Infinite Well Expectation values

In summary, for this problem, the goal is to show that a particle in an infinite potential well in the nth energy level obeys the uncertainty principle and to determine which state comes closest to the limit of the uncertainty principle. The wavefunction for energy n is given by \psi(x) = \sqrt{\frac{2}{a}}\sin(n\pi x/a}), and the uncertainty principle is \sigma_x\sigma_p >= \hbar/2. The calculation involves finding <x>, <x^2>, <p>, and <p^2>. After some calculations, it is determined that <p> is equal to 0 and <p^2> is \frac{h^2n^2\pi
  • #1
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Homework Statement


I need to show that a particle in an infinite potential well in the nth energy level, obeys the uncertainty principle and also show which state comes closest to the limit of the uncertainty principle.

This means i have to calculate <x>, <x^2>, <p> and <p^2>

Homework Equations


The wavefunction for Energy n: [tex] \psi(x) = \sqrt{\frac{2}{a}}\sin(n\pi x/a}) [/tex]

uncertainty principle: [tex]\sigma_x\sigma_p >= \hbar/2[/tex]

The Attempt at a Solution


I have calculated <x> and <x^2> and <p> and <p^2> :

[tex]<x> = \frac{2}{a}\int_0^a x\sin^2(\frac{n \pi x}{a})dx = a/2 +a/(4n^2\pi^2) [/tex][tex]<x^2> = \frac{2}{a}\int_0^a x^2\sin^2(\frac{n \pi x}{a})dx = a^2/3 - a^3/(4n^2\pi^2)[/tex]

Im not so sure about this one. I did integration by parts twice to obtain it.

Heres what gets me:

<p> comes out to be 0

<p^2> come out to be [tex] \frac{h^2n^2\pi^2}{a^2}[/tex], which seems to be correct.

I can find the sigmas easily, the expectation values, while I know how to find them, I keep screwing up, so could someone please tell me if <p> is supposed to be zero? Also, if anything about the rest of the problem looks incorrect, please tell me. Thank you for your help. I'm trying to teach myself this material and this forum has been indispensable. I really appreciate the help.
 
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  • #2
The <p> and <p^2> are right. The <x> and <x^2> seem to have a problem.

I got [tex]<x>=\frac{a}{2}[/tex] and [tex]<x^2>=\frac{a^2}{3}-\frac{a^2}{2{\pi}^2n^2}[/tex]
 
  • #3
Thank you for your help. I fixed my mistake, which had nothing to do with physics just calculation and distributing! Thanks alot! I'm sure I'll be back with more problems though:yuck:
 

What is an infinite well expectation value?

An infinite well expectation value is a concept in quantum mechanics that represents the average value of a physical quantity, such as position or energy, in an infinite potential well. This value is calculated using the wave function of the particle and its corresponding operator.

How is the infinite well expectation value calculated?

The infinite well expectation value is calculated by taking the integral of the wave function squared multiplied by the corresponding operator over the entire well. This integral represents the average value of the physical quantity within the well.

What is the significance of the infinite well expectation value?

The infinite well expectation value helps to predict the most probable values of physical quantities in the infinite well. It also provides insight into the behavior of quantum particles within such a well.

Can the infinite well expectation value be negative?

No, the infinite well expectation value cannot be negative as it represents an average value and is always greater than or equal to zero.

How does the infinite well expectation value change with different wave functions?

The infinite well expectation value can change with different wave functions as the shape and amplitude of the wave function affect the probability of finding the particle at a certain position within the well. Therefore, the expectation value will change accordingly.

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