentum operator L through the equation L = S/ħ, where ħ is the reduced Planck's constant. This relationship allows us to calculate the spin of a particle using the components of L.

4. What is the commutation relationship between the components of L?

The components of the angular momentum operator L do not commute with each other, meaning that their order of multiplication affects the result. This is described by the commutation relationships [Lx,Ly] = iħLz, [Ly,Lz] = iħLx, and [Lz,Lx] = iħLy. These relationships are fundamental to the principles of quantum mechanics and have important implications for the measurement of angular momentum.

5. How are the components of L used in solving the Schrödinger equation?

The components of the angular momentum operator L are incorporated into the Schrödinger equation to describe the rotational motion of a particle or system of particles. In particular, the total angular momentum operator J, which is the sum of the orbital angular momentum operator L and the spin operator S, appears in the equation as an operator acting on the wave function. Solving the Schrödinger equation with the components of L allows us to determine the energy levels and wave functions of a particle in a given potential.

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