Heisenberg Uncertainty Principle & Wave Function

[Sum Guy]

Considering how Heisenberg's uncertainty principle is applied to a top-hat wave function:

This hyperphysics page shows how you can go about estimating the minimum kinetic energy of a particle in a 1,2,3-D box: http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/uncer2.html

You can also investigate a particle in a box via the following treatment: http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/schr.html#c2

I'm having difficulty reconciling these two things? The first I would think of as applying the uncertainty principle to a top hat function, yet in the second link (the same scenario I think) we clearly have sinusoidal (i.e. non top hat) wavefunctions. How can you bring these two things together? Can you in general apply the uncertainty principle to a top hat function?

 

[ZapperZ]

I'm not sure I understand how you are interpreting those web pages.

The first one is a square potential, not a "top hat". The secon one is a1D infinite square well. These represent the POTENTIAL profile. BOTH will result in "sinusoidal" wavefunctions.

Zz.

 

[vanhees71]

To use the infinite box potential to "demonstrate" the uncertainty relation between position and momentum is quite common in the introductory QT textbook literature, but it's nevertheless one of the sins in physics didactics one should avoid. The reason is that for this problem no proper momentum operator exists. So it doesn't make sense to talk about a momentum probability distribution in this case either. The reason is pretty formal concerning the self-adjointness of operators in the Hilbert space $L^2([-L/2,L/2])$ with "rigid boundary conditions" $\psi(-L/2)=\psi(L/2)=0$.

The second link is however correct since the Hamiltonian
$$\hat{H}=-\frac{\hbar^2}{2m} \partial_x^2$$
is indeed self-adjoint in this space and thus energy is properly defined for the problem.