Finding minimum potential

In summary, the problem asks to find the minimum value of the potential depth, V_0, for which there exists a bound state in a particle moving in a 3-D potential well. The wave function for the bound state with the lowest energy is spherically symmetric and the radial wave satisfies a linear ordinary differential equation. The function u(r) must be regular (smooth) at the origin, and thus must have a Taylor series that proceeds in even powers of r. The solution for bound states must vanish for r > r_0.
  • #1
noblegas
268
0
finding minimum potential
1. Homework Statement

A particle of mass m moves in 3-d in the potential well

LaTeX Code: V(r)=-V_0 at LaTeX Code: r<r_0

where LaTeX Code: V0 and LaTeX Code: r_0 are positive constants. If there exists a state in which the particle is bound to the potential well, the wave function for the bound state with the lowest energy is spherically symmetric and the radial wave satisfies equations

LaTeX Code: -h-bar^2/2m*(d^2/dr^2)*u(r)+V(r)*u(r)=Eu(r)

LaTeX Code: u=\\varphi*r

Find the minimum value of the depth LaTeX Code: V_0<BR> for which there exists a bound state. (recall that the radial function satisfies the condition u(0)=0 , because LaTeX Code: \\varphi (r)= u(r)/r has to be regular at the origin

2. Homework Equations LaTeX Code: -h-bar^2/2m*(d^2/dr^2)*u(r)+V(r)*u(r)=E*u(r)
LaTeX Code: r^2=(x^2+y^2+z^2)3. The Attempt at a Solution

I don't know what they mean when they state ' LaTeX Code: \\varphi (r)= u(r)/r has to be regular at the origin'; I don't know why they want you to find a minimum value for V_0 since it is already given in the problem

Should I apply separation of variables where LaTeX Code: u=R(r)*THETA(\\vartheta)*\\Phi(\\phi)
and transform should i differentiate r^2 with respect to x?
Report Post Edit/Delete Message
 
Last edited:
Physics news on Phys.org
  • #2
First, let me clean up the tex:

Potential is [tex] V(r) = -V_0 [/tex] for [tex] r < r_0 [/tex]

[tex] u(r) [/tex] satisfies the equation

[tex] -\hbar^2/2m \frac{d^2 u}{d r^2} + V(r) u(r) = E u(r)[/tex] where [tex] u(r) \equiv \varphi(r) r[/tex]

What they mean by regular at the origin is "smooth at the origin". For example the linear function [tex] f(r) = r [/tex] is not smooth at the origin when viewed as a function of [tex] x, y, [/tex] and [tex] z [/tex].

To see this write: [tex] f(r) = r = (x^2 + y^2 + z^2)^{1/2} [/tex]. Let's see what this function looks like along the [tex]x[/tex] axis where [tex]y=z=0[/tex]. Along this axis [tex]f(x) = (x^2)^{1/2} = |x|[/tex]. This function has a cusp at [tex]x=0[/tex] and so is not smooth. On the other hand [tex]f(r)=r^2[/tex] is smooth. But [tex] f(r) = r^3 [/tex] is not smooth. One thus concludes that smooth functions of "r" must have a Taylor series at the origin that proceeds in even powers of "r".

Now as far as the solution of the equation is concerned. It is a linear ordinary differential equation so you should be able to solve it yourself. You are looking for bound states, i.e., solutions for which the wavefunction vanishes (please check this; I am not a physicist) for
[tex] r > r_0 [/tex].

Good luck
 
Last edited:
  • #3
ignore this post. i posted a more clearer post.
 

1. What is the concept of "minimum potential"?

The concept of "minimum potential" refers to the lowest possible value that a system can achieve within a given set of conditions. In science, this is often used in reference to finding the most stable or energy-efficient state of a system.

2. Why is it important to find the minimum potential of a system?

Finding the minimum potential of a system is important because it allows scientists to understand the most stable or optimal state of the system. This information can then be used to predict how the system will behave under different conditions.

3. How is the minimum potential of a system determined?

The minimum potential of a system is typically determined through mathematical calculations and simulations. This involves analyzing the different variables and parameters that affect the system and finding the combination that results in the lowest potential value.

4. Can the minimum potential of a system change over time?

Yes, the minimum potential of a system can change over time as the conditions and variables affecting the system change. For example, if there is an external force or energy input, the minimum potential may shift to a different state.

5. What are some real-world applications of finding minimum potential?

Finding minimum potential has many real-world applications, such as in engineering, chemistry, and physics. It is used to optimize processes and systems, design efficient structures, and predict the behavior of materials and substances. One example is in designing energy-efficient buildings by finding the minimum potential of heat loss and implementing measures to reduce it.

Similar threads

  • Advanced Physics Homework Help
Replies
4
Views
931
  • Advanced Physics Homework Help
Replies
1
Views
2K
  • Advanced Physics Homework Help
Replies
1
Views
2K
  • Advanced Physics Homework Help
Replies
7
Views
2K
  • Advanced Physics Homework Help
Replies
3
Views
919
  • Advanced Physics Homework Help
Replies
17
Views
2K
  • Advanced Physics Homework Help
Replies
7
Views
1K
  • Advanced Physics Homework Help
Replies
11
Views
2K
  • Advanced Physics Homework Help
Replies
7
Views
2K
  • Advanced Physics Homework Help
Replies
2
Views
1K
Back
Top