# Solving Integration of Wave Equation for x|Psi|^2

• diewlasing
In summary: This can't be right since it does not have the dimensions of a length. Even worse, all the terms don't have the same dimensions. So check your calculation carefully.
diewlasing
where the wave equation is Psi_n = sqrt(2/a)sin(n*pi*x/a). When you do the integral of -inf to +inf of x|Psi|^2, the CRC handbook works it out to be:

(x^2)/4 - (xsin(2ax))/4a - cos(2ax)/(8a^2).

And I know the solution works out to be a/2 somehow but I don't know how to get it. I worked it down to:

ax - 2sin(2ax) - 2/ax = 0. I don't know if this is the right track. Can someone shed some light on this?

diewlasing said:
where the wave equation is Psi_n = sqrt(2/a)sin(n*pi*x/a). When you do the integral of -inf to +inf of x|Psi|^2, the CRC handbook works it out to be:

(x^2)/4 - (xsin(2ax))/4a - cos(2ax)/(8a^2).

And I know the solution works out to be a/2 somehow but I don't know how to get it. I worked it down to:

ax - 2sin(2ax) - 2/ax = 0. I don't know if this is the right track. Can someone shed some light on this?

Don't integrate from minus inifnity to plus infinity. The wavefunction you give is valid only inside the well. Outside the well, the wavefunction is zero. So integrate over the width of the well only.

right my fault, but the integral works out to be:

(x^2)/4 - (xsin(2ax))/4a - cos(2ax)/(8a^2)

My question is how does that simplify to a/2?

diewlasing said:
right my fault, but the integral works out to be:

(x^2)/4 - (xsin(2ax))/4a - cos(2ax)/(8a^2)

My question is how does that simplify to a/2?

This can't be right since it does not have the dimensions of a length. Even worse, all the terms don't have the same dimensions. So check your calculation carefully.

Where did all the factors of $$n\pi$$ go? Those will help when simplifying

## 1. What is the wave equation and why is it important?

The wave equation is a mathematical model that describes the behavior of waves in various physical systems, such as sound, light, and water. It is important because it allows us to understand and predict the behavior of waves in these systems, which is crucial in many scientific and engineering applications.

## 2. How is the wave equation solved for x|Psi|^2?

The wave equation for x|Psi|^2 is solved using integration, which involves finding the area under the curve of the wave function. This allows us to determine the probability of finding a particle at a specific position x.

## 3. What information can be obtained from solving the integration of the wave equation for x|Psi|^2?

Solving the integration of the wave equation for x|Psi|^2 allows us to determine the probability of finding a particle at a specific position x, and also provides information about the energy and momentum of the particle.

## 4. What are some common methods for solving the integration of the wave equation for x|Psi|^2?

Some common methods for solving the integration of the wave equation for x|Psi|^2 include using Fourier transforms, separation of variables, and numerical methods such as finite difference or finite element methods.

## 5. Are there any real-world applications for solving the integration of the wave equation for x|Psi|^2?

Yes, there are many real-world applications for solving the integration of the wave equation for x|Psi|^2. This includes understanding the behavior of particles in quantum mechanics, calculating the probability of electron transfer in chemical reactions, and predicting the behavior of electromagnetic waves in communication systems.

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