Central potential in quantum mechanics

In summary, the conversation is about a question regarding central potentials and finding physically acceptable solutions for a specific equation. The participants discuss the terms in the equation and potential, and mention a constraint that excludes non-physical solutions. One person solves the problem themselves and asks for help with another problem.
  • #1
Bavon
5
0
Hi all,
I'm new to his forum. I'm having trouble with the following question on central potentials. It's an exercise from (Bransden and Joachain, Introduction to quantum mechanics).

Homework Statement


Suppose V(r) is a central potential, expand around r=0 as [tex]V(r)=r^p(b_0+b_1r+\ldots).[/tex] When p=-2 and [tex]b_0<0[\tex], show that physically acceptable solutions exist only when [tex]b_0>-\frac{\hbar}{8\mu}[/tex]


Homework Equations


R(r) is the radial component of the wave function
[tex]u(r)=r^{-1}R(r)=r^s\sum{c_kr^k}[/tex] is a solution of
[tex]-\frac{\hbar^2}{2\mu}\frac{d^2u(r)}{dr^2}+V_{eff}u(r)=Eu(r)[/tex]


The Attempt at a Solution


The effective potential is [tex]r^{-2}(b_0+\frac{l(l+2)\hbar^2}{2\mu}+b_1r+\ldots)[/tex]

When p>-2, the case that is discussed in the textbook, they compare the lowest order terms in the radial Schroedinger equation. For p=-2, I get:

The lowest order term of [tex]\frac{d^2u}{dr^2}=s(s-1)r^{s-2}\sum{c_kr^k}[/tex]
The lowest order term of [tex]V_{eff}(r)u(r)=(b_0+\frac{l(l+2)\hbar^2}{2\mu})r^{s-2}\sum{c_kr^k}[/tex]

The difference is apparently that [tex]b_0[/tex] appears in the lowest order terms.

Now I need some constraint that excludes non-physical solutions. For p>-2, that is u(0)=0. But for [tex]b_0<0[/tex] that can't be used. Because the potential is attracting, I think the probability of finding a particle at the origin should be positive. The problem is I can't think of any constraint that should be imposed, that could limit the allowed values of [tex]b_0[/tex].

Any hints would be greatly appreciated.
 
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  • #2
I solved this myself. It was so easy after all!
 
  • #3
Dear Bavon
Would u please help me to solve this problem too?
thanks
I am looking forward for ur reply. It is an emergency situation!
 
Last edited:
  • #4
If I decyphered my notes from almost 3 years ago correctly, I constructed an equation in [tex]s[/tex], and then expressed that it should have real solutions.
 
  • #5
How can I undrestand that an equation have real solutions?
thanks
 

1. What is a central potential in quantum mechanics?

A central potential in quantum mechanics is a type of potential energy that depends only on the distance between two particles. It is a spherically symmetric potential, meaning that it has the same value at any given distance from the center. Examples of central potentials include the gravitational potential and the Coulomb potential.

2. How is a central potential different from other types of potentials?

A central potential is different from other types of potentials in that it only depends on the distance between two particles, while other potentials may also depend on other factors such as direction or time. Additionally, a central potential is spherically symmetric, while other potentials may have different values at different angles or distances from the center.

3. What is the role of central potentials in quantum mechanics?

Central potentials play a crucial role in understanding the behavior of particles at the quantum level. They are used to describe the interactions between particles in a system, and they are a key component in many quantum mechanical models and equations.

4. How are central potentials solved in quantum mechanics?

Central potentials are solved using mathematical techniques such as the Schrödinger equation and the Born-Oppenheimer approximation. These methods allow for the calculation of the energy levels and wavefunctions of particles in a central potential.

5. What are some real-world examples of central potentials?

Some real-world examples of central potentials include the interaction between an electron and a proton in an atom, the gravitational interaction between planets, and the interaction between a nucleus and an electron in a molecule. These interactions can be described by central potentials and are essential in understanding many physical phenomena.

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