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Group Work 4: Particle in a Box Probability

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    Work in groups on these problems. You should try to answer the questions without referring to your textbook. If you get stuck, try asking another group for help.

    The Particle in a 1D Box - A Quantum Trapped Particle

    The probability of finding a 1-D quantum mechanical particle in a range from \(x\) and \(x+dx\) is

    \[ \int_x^{x+dx} \psi _n^*(x)\psi_n(x)dx \label{W.1}\]

    where \(\psi_n(x)\) are the eigenfunctions of \(\hat{H}\) (i.e., wavefunctions) with \(\psi_n^*(x)\) representing the complex conjugate of \(\psi_n(x)\). The product

    \[\psi _n^*(x)\psi_n(x) \label{W.3}\]

    is called the probability density (within the Born interpretation of wavefunctions) for the \(n^{th}\) wavefunction. If a different wavefunction is used, then a different probability density is applicable. When \(\psi _n^*(x)\psi_n(x)\) is integrated over a finite length (for 1-D system) or volume (for a 3-D system), then the probability of finding a particle in that length of volume can be quantified. If \(dx\) is small, then Equation \(\ref{W.1}\) can be approximated:

    \[ \int_x^{x+dx} \psi _n^*(x)\psi_n(x)dx \approx \psi _n^*(x)\psi_n(x) \Delta x\]

    where \(\Delta x \approx dx\). 

    The eigenfunctions obtains for the particle in a box is

    \[\psi_n(x)=B \sin{\dfrac{n\pi x}{a}} \label{W.4}\]

    where

    • \(n\) is the quantum number and
    • \(a\) is the length of the box.
    • \(B\) is a constant (to be determined later)
     

    Q1

    Given the definitions above, what is the most simplified expression for the probability density for the particle in a box?

     

     

    We express the probability of finding a particle in a certain range, for example, \(b\) to \(c\), by integrating the probability density (Equation W.3) over that range, 

    \[P=\int_{b}^{c}\psi_n^*(x)\psi_n(x)dx \label{W.5}\]

    Using the particle in a box eigenfunctions (Equation \(\ref{W.4}\)), what is a general expression for the probability of finding the particle in a box of length \(a\) for any quantum level \(n\)?

     

     

    Intuitively, if the particle is in the box, what is the probability of finding the particle in that box? (Do not overthink this one)

     

     

    Q2

    The general solution for the integral of \(\sin^2\) over all space (using half angle trigonometric identity) is:

    \[\int \sin^2 (kz) dz=\dfrac{1}{2}z-\dfrac{1}{4k} \sin(2kz) \label{W.6}\]

    Using Equation \(\ref{W.4}\) and your expression for the probability of finding the particle in the box, find the numeric value for the probability that the particle is in the box. Does your answer make sense? What do you think it should be?

     

     

     

     

    Q3

    When the probability of finding a particle in the entire region to which it is confined is equal to one, the eigenfunction or wavefunction, \(\psi_n(x)\), is normalized. 

    For a ID system this means that

    \[P=\int_{-\infty}^{\infty} \psi_n^*(x)\psi_n(x)dx=1 \label{W.7a}\]

    For a 3D system that means

    \[P=\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \psi_n^*(x)\psi_n(x)dx\,dy\,dz=1 \label{W.7b}\]

    What must \(B\) equal to normalize \(\psi_n(x)\) in Equation \(\ref{W.4}\)?

     

     

     

     

    What is the normalized eigenfunction for the particle in a box?

     

     

     

     

    Using the normalized eigenfunction, what is the probability of finding the particle in the box, ie.., between \(0\) and the length of the box \(L\) (use expression in Q1)? Does your answer make physical sense?