Particle in a potential- variation method

In summary, the author suggests that a function that is appropriate for solving a given problem is one that is normalizable and has no nodes. The variation method can be used to determine an approximate value of the ground state energy.
  • #1
Rorshach
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0

Homework Statement


Okay, I have no idea about the method they want me to solve it with. What in this case is the indicator that a function is appropriate?
A particle mass m affects a potential of the form ##V(x)=V_0 \frac{|x|}{a}## where ##V_0## and ##a## are positive constants.

a) Draw a sketch of the ground state wave function and indicate the characteristics of this function.

b) Use the variation method to determine an approximate value of the ground state energy. Use functions selected from the list below. Some of these functions are unsuitable while others are less good. Give reasons for all if they are worse or better! Then do the calculation with the function that you find most suitable.

##Nsin(\alpha x)exp(-\alpha |x|)##
##Nexp(-\alpha x^2)##
##\frac{N}{(x^2+\alpha^2)}##
##Nexp(-\alpha x^2)##
##\frac{N}{\sqrt{|x|+\alpha}}##


Homework Equations





The Attempt at a Solution



 
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  • #2
You've never learned about the variational method? Just look it up.

In quantum I don't think there was any way to choose an "appropriate" trial function, but with a given trial function you just normalize it and then minimize the expectation value of the Hamiltonian with respect to the variational parameter (alpha).
 
  • #3
Rorshach said:
What in this case is the indicator that a function is appropriate?


The wavefunction should be normalizable, for example. Should the ground state have any "nodes" (places where ψ = 0 other than at infinity)?

Your 2nd and 4th wavefunctions look identical to me.

For a rough sketch of the ground state, note that your potential is very roughly like a harmonic oscillator potential. So, roughly, you can expect a graph of the ground state wavefunction to have the general appearance of the ground state of the harmonic oscillator.
 
  • #4
The ground state should be flat, without any nodes, right? You are right, the second function was supposed to be ##Nxexp(-\alpha x^2)## I decided that fourth function is the one I should use, so I normalized it and the constant is equal to ##N=\sqrt[4]{\frac{2}{\pi}\alpha}##. The next thing according to the book is the equation that looks more or less like this: ##\int_{-\infty}^{+\infty} exp(-2\alpha x^2)(-\frac{2\hbar^2 \alpha^2 x^2}{m}+\frac{\hbar^2 \alpha}{2m}+V_0 \frac{|x|}{\alpha}) \,dx##
http://www.wolframalpha.com/input/?...|x|/a))),[x,-inf,+inf]&a=*C.V-_*RomanNumeral-
But I have a problem with solving this integral. How should it look like?
 
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  • #5
you have to setup a variation integral for the energy and then minimize it with respect to alpha to get alpha.Then you will have wave function and energy both.
 
  • #6
Rorshach said:
The ground state should be flat, without any nodes, right? You are right, the second function was supposed to be ##Nxexp(-\alpha x^2)## I decided that fourth function is the one I should use, ...

I haven't worked it out, but why not also investigate the third function and compare with result for fourth function?
 

FAQ: Particle in a potential- variation method

1. What is the "Particle in a potential- variation method"?

The "Particle in a potential- variation method" is a numerical method used in quantum mechanics to solve the Schrödinger equation for a particle in a given potential. It involves varying the potential and solving for the eigenvalues and eigenfunctions at each step, in order to find the ground state energy and wavefunction of the system.

2. How does the particle's potential affect its behavior?

The particle's potential determines the allowed energy levels and corresponding wavefunctions of the particle. A deeper potential well results in lower energy levels and a more confined particle, while a shallower potential well allows for higher energy levels and a less confined particle.

3. What is the significance of the ground state energy in this method?

The ground state energy is the lowest energy level that a particle can have in a given potential. It serves as a reference point for all other energy levels and plays a crucial role in understanding the behavior of the particle in the potential.

4. Is there a limit to the complexity of potentials that can be solved using this method?

There is no theoretical limit to the complexity of potentials that can be solved using the "Particle in a potential- variation method." However, as the potential becomes more complex, the calculations become more computationally intensive and may require more advanced numerical techniques.

5. How does this method compare to other numerical methods in quantum mechanics?

The "Particle in a potential- variation method" is a relatively simple and efficient method for solving the Schrödinger equation, especially for one-dimensional systems. However, it may not be suitable for more complex systems, in which case other methods such as the finite difference or finite element methods may be more appropriate.

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