Quantum spherical harmonic oscillator:eigenfunctions

In summary, the conversation involves finding eigenfunctions common to the Hamiltonian, Lz, and L^2 operators in the problem of a spherical symmetric harmonic oscillator. The first step is to use spherical coordinates to express the Hamiltonian, and then to solve the eigenfunctions equation by making a substitution and solving the resulting radial equation. Possible methods include using series solutions and looking for solutions in the form of spherical Bessel and Neumann functions. One possible approach is to use the Laguerre ODE, with the solution being in the form of Laguerre polynomials. Another method involves making a substitution and solving a particular ODE with solutions in the form of a power series. More details and suggestions are needed for further progress.
  • #1
folgorant
29
0
Hi to everybody of PF community!

I have some troubles to find eigenfunctions common to [tex] H, L_{z}, L^2 [/tex] in the problem of spherical simmetric harmonic oscillator.

I start with the Hamiltonian [tex] H=\frac{\textbf{p}^2}{2 \mu} - \frac{1}{2}k\textbf{x}^2[/tex] that in spherical coordinates become

[tex] H=\frac{- \hbar ^2}{2 \mu} (\frac{\partial^2}{r \partial r^2} r - \frac{L^2}{r^2 \hbar^2}) - \frac{k r^2}{2} [/tex]

now,the eigenfunctions equation is: [tex] H \psi = E \psi [/tex]

i know the angular part of the problem that are the spherical armonics...so it remain to solve the radial equation

[tex] (\frac{- \hbar ^2}{2 \mu} \frac{\partial^2}{r \partial r^2} r + \frac{\hbar^2 l(l+1)}{2 \mu r^2 } - \frac{k r^2}{2}) R(r) = E R(r) [/tex]

then i don't know if to try with the hydrogen-like method (but potential is different) or with the armonic 1D oscillator (but I don't know whato to do with the centrifugal term)...so, please give me an help!
 
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  • #2
Some of the community might be more familiar with these things, but my understanding is that it's normal to look for series solutions. The requirement that the solution has a finite area under it usually constrains the allowed eigenvalues, and gives you some series of polynomials. I forget, but I think you should get Legendre polynomials?
 
  • #3
Make the substitution [itex] u(r) = r R(r) [/itex] and you should find that the equation reduces to

[tex] \frac{d^2 u}{dr^2} = \displaystyle \left[ \frac{l(l+1)}{r^2} - k^2 \right] u [/tex]

For [itex] l=0 [/itex] this will reduce to the simple harmonic oscillator. For [itex] l \neq 0 [/itex] you will find solutions in the form of spherical Bessel and Neumann functions. You can look up the ode's that correspond to those functions and they'll show you how to solve them.
 
  • #4
Hi! The radial solution to:

[tex] (\frac{- \hbar ^2}{2mr^2} \frac{d}{dr}(r^2\frac{d}{dr}) + \frac{\hbar^2 l(l+1)}{2 m r^2 } + U(r))R(r) = E R(r) [/tex]


by using [itex] u(r) = r R(r) [/itex]


and [itex] U(r) = \frac{1}{2}m\omega ^2 r^2 [/itex]

and imposing BC:
[itex] u(0) = 0 [/itex]
[itex] u(r) = 0 , r \rightarrow \infty [/itex]

is the radial Laguerre equation:

[tex] u_{kl}(r) = r^{l+1}e^{-\nu r^2}L^{l+1/2}_k(2\nu r^2) [/tex]

Where L is Laguerre polynomial and
[tex] \nu = m\omega / 2\hbar [/tex]

Source: Nuclear Shell Model, Kris Heyde, Springer 1994
 
Last edited:
  • #5
if I make the substitution : [tex]u(r)=rR(r)[/tex]

it become:

[tex] (\frac{- \hbar ^2}{2 \mu} \frac{\partial^2}{\partial r^2} + \frac{\hbar^2 l(l+1)}{2 \mu r^2 } - \frac{k r^2}{2}) u(r) = u R(r) [/tex]

...isn't it??

so... after find the solution to be [tex] u_{kl}(r) = r^{l+1}e^{-\nu r^2}L^{l+1/2}_k(2\ny r^2) [/tex]
...I have to divide by a factor r to obtain R(r)??
like [tex] R_{kl}(r) = r^{l}e^{-\nu r^2}L^{l+1/2}_k(2\ny r^2) [/tex] ?

and another my question is:where I could find the solution to that ode?

thanks
 
  • #6
malawi: ops...I see now the "source" of solution you give to me
 
  • #7
Search for Laguerre ODE
 
  • #8
but...is my post n°5 correct?
 
  • #10
[tex] (\frac{- \hbar ^2}{2 \mu} \frac{\partial^2}{r \partial r^2} r + \frac{\hbar^2 l(l+1)}{2 \mu r^2 } - \frac{m \omega^2 r^2}{2}) R(r) = E R(r) [/tex]

[tex] u(r)=rR(r) [/tex]

[tex] \frac{\partial^2}{r \partial r^2} r = \frac{\partial}{r^2 \partial r}(r^2 \frac{\partial}{\partial r})[/tex]

[tex] u_{kl}(r) = r^{l+1}e^{-\nu r^2}L^{l+1/2}_k(2\nu r^2) [/tex]

[tex] \nu = m\omega / 2\hbar [/tex]

[tex] u_{kl}(r) = r^{l+1} e^\frac{-2m\omega r^2}{2\hbar} L^{l+1/2}_k(\frac{m\omega r^2}{\hbar}) [/tex]

[tex] R_{kl}(r) = r^{l} e^\frac{-2m\omega r^2}{2\hbar} L^{l+1/2}_k(\frac{m\omega r^2}{\hbar}) [/tex]

[tex] R_{10}(r) = re^\frac{-m \omega r^2}{2\hbar} [/tex]

Continue in next post……
 
  • #11
...continue:

and the next is what I found with another method:

[tex] (\frac{- \hbar ^2}{2 \mu} \frac{\partial^2}{r \partial r^2} r + \frac{\hbar^2 l(l+1)}{2 \mu r^2 } - \frac{m \omega^2 r^2}{2}) R(r) = E R(r) [/tex]

[tex] \rho=\alpha r [/tex]

[tex] \alpha=\sqrt{m \omega/ \hbar} [/tex]

[tex] dr=\frac{d \rho}{\alpha}[/tex]

[tex] \frac{2 \mu E}{\hbar^2 \alpha^2} = \frac{2 E}{\hbar \omega} = \lambda [/tex]

[tex] \frac{d^2}{r dr^2}r = \frac{d^2}{dr^2} + \frac{2d}{rdr} [/tex]

[tex] (\frac{d^2}{d \rho^2} + \frac{2 d}{\rho d \rho} - \frac{l(l+1)}{\rho^2} - \rho^2 + \lambda) R(\rho) = 0 [/tex]

that is a particular ODE : [tex] (\frac{d^2}{d \rho^2} + p(\rho)\frac{d}{d \rho} +q(\rho)) R(\rho) = 0 [/tex]

and solutions are [tex] R(\rho)=\rho^{\beta} \sum^{0}_{\infty}c_{n} \rho^n [/tex] where beta is a constant and Cn is a function both dependents from the powers series of functions p(x) and q(x). The procedure is very long to write in latex, I did it in my exercise-book and the resul is different : [tex]R_{10}= \frac{-2E}{6 \hbar \omega}\rho^2[/tex]
If anybody wants to see the procedure I'll write it ,also if it will be an hard work...

anyway...any suggest??
 

1. What is a quantum spherical harmonic oscillator?

A quantum spherical harmonic oscillator is a mathematical model used to describe the behavior of a particle confined to a three-dimensional spherical potential well. It takes into account both the particle's position and momentum, and is often used in quantum mechanics to study the properties of atoms and molecules.

2. What are eigenfunctions in the quantum spherical harmonic oscillator?

Eigenfunctions in the quantum spherical harmonic oscillator are solutions to the Schrödinger equation that represent the energy states of the system. They describe the probability distribution of finding the particle at a certain position and momentum within the potential well.

3. How are eigenfunctions related to energy levels in the quantum spherical harmonic oscillator?

The eigenfunctions of the quantum spherical harmonic oscillator are directly related to the energy levels of the system. Each energy level has a corresponding eigenfunction, and the higher the energy level, the more nodes (points of zero probability) the eigenfunction has.

4. Can the eigenfunctions of the quantum spherical harmonic oscillator be visualized?

Yes, the eigenfunctions of the quantum spherical harmonic oscillator can be visualized using mathematical plots known as spherical harmonics. These plots show the shape of the probability distribution for each energy level, and they become more complex as the energy level increases.

5. How are the eigenfunctions of the quantum spherical harmonic oscillator used in real-world applications?

The eigenfunctions of the quantum spherical harmonic oscillator are used in applications such as quantum chemistry and solid-state physics. They help scientists understand the behavior of atoms and molecules, and they are also used in the development of new technologies such as quantum computers.

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