Spin-3/2 particle and degeneracy in excited state

[ShayanJ]

Homework Statement

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

Homework Equations

$\Delta x \Delta p \geq \frac{\hbar}{2}$

The Attempt at a Solution

My problem is, I don't know what to use for $\Delta x$! 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 $\alpha(\frac{\hbar^2}{gm^2})^{\frac 1 3}$(where $\alpha$ is a dimensionless constant) and this gives $\Delta p \geq \frac{1}{2\alpha} (m^2 \hbar g)^{\frac 1 3} \Rightarrow E_0=\frac{1}{8\alpha^2} (m \hbar^2 g^2)^{\frac 1 3}$.
But the problem is, this method can give any multiple of $(m \hbar^2 g^2)^{\frac 1 3}$!
What should I do?
Thanks

 

[TSny]

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

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

 

[ShayanJ]

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

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

Thanks man, it was very helpful.