Suppose we have a particle moving freely in a box.

  1. We can model this as a Quantum state .
  2. The Wave function is on the boundary of the domain.
  3. Since we are working with a free particle, the Hamiltonian is given by , where
    1. is the particle’s mass.
    2. is the Momentum Operator.


Using the Time-independent Schrodinger equation, we get

Using the ansatz , we can split the equation above into independent 2nd order differential equations , which have the generic solution . Plugging in the boundary condition gives us the constraint , simplifying our solution to . Plugging in the boundary condition we get

Plugging the result above we get . Since needs to be Normalizable, we have

Since the Global Phase of does not impact the solution, we can assume that , giving us that . Hence the solution is given by . Since is normalised, the Energy associated with this solution is given by .

Where we needed to lookup the Table of integrals to find .

  • Correct the above ⏫