Variational Method and Bound States

  • Thread starter Einj
  • Start date
  • #1
470
58

Homework Statement


Consider a potential function [itex] V(x)[/itex] such that:
$$
\begin{cases}
V(x)\leq 0\text{ for }x\in[-x_0,x_0] \\
V(x)=0 \text{ for }x\not\in[-x_0,x_0]
\end{cases}
$$
Show, using the variational method that:

(a) In the 1-dimensional case [itex]\lambda^2V(x)[/itex] always possesses at least one bound state.
(b) In the 3-dimensional spherically symmetric case, [itex]V(|\vec r|)[/itex], it possesses no bound states if [itex]\lambda^2[/itex] is made sufficiently small.



2. The attempt at a solution
The idea is to use the variational method, i.e.:
$$
\frac{\langle\psi|H|\psi\rangle}{\langle \psi|\psi\rangle}\geq E_{ground},
$$
to show that the average value of the energy is negative and hence the ground state energy is negative.

In my first attempt I used a gaussian trial function:
$$
\psi(x)=\left(\frac{2a}{\pi}\right)^{1/4}e^{-ax^2}.
$$
However the problem is that it turns out that the average value of the kinetic energy is [itex]\hbar^2a/2m[/itex] and so, in order to determine whether the average energy is negative or not we need to know [itex]a[/itex] explicitly. However, this is not possible since we have no insight on the actual shape of the potential.

It seems to me that this thing will turn out to be a problem for every trial function. How can I do that?

Thank you

Homework Statement





Homework Equations





The Attempt at a Solution

 

Answers and Replies

  • #2
Orodruin
Staff Emeritus
Science Advisor
Homework Helper
Insights Author
Gold Member
17,158
6,966
In the 1D case the kinetic energy is proportional to ##a##. The potential energy is negative and goes as ##\sqrt{a}## when ##a## is small. Hence, if ##a## is small enough, the energy in that state will be negative, implying the existence of at least one bound state.

In the 3D case, the normalisation of the wave function gets additional factors of ##a## and the argument changes.
 

Related Threads on Variational Method and Bound States

  • Last Post
Replies
0
Views
1K
Replies
6
Views
966
  • Last Post
Replies
1
Views
1K
Replies
3
Views
3K
  • Last Post
Replies
4
Views
12K
  • Last Post
Replies
0
Views
970
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
1
Views
2K
Replies
3
Views
2K
Replies
1
Views
1K
Top