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Quantum uncertainty after passing through a slit

  1. Sep 29, 2013 #1
    1. The problem statement, all variables and given/known data

    A given particle is confined to a certain potential. At a given instant that potential is turned off and the particle is accelerated by gravity. In the initial instant t = 0, when the potential is turned off, it has an initial speed of vz = 1 m/s.

    In the instant t = 0 the particle goes through a slit with width y = 1 mm = 10^-3 m in the y axis. How much time does it take for the spatial uncertainty associated with y to double? Consider the law of propagation of uncertainty for the calculation of the spatial uncertainty of y at a given instant. Consider the particle a proton.

    2. Relevant equations

    [;\sigma_{x}\sigma_{p} \geq \frac{\hbar}{2};]

    [;\mathbf{p} = m\mathbf{v};]

    [;v = v_0+at \,;]

    [;\left |\Delta X\right |=\left |\frac{\partial f}{\partial A}\right |\cdot \left |\Delta A\right |+\left |\frac{\partial f}{\partial B}\right |\cdot \left |\Delta B\right |+\left |\frac{\partial f}{\partial C}\right |\cdot \left |\Delta C\right |+\cdots;]

    3. The attempt at a solution

    I understood that, as the particle passes through the slit, we have [;\Delta y\right;] equal to the slit's width.

    But, from that, I don't really get how to proceed, considering the particle is moving along the z axis. I can relate the errors along the z axis, but since there's no momentum along the y axis, I am kind of lost.

    How am I supposed to use the propagation of uncertainty?

    Thank you in advance. I don't expect to be just given the solution, I just want a little push forward. (I am new here, so please tell me if I'm doing anything wrong!)
     
  2. jcsd
  3. Sep 29, 2013 #2
    I'm not sure I understand this problem entirely but...

    To start, ignore the z axis. What does the uncertainty in y position tell you about the uncertainty in y momentum?
     
  4. Sep 29, 2013 #3
    I'm not sure either, actually!

    And well, according to the Uncertainty principle:

    [;\sigma_{py} \geq \frac{\hbar}{2\sigma_{y}};]

    But since they mention the propagation, and even with this info, I'm really lost.
     
  5. Sep 29, 2013 #4
    Ok, that's right. (I think you should check that it's really supposed to be an initial v_z and not an initial v_y)

    If you have a range of potential initial y positions, and a range of potential y velocities, how would you compute the range of potential y positions in the future?
     
  6. Sep 29, 2013 #5
    (Confirmed! It really is an initial v_z.)

    So, taking this:

    [;\sigma_{py} \geq \frac{\hbar}{2\sigma_{y}};]

    [;\sigma_{y_0} = 10^{-3};]

    [;\sigma_{py_0} \geq 5.275 \times 10^{-32} ;]

    Which means... (dividing by the proton mass)

    [;\sigma_{vy_0} \geq 3.153 \times 10^{-5};]

    Do I propagate to [;\sigma_{y};], using the info I have from [;\sigma_{vy_0};] and [;\sigma_{y_0};]? (using a normal position equation, like [; x = x_0 + v_0 \times t + 0.5 \times a \times t^2 ;])
     
    Last edited: Sep 29, 2013
  7. Sep 29, 2013 #6
    Yeah that's the idea. Basically, what I'm thinking is: what's the furthest in one direction it could be? what's the furthest in the other direction? Presumably the particle is somewhere in between these two points.

    Also I figured out why the v_z matters: if there was no v_z, then the particle would still be in the slit, and the y position uncertainty would never change.
     
  8. Sep 29, 2013 #7
    That's the uncertainty, no? For example, in the y direction, at t = 0, the particle could be anywhere within the slit, from one border to the other.

    The thing is, now that I propagated (using the partial derivatives of y in respect to y_0 and vy_0), I have a few questions: are the initial position / velocity null? I now have an expression for [;\sigma_{y};] with respect to t, y_0, vy_0, and a. Considering none of those apart from t really apply to the y axis in this problem, I'm kind of confused. I got an expression similar to:

    [;\sigma_{y}^{2}(t) = (1 + v_y_0 \times t + 0.5 \times a \times t^2)^{2} \times (\sigma_{y_0})^{2} + (y_0 + t + 0.5 \times a \times t^{2})^{2} \times (\sigma_{v_y_0})^{2} ;]

    Am I making a complete mess of things or am I going the right way? I'm definitely understanding the problem better, but the way to get there I'm quite not grasping yet.
     
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