Spin-3/2 particle and degeneracy in excited state

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Homework Statement



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?

Homework Equations



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

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
 

Answers and Replies

  • #2
TSny
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  • #3
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I'm not sure exactly what the questioner had in mind. But it might be similar to the fairly well-known example of using the uncertainty principle to estimate the ground state energy of the hydrogen atom.

See

http://quantummechanics.ucsd.edu/ph130a/130_notes/node98.html

http://www.pha.jhu.edu/~rt19/hydro/node1.html [Broken]

http://www.uio.no/studier/emner/matnat/astro/AST1100/h06/undervisningsmateriale/lecture-2.pdf

Thanks man, it was very helpful.
 
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