Particle in a cylindrically symmetric potential (Quantum mechanics)

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The discussion revolves around solving a Hamiltonian operator in a cylindrically symmetric potential, focusing on the commutation with angular momentum and momentum operators. The participants analyze the eigenvalue equations for the angular components, emphasizing that the wave function must be continuous, leading to the conclusion that the azimuthal quantum number must be an integer. The challenge lies in determining the conditions for the wave function in the z-direction, where no boundary conditions restrict the value of k. The Hamiltonian is expressed as a sum of components, allowing for independent treatment of the radial and axial motions, ultimately clarifying the energy contributions from both parts. This understanding enables the completion of the exercise related to the quantum system.
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Homework Statement
Let ##\rho , \phi , z## be the cylindrical coordinates of a spinless particle (##x=\rho \times \cos(\phi) ,y=\rho \times \sin(\phi) ; \rho \geq 0 , 0 \leq \phi \lt 2\pi##. Assume that the potential energy of this particle depends only on ##\rho##, and not on ##\phi## and z.
a. Write, in cylindrical coordinates, the differential operator associated with the Hamiltonian.Show that H commutes with Lz, and Pz.Show from this that the wave functions associated with the stationary states of the particle can be chosen in the form:
##\phi_{n,m,k}(\rho,\phi,z)=f_{n,m}(p) e^{im\phi} e^{ikz}##
Relevant Equations
See below
Hi, everyone.
Please check the following questions (extracted of the cohen Tanpoudji)
jhh.png

for the first question, here my Hamiltonian operator.

jhh.png

It's easy to see that it commutes with Lz and Pz.
Now we can determine a common eigenvector basis for these 3 operators.
For the angular part we need to solve
Lzg(théta,phi)=m##\hbar##g(théta,phi)(1)
Pzg(théta,phi)=k##\hbar##g(théta,phi)(2)
The resolution of the differential equation (1) and (2) gives us the angular part we see on the homework statement.
for the part ##e^{im*\phi}##, the wave function needs to be continuous in all the space so ##e^{2im*\pi}##=1,consequently, m need to be an integer
for the part ##e^{ikz}## I can't determine a condition and I don't know which values it can take.
So if someone have an hint thanks in advance !
 
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In z direction the state is free wave which has momentum eigenvalue and energy eigenvalue.
E=\frac{k^2\hbar^2}{2m}
 
If the wave function was ##e^{ikz}## and if I injected this in the schrödinger equation, yes I will obtain this result. But the wavefunction is ##e^{ikz}e^{im\phi}f(\rho)## and if I write the eigenvalue equation of the Hamiltonian I will not obtain this, so I don't understand why we can say that. Could you enlighten me please ?:biggrin:
 
Stylord said:
For the part ##e^{ikz}##, I can't determine a condition, and I don't know which values it can take.
So if someone have an hint thanks in advance!
There is no boundary condition that restricts the value of ##k##.
 
Stylord said:
If the wave function was eikz and if I injected this in the schrödinger equation, yes I will obtain this result. But the wavefunction is eikzeimϕf(ρ) and if I write the eigenvalue equation of the Hamiltonian I will not obtain this, so I don't understand why we can say that. Could you enlighten me please ?:biggrin:
Hamiltonian is written as sum of
H=H(\rho,\phi)+H(z)
Wavefunction of energy eigenstate is written as product of
\psi=\eta(\rho,\phi)\xi(z)
H\psi =\xi H(\rho,\phi)\eta +\eta H(z)\xi=(E(\rho,\phi)+E(z))\xi \eta=(E(\rho,\phi)+E(z))\psi
Round quantum well in ##\rho,\phi## and free movement in z are independent. Sum of their energy is the system energy, e.g. ground state of the well and kinetic energy of ##\frac{p_z^2}{2m}##.
 
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Ohhh it's a very good idea ! Now I Can finish the exercice, Thank you !
 
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