- #1

- 2,810

- 604

## 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