Question 41.5: A Ground-State Energy Free Particle Take the limit of E1 = h......
A Ground-State Energy Free Particle
Take the limit of E_1 = h^2 /8mL^2 (Eq. 41.16 with n = 1) to find the zero-point energy of a free particle.
E_n=\left(\frac{h^2}{8 \pi^2 m}\right)\left(\frac{n^2 \pi^2}{L^2}\right)=n^2\left(\frac{h^2}{8 m L^2}\right) \quad \quad (41.16)INTERPRET and ANTICIPATE
The key to this problem is knowing what limit we must take in order to go from the energy expression for a trapped particle to that for a free particle. There are only two parameters we can change: m and L. Changing the mass does not set a particle free, but increasing the width of the well does. An infinitely wide well means that the particle can go anywhere.
SOLVE
Take the limit as L → ∞.
CHECK and THINK
When a particle is free (not part of any system), it can have any energy value. So its minimum energy is zero and it may be at rest. Only a confined particle is required to have nonzero energy and to be in motion.