Solving the Schrodinger Equation Along a Nanowire
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?
I need step-by-step solution please.
Solution Preview
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 ...
Solution Summary
We solve the Schrodinger equation along a nanowire with a square cross section of given dimensions to determine the speed of an electron in the 1,1 mode which would give it the same energy as a stationary electron in the 1,2 mode.