Easy integration by parts but getting wrong answer. Help

In summary, the student is trying to find the speed of the cats in an evacuated lecture hall. They estimate that the maximum speed of the cats is equal to the speed of light.
  • #1
Bacat
151
1

Homework Statement



I am actually looking for the expectation of x for the wavefunction that is [tex]\Psi (x) = \sqrt{\frac{2}{L}}Sin(\frac{\pi x}{L})[/tex] for [tex]0 < x < L[/tex].

To do this I need to find the solution to this integral:

[tex]f = \int_0 ^L \! \Psi^* x \Psi \, dx = \int_0^L \! x*(\sqrt{\frac{2}{L}}Sin(\frac{\pi x}{L}))^2 \, dx[/tex]


The Attempt at a Solution



Using Mathematica, I know that the answer to the integral is [tex]\frac{L^2}{4}[/tex].

However, when I attempt the solution by integration by parts, I get 0. Help!

[tex]f = \frac{2}{L} \int_0^L \! xSin^2(\frac{\pi x}{L}) \, dx = \frac{2}{L} \left[ uv - \int v du \right][/tex]

Let [tex]u=x[/tex], [tex]du=dx[/tex], [tex]dv=Sin^2(\frac{\pi x}{L}) dx[/tex]

Then [tex]v=\int dv = \int_0^L Sin^2(\frac{\pi x}{L}) dx = \frac{1}{2} \int_0^L (1-Cos(\frac{\pi x}{L}) dx)[/tex]

[tex]v = \frac{L}{2} - Sin^2(\pi) = \frac{L}{2}[/tex]

Then [tex]f = \frac{2}{L}\left[uv - \int v du\right] = \frac{2}{L}\left[x\frac{L}{2}\right|_{0}^{L} - \int_0^L \frac{L}{2} dx\right] = \frac{2}{L}\left[ \frac{L^2}{2} - \frac{L^2}{2}\right] = 0.[/tex]

Where have I made an error?
 
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  • #2
[itex]v[/itex] is supposed to be the antiderivative of [itex]dv[/itex], not the definite integral from 0 to L of [itex]dv[/itex]!
 
  • #3
guess who goes to U of M and is in Prisca's class...am i right?

in addition to the reply above...

check your power reduction of the sine squared term again, and you should notice right away what you did wrong...hint, what happens to the angle when you go from the squared trig function to a first power trig function?
 
  • #4
Ah, thank you gabbagabbahey!

And Dahaka14, no. I live in Houston. Oddly enough, the curriculum for quantum mechanics is pretty much the same wherever you go.
 
  • #5
Bacat said:
Oddly enough, the curriculum for quantum mechanics is pretty much the same wherever you go.

Only once you measure it. Before measurement there is a non-zero probability that the curriculum consists entirely of exorcising the vengeful spirits of cats which are both alive and dead simultaneously.
 
  • #6
Haha!

Just for fun, here is one of the other problems that were part of this homework (I solved it already):

Suppose a lecture hall is evacuated and (Schrodinger) cats are projected with speed [tex]\nu[/tex] at the two doors leading out of the lecture hall in a double-slit experiment. Assume that in order for the interference fringes to be observed as the cats pile up against a distant wall the wavelength of each cat must be greater than 1 meter. Estimate the maximum speed of each cat. If the distance between the front of the lecture hall to the wall is 30 meters, how long will it take to carry out the experiment? Compare this time with the age of the universe, roughly [tex]10^{10}[/tex] years. Assume the cats each weight 1 kg.
 

1. What is the integration by parts method?

The integration by parts method is a technique used in calculus to find the integral of a product of two functions. It is based on the product rule for derivatives and allows for the integration of functions that cannot be easily integrated using other methods.

2. How do I apply integration by parts?

To apply integration by parts, you need to choose which function will be the "u" and which will be the "dv" in the formula. Then, you use the formula u dv = uv - v du to find the integral of the product. This process may need to be repeated multiple times, depending on the complexity of the function.

3. Why am I getting the wrong answer when using integration by parts?

There are a few common mistakes that can lead to the wrong answer when using integration by parts. These include incorrect choice of "u" and "dv", not distributing the negative sign properly, and forgetting to apply the formula multiple times if necessary. It is important to carefully check each step and correct any errors that may have been made.

4. How can I improve my understanding of integration by parts?

Practicing with a variety of problems and seeking guidance from a tutor or teacher can help improve your understanding of integration by parts. It is also helpful to review the basic principles of calculus and the product rule for derivatives.

5. Are there any alternative methods to integration by parts?

Yes, there are other methods for integrating functions such as substitution and partial fractions. It is important to understand and be able to use multiple integration techniques to solve different types of problems.

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