Consider an electron in an infinitely deep one-dimensional potential well of thickness L with zero potential energy at the bottom of the well. The normalized eigenfunction solutions to this problem can be written as the ket vectors ||(Phi)n where n is the quantum number of the state in the usual notation. We will consider energies in terms of E1, the eigenenergy of the n=1 state. H^ is the Hamiltonian operator for the electron in this well.

Evaluate the following matrix elements of the Hamiltonian in this basis,||(Phi)n,in units of E1, the eigenenergy of the n=1 state. For example, if you think the answer to some question is 10E1, then enter the number 10.

The matrix element H11 in units of E1 is?

Question

Consider an electron in an infinitely deep one-dimensional potential well of thickness L with zero potential energy at the bottom of the well. The normalized eigenfunction solutions to this problem can be written as the ket vectors ||(Phi)n> where n is the quantum number of the state in the usual notation. We will consider energies in terms of E1, the eigenenergy of the n=1 state. H^ is the Hamiltonian operator for the electron in this well.

Evaluate the following matrix elements of the Hamiltonian in this basis,||(Phi)n>,in units of E1, the eigenenergy of the n=1 state. For example, if you think the answer to some question is 10E1, then enter the number 10.

The matrix element H11 in units of E1 is?

surry99 · Answer

What have you tried so far to solve this?

anonymous · Answer

So you know that the energy eigenfunctions are the eigenfunctions of the Hamiltonian, right ? Also we are assuming that the eigenfunctions are normalised - so the matrix element H11 is pretty obvious, you can pretty much just write it down in one line.