Uncertainty in Energy an ground state energy

[Chiborino]

I'm asked to show that in the ground state, a particle trapped in a potential well approximately equal to its lowest allowed energy.

I know the expression for the energy is h^2/8mL^2 for a finite well, and ΔEΔt~h/2π. But I'm at a loss as to how I'm supposed to even begin to start this.

Attempts at a solution: I played around with Heisenberg's uncertainty principle, the energy uncertainty principle, and the Schrodinger equation for minimum allowed energy that I listed above, but didn't really get anywhere meaningful.

 

[jbsteele]

Any help would be appreciated. Answer: The particle in the potential well must have an energy equal to its lowest allowed energy. This is because according to the Heisenberg Uncertainty Principle, the product of the uncertainty in position and momentum for a particle in its ground state must be equal to or greater than Planck's constant h divided by 4π. Therefore, the particle must occupy the lowest energy state in order to satisfy this principle. Furthermore, the energy uncertainty principle states that the uncertainty in energy multiplied by the time it takes to measure this energy must be greater than or equal to Planck's constant h divided by 2π. In other words, the particle must remain in its lowest energy state for a sufficient amount of time to measure its energy. Since the particle is trapped in a potential well, its energy cannot change, so the particle must remain in its lowest energy state.