Determining Neutron Wavelength with mass alone

[atomicpedals]

Homework Statement

Consider a particle of mass m in the potential

V(z)=mgz z>0, infinity if z<0

Show that the wavelength of a neutron (m~1000MeV/c^2) is on the order of 10um.

Homework Equations

p=(2m(E-mgz))^1/2
$\lambda$=h/p

The Attempt at a Solution

The most direct solution would seem to simply be

h/(2m(E-mgz))^1/2 (with appropriate substitutions)

however, that still leaves the factors of g and z to be resolved... I could take z as 1, but there's still g with a units problem (m/s/s vs MeV and MeV/c^2).

Any suggestions?

 

[vela]

Try drawing a diagram of the potential and then sketch what the ground-state wave function will look like. That should give you an idea for a relationship between the wavelength and the energy of the neutron.

 

[atomicpedals]

So, for z>0 the potential will increase linearly (holding m and g constant).