Day 4 In-Class Activity
Infinite Square Well Solutions
By defining the position and momentum operators,
By doing this, we are using the “position representation” of the state vector,
Consider a 1D potential given by,
- Question: Let’s define
as McIntyre does. Inside the well, we have the differential equation: . Determine the general solution .- Discussion: How does your solution differ from McIntyre’s?
- Question: You are solving a “boundary value problem”, which is characterized by knowing the value of or the first derivative of a function at locations that “bound” the problem. For us, we need the wavefunction to vanish at the boundary. What happens to
? What condition does this put on ?- Discussion: The position representation is continuous, but only certain functions “fit”. How might you use symmetry to guess what functions would work? Why would this process be called “establishing the quantization condition”?