The Schrödinger equation for a free particle (no potential energy) is
.
Part A 

What is the most general solution of the timeindependent Schrödinger equation? 
ANSWER:
Part B 

What are the energy levels of this free particle associated with a wave number ? Express your answer in terms of wave number , mass , and Planck's constant divided by : .
ANSWER: 
 =  


Part C 

To normalize this wave function, you must calculate the integral . What is the value of this integral?
ANSWER: 
=  


Thus, a free particle wave function is unnormalizable. This is due to the fact that a free particle wave function has no boundaries and thus is
unlocalized. This means that there is the same probability of finding a particle anywhere in the universe. You can think of this in a different way: Since there are no boundaries, there is no potential energy present, and thus a free particle's energy is not constrained. This means that a free wave has a continuous spectrum of frequencies.
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