Particle in a one dimensional box: probability of observing px between p and dp

In summary, the probability of observing a value of p_x between p and dp is given by: \frac{4|N|^2 s^2}{L(s^2-b^2)^2}[1-(-1)^n \cos(bL)]dp, where s=n\pi L^{-1} and b=p\bar{h}^{-1}.
  • #1
Dunhausen
30
0

Homework Statement


Show that for a particle in a one dimensional box of length L, the probability of observing a value of [tex]p_x[/tex] (recall [tex]\hat{p}_x[/tex] is Hermitian and that [tex]\Psi[/tex] is not an eigenfunction of [tex]\hat{p}_x[/tex]) between p and dp is:

[tex]\frac{4|N|^2 s^2}{L(s^2-b^2)^2}[1-(-1)^n \cos(bL)]dp[/tex]
where [tex]s=n\pi L^{-1}[/tex] and [tex]b=p\bar{h}^{-1}[/tex] and the constant N so that the integral of this result from - infinity to + infinity is one.

Evaluate this result from
[tex]p=\pm \frac{nh}{2L}[/tex]

What is the significance of this choice for p?

Homework Equations


[tex]p[/tex]
[tex]p[/tex]
[tex]\Psi = \left(\frac{2}{L}\right)^{1/2} \sin \frac{n\pi x}{L}[/tex]

The Attempt at a Solution


I tried
[tex]
<p_x> = \int_p^{p+dp} \Psi^* \frac{\bar{h}}{i} \frac{d}{dx}\Psi
= \frac{4\pi\bar{h}}{i L^2} \int_p^{p+dp} \sin \frac{n\pi x}{L} \cos \frac{n\pi x}{L} [/tex]
but it doesn't seem to be taking me where I want to go, and I wasn't sure whether integrating from p to p+dp was kosher. The source of the (-1)^n term in the final result is a bit baffling.

Any suggestions?
 
Last edited:
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  • #2
Are you familiar with bra-ket notation, aka Dirac notation?
 
  • #3
My course doesn't use it (so far) but if you prefer to write that way I can try to muscle through it with reference to the wikipedia article. :p
 
  • #4
No, that's okay. It's just that the notation suggests a certain way to explain things.

First, you should recall that the probability of finding a particle between x and x+dx is given by

[tex]P(x \le X \le x+dx) = \psi^*(x)\psi(x) dx[/tex]

Note that you're not doing an integral and you're not throwing the operator [itex]\hat{x}[/itex] in there anywhere. Remember that the complex amplitude of finding the particle at x is just the wavefunction evaluated at x. The probability (density) is then the square of the absolute value of the amplitude.

Now you're being asked to find the probability that the momentum of the particle is between p and p+dp. It's the same problem except this time you need the complex amplitude of finding the particle with momentum [itex]p[/itex], i.e.

[tex]P(p \le p_x \le p+dp) = \psi_p^*(p)\psi_p(p) dp[/tex]

In other words, you need to find the momentum representation [itex]\psi_p(p)[/itex] of the particle's state. How to do this should be in your notes and textbook.
 
  • #5
Thanks vella! That makes a lot of sense to me. :)

In other words, you need to find the momentum representation LaTeX Code: \\psi_p(p) of the particle's state. How to do this should be in your notes and textbook.
You would think that! But it really isn't. (fyi this is a physical chemistry course so it doesn't quite fit the normal physics curriculum) If I know my professor, there is another way to do this problem which has something to do with properties of the Hermitian, but I'm just happy to solve it by any means necessary.

Anyway I found http://www.google.com/url?sa=t&source=web&ct=res&cd=3&ved=0CBgQFjAC&url=http%3A%2F%2Fwww.ecse.rpi.edu%2F~schubert%2FCourse-ECSE-6968%2520Quantum%2520mechanics%2FCh03%2520Position%26momentum%2520space.pdf&ei=ajiCS_CxLYL0sgPD-OX6Aw&usg=AFQjCNH5tdKEyUg9XDGzKPunWoPaaBM83g&sig2=9CzZhXrFYsgNpO2puHyOuQ [Broken] article which described how to go into "momentum space" using a Fourier transform:

[tex]
\Phi(p) = \int_\infty^\infty \Psi(x) e^{-ipx/h} dx
[/tex]

So I put that into Wolfram|Alpha (integrating from 0 to q, where q is the length of the square well) and got this

http://img199.imageshack.us/img199/7875/63803268.gif [Broken]

I want to say it seems it will give me the right (or very similar) coefficients to that shown in my topic post, although I'm still not sure where that (-1)^n term comes from unless I have to write things out as their infinite series.

Anyway, I will hack away at it, but if anyone has any other insights, they are more than appreciated. :)
 
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  • #6
That's exactly what you want to do. Remember that Wolfram Alpha doesn't know that n is an integer, so it doesn't simplify the result as much as it could, in particular the sine and cosine terms.
 
  • #7
vela said:
Remember that Wolfram Alpha doesn't know that n is an integer, so it doesn't simplify the result as much as it could, in particular the sine and cosine terms.
Ah, yes! That was the clue I needed to finish this! The LaTeX here is being pretty buggy (and it's a long problem) so I'll post a scan of my solution later.

Thank you again for your help! I would have spent many fruitless hours doing the wrong thing otherwise.
 

1. What is a "particle in a one dimensional box"?

A particle in a one dimensional box is a theoretical model used in quantum mechanics to study the behavior of a particle confined to a one-dimensional space. The particle is assumed to be free to move within the box, but cannot escape its boundaries.

2. How is the probability of observing a particle's momentum (px) between two values (p and dp) calculated in this scenario?

In this scenario, the probability of observing a particle's momentum between two values (p and dp) is calculated using the wave function, which describes the probability amplitude of the particle at a given location and time. The square of the wave function gives the probability density, which can be used to calculate the probability of finding the particle in a certain range of momentum values.

3. What is the significance of the "box" in this model?

The "box" in this model represents the boundaries that confine the particle to a one-dimensional space. This confinement leads to quantized energy levels and allows for the study of the particle's behavior and probabilities within the box.

4. How does the probability of observing a particle's momentum change as the size of the box is altered?

The probability of observing a particle's momentum changes as the size of the box is altered. As the box size decreases, the energy levels of the particle increase, leading to a narrower range of possible momentum values and a higher probability of observing a specific momentum value. Conversely, as the box size increases, the energy levels decrease and the probability of observing a specific momentum value decreases as well.

5. Can this model be applied to real-world scenarios?

Yes, this model can be applied to real-world scenarios, such as the behavior of electrons in a semiconductor or the vibrational modes of molecules. However, it is important to note that this model is a simplified representation and does not take into account other factors that may affect the particle's behavior in a real-world system.

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