8.*A particle of massmis in a 1 dimensional box of length L. The particle is in the ground
state. A measurement is made of the particle’s momentum. Find the probability that the
value measured is betweenp 0 andp 0 +dp.- A particle of massmis in a constant potentialV(x) =−V 0 for all x. What two operators
commute with the Hamiltonian and can therefore be made constantsof the motion? Since these
two operators do not commute with each other, there must be twoways to write the energy
eigenfunctions, one corresponding to each commuting operator.Write down these two forms
of the eigenfunctions of the Hamiltonian that are also eigenfunctions of these two operators. - A particle is confined to a ”box” in one dimension. That is the potential is zero forxbetween
0 andL, and the potential is infinite forxless than zero orxgreater thanL.
a) Give an expression for the eigenfunctions of the Hamiltonian operator. These are the
time independent solutions of this problem. (Hint: Real functions willbe simplest to use
here.)
b) Assume that a particle is in the ground state of this box. Now one wall of the box is
suddenly moved fromx=Ltox=W whereW > L. What is the probability that the
particle is found in the ground state of the new potential? (You may leave your answer
in the form containing a clearly specified integral.)
A particle of massmis in a 1 dimensional box of lengthL. The particle is in the ground state.
The size of the box is suddenly expanded to length 3L. Find the probability for the particle to
be in the ground state of the new potential. (Your answer may include an integral which you
need not evaluate.) Find the probability to be in the first excited state of the new potential.