Question: A quantum particle is inside a one-dimensional infinitely deep potential well.
Find the wavefunction that satisfies Schrödinger’s equation.
Answer:
First assume that
can be represented as
. This means that Schrödinger’s equation can be manipulated as followed:
Since the this equation holds for all
and
, both sides of the equation must equal a constant.
Time Component
Next, we will solve for the time component of the above equation.
Time-Independent Component
The time-independent component of the Schrödinger equation would be:
The solution to this problem can be easily solved for the case
and
. Since
, the only way to solve
is to have
.
For the case when
, we know that
. Therefore, the time-independent equation will become:
This equation should look rather familiar because it is the differential equation describing a simple harmonic oscillator. Therefore, the general solution to this differential equation would be:
Therefore, the solution would be:
If we add a requirement that
needs to be piecewise-smooth, we can add boundary conditions
and
to the problem:
Therefore, when
, we can see that
because
Now that we know
, we can apply the second boundary condition
to the remaining portion of the equation.
This implies
Therefore, we can see that the general solution can be reduced to:
The last step is to normalize this wave-function
Therefore
Therefore, the final solution would be

where
