How Do You Determine the Value of A in a Wave Function?

In summary, the conversation was about finding the value of A in a wave function given certain information and equations. It was determined that the wave function needed to be squared and then integrated using the standard Gaussian integral in order to solve for A.
  • #1
birdgirl
6
0
[SOLVED] Wave Function Solution

Homework Statement



An electron is found to be in a state given by the wave function http://rogercortesi.com/eqn/tempimagedir/eqn7955.png

Find the value of A.

Homework Equations



The normalization of the wave function: http://rogercortesi.com/eqn/tempimagedir/eqn9043.png

where [tex]\psi[/tex]* is found by replacing i with -i.

The Attempt at a Solution



Since the wave function is completely real, the normalization is simply the wavefunction squared, then
eqn8340.png


I know that the integral of the normalization from negative infinity to infinity is equal to one, when integrated with respect to dx, but I have no idea as to how to solve this integral and find A. Is there a simplification that I am missing, or is it just a really difficult integral?

This is my first time posting with an equation editor, so if the programming does not come out I will repost with pictures. Thanks so much for your help.
 
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  • #2
No need for pictures. A simple correction to your syntax (you need to surround equations with [ tex ] ... [ /tex ] does the trick.

birdgirl said:

Homework Statement



An electron is found to be in a state given by the wave function [tex] \psi (x) = Ae^{-[{(x-a)} / {2 \epsilon }]^2} [/tex]

Find the value of A.

Homework Equations



The normalization of the wave function: [tex] | \psi |^2 = \psi^{*} \psi [/tex]

where [tex] \psi^{*} [/tex] is found by replacing i with -i.

The Attempt at a Solution



Since the wave function is completely real, the normalization is simply the wavefunction squared: [tex] \psi (x) = (Ae^{-[{(x-a)} / {2 \epsilon }]^2})^2 [/tex]

Then [tex] \psi (x) = A^2 e^{-{(x-a)^2} / {2 \epsilon^2 }} [/tex]

I know that the integral of the normalization from negative infinity to infinity is equal to one, when integrated with respect to dx, but I have no idea as to how to solve this integral and find A. Is there a simplification that I am missing, or is it just a really difficult integral?

This is my first time posting with an equation editor, so if the programming does not come out I will repost with pictures. Thanks so much for your help.
 
  • #3
Just a correction to your terminology:

[tex] |\psi|^2 [/tex]

is not the "normalization" of the wavefunction. It is the modulus squared of the wavefunction. The normalization is not a "thing". It's a condition. When this condition is satisfied

[tex] \int |\psi|^2 \, dx = 1 [/tex]

we say that the wavefunction is normalized. Hence, the above equation is called the normalization condition. Stated in words, the normalization condition says that the integral of the modulus squared of the wavefunction over all space is unity.
 
  • #4
Also remember that

[tex] (e^a)^2 = e^{2a} [/tex]

So your squared wavefunction should be

[tex] |\psi(x)|^2 = A^2 e^{-2[(x-a)/2\epsilon]^2} [/tex]
 
  • #5
A hint for solving the integral is that is you substitute y = x - a, it looks like you end up with a standard Gaussian integral.
 
  • #6
Thanks, I forgot to square the wavefunction psi of my last equation, so with simplification I get

[tex] |\psi(x)|^2 = A^2 e^{-(x-a)^2/2\epsilon^2} [/tex]
 
  • #7
The standard Gaussian integral which cepheid stated is

[tex]\int_{-\infty}^\infty e^{-y^2}\,d\,y=\sqrt\pi[/tex]
 
  • #8
Hello! It did turn out to be Gaussian integration, which we hadn't been taught yet. :) Thanks for your help!
 

Related to How Do You Determine the Value of A in a Wave Function?

What is a wave function solution?

A wave function solution is a mathematical expression that describes the behavior and properties of a quantum mechanical system. It represents the probability of finding a particle in a particular state at a given time.

How do you find the wave function solution for a system?

The wave function solution for a system can be found by solving the Schrödinger equation, which is a differential equation that describes the evolution of a quantum system over time. This involves using mathematical techniques such as separation of variables and boundary conditions.

What is the significance of finding a wave function solution?

Finding a wave function solution allows us to understand the behavior of quantum systems and make predictions about their properties, such as energy levels and probabilities of measurement outcomes. It also helps us to develop new technologies, such as quantum computers.

What are the limitations of the wave function solution?

The wave function solution is limited in its ability to accurately predict the behavior of macroscopic objects, as it only applies to quantum systems. It also does not take into account the effects of external forces or interactions with other particles.

Can the wave function solution be experimentally verified?

Yes, the predictions made by the wave function solution can be tested and verified through experiments. This is an important aspect of the scientific method and helps to validate the accuracy and usefulness of the solution.

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