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Spin-3/2 particle and degeneracy in excited state

  1. Jan 28, 2015 #1

    ShayanJ

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    Gold Member

    1. The problem statement, all variables and given/known data

    Consider a particle with mass m in the following 1D potential:
    [itex]
    V(x)=\left\{ \begin{array}{lr} mgx \ \ \ x>0 \\ \infty \ \ \ \ \ \ \ x\leq 0 \end{array} \right.
    [/itex]
    What is its minimum energy calculated using the uncertainty relation?

    2. Relevant equations

    [itex]
    \Delta x \Delta p \geq \frac{\hbar}{2}
    [/itex]

    3. The attempt at a solution

    My problem is, I don't know what to use for [itex] \Delta x [/itex]! I see no length scale I can use. The only thing that came into my mind was making a constant with the dimension of length using the available constants and so I got [itex] \alpha(\frac{\hbar^2}{gm^2})^{\frac 1 3} [/itex](where [itex] \alpha[/itex] is a dimensionless constant) and this gives [itex] \Delta p \geq \frac{1}{2\alpha} (m^2 \hbar g)^{\frac 1 3} \Rightarrow E_0=\frac{1}{8\alpha^2} (m \hbar^2 g^2)^{\frac 1 3}[/itex].
    But the problem is, this method can give any multiple of [itex] (m \hbar^2 g^2)^{\frac 1 3}[/itex]!!!
    What should I do?
    Thanks
     
  2. jcsd
  3. Jan 28, 2015 #2

    TSny

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    Homework Helper
    Gold Member

    Last edited by a moderator: May 7, 2017
  4. Jan 28, 2015 #3

    ShayanJ

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    Gold Member

    Thanks man, it was very helpful.
     
    Last edited by a moderator: May 7, 2017
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