Eigenvectors eigenvalues and constant of motion

[umagongdi]

Homework Statement

a.) The motion of a particle in the 3-dimensional space is described by the Hamiltonian H = H_x+H_y+H_z, where

H_x=1/2*(p_x²+x²), H_y=1/2*(p_y²+y²), H_z=1/2*(p_z²+z²)

Check that the standard angular momentum operators L_x, is a constant of motion.

b.) By knowing that the ground state wavefunction for H_x is proportional to e^-x2/2, write the wavefunction Y₀(x,y,z) representing the ground state for H (you are not required to fix the normaliszation of the wavefunctions in this problem).

Homework Equations

The Attempt at a Solution

a.) Do you need to check if L and H commute?
b.) I really don't have a clue any tips?

 

[G01]

Homework Statement

a.) The motion of a particle in the 3-dimensional space is described by the Hamiltonian H = H_x+H_y+H_z, where

H_x=1/2*(p_x²+x²), H_y=1/2*(p_y²+y²), H_z=1/2*(p_z²+z²)

Check that the standard angular momentum operators L_x, is a constant of motion.

b.) By knowing that the ground state wavefunction for H_x is proportional to e^-x2/2, write the wavefunction Y₀(x,y,z) representing the ground state for H (you are not required to fix the normaliszation of the wavefunctions in this problem).

Homework Equations

The Attempt at a Solution

a.) Do you need to check if L and H commute?
b.) I really don't have a clue any tips?

a.) Do you need to check if L and H commute?

Can you relate that commutator to the equations of motions. HINT: Look up the Heisenberg equation of motion.

b.) I really don't have a clue any tips?

Assume the wave function can be separated in its variables.

 

[umagongdi]

Assume the wave function can be separated in its variables.

Oh i think i get it now thanks. You can just separate the wave function like this?

Y₀(x,y,z)=e^-x2/2+e^-y2/2+e^-z2/2