- A nucleus has a radius of 4 Fermis.
Use the uncertainty principle to estimate the kinetic energy for a neutron localized inside the nucleus.
Do the same for an electron.
**Answer**

Try non-relativistic formula first and verify approximation when we have the energy.

This is much less than 940 MeV so the non-relativistic approximation is very good.The electron energy will be higher and its rest mass is only 0.51 MeV so it WILL be relativistic. This makes it easier.

*****Assume that the potential for a neutron near a heavy nucleus is given by for Fermis and for Fermis. Use the uncertainty principle to estimate the minimum value of needed for the neutron to be bound to the nucleus.- Use the uncertainty principle to estimate the ground state energy of Hydrogen.
**Answer**

(We could have replaced equally well.) Minimize.

*****Given the following one dimensional probability amplitudes in the position variable x, compute the probability distribution in momentum space. Show that the uncertainty principle is roughly satisfied.- a
- for , otherwise .
- b
- c

- Use the Heisenberg uncertainty principle to estimate the ground state energy for a particle of mass in the potential .
*****Find the one dimensional wave function in position space that corresponds to .*****Find the one dimensional wave function in position space that corresponds to for , and otherwise.*****Assume that a particle is localized such that for and that elsewhere. What is the probability for the particle to have a momentum between and ?- A beam of photons of momentum is incident upon a slit of
width . The resulting diffraction pattern is viewed on screen which is
a distance from the slit.
Use the
**uncertainty principle**to estimate the width of the central maximum of the diffraction pattern in terms of the variables given. *****The wave-function of a particle in position space is given by . Find the wave-function in**momentum space**. Is the state correctly normalized? Explain why.*****A particle is in the state . What is the probability for the particle to have a momentum between and ?- A hydrogen atom has the potential . Use the uncertainty principle to estimate the ground state energy.
*****Assume that for and elsewhere. What is ? What is the probability to find the particle between and ?- The hydrogen atom is made up of a proton and an electron bound together by the Coulomb potential, . It is also possible to make a hydrogen-like atom from a proton and a muon. The force binding the muon to the proton is identical to that for the electron but the muon's mass is 106 MeV/c. Use the uncertainty principle to estimate the energy and the radius of the ground state of muonic hydrogen.
*****Given the following one dimensional probability amplitudes in the momentum representation, compute the probability amplitude in the position representation, . Show that the uncertainty principle is satisfied.- (a)
- for , elsewhere.
- (b)
- (c)

*****Assume that . What is ? What is ? What is ?

Jim Branson 2013-04-22