Given a Nanowire with cross sectional dimensions of 10 nm x 10 nm, what momentum would an
electron in the ground state need in order to possess the same energy as a stationary electron
(zero momentum) in the n=1,2 state?
Note that, when it says n=1,2 it means nx=1 and ny=2... In reality its not that restrictive, but it was pointed out to me that the wording was confusing. I am not looking for two solutions, one for n=1 and one for n=2. This is a problem dealing with a mixed system where two dimensions are governed by quantum mechanics and the third by classical.
I need the step-by-step solution please.
To solve this problem, we need to solve the time-independent Schrodinger equation for an electron in a nanowire. This equation says
(1) H psi(x, y, z) = -hbar^2/2m laplacian(psi(x,y,z)) + V(x,y,z) psi(x,y,z) = E psi(x,y,z),
where m is the mass of the electron, H = -hbar^2/2m laplacian + V is the Hamiltonian of the electron, V(x,y,z) = 0 is its potential (which is zero in this case because the electron is free inside the nanowire), psi(x,y,z) is its wavefunction, and E is its energy. We solve (1) by separation of variables. We let
psi(x,y,z) = X(x) Y(z) Z(z)
where the z-axis points along the direction of the nanowire and the x and y axes point along the sides of the square cross section, with x = y = 0 at one of the corners. ...
We solve for the momentum of an electron in a nanowire of given dimensions in the 1,1 mode which has the same energy as a stationary electron in the 1,2 mode.