An electron in an infinite square well of width is prepared at in (x,0)= 1 2 (…

PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Wave mechanics in one dimension · The formalism of quantum mechanics

The Structure of Quantum Mechanics, Ch The wave function is not the state

Problem

An electron in an infinite square well of width

L=0.50\,\mathrm{nm}

is prepared at

t=0

\Psi(x,0)=\frac{1}{\sqrt2}\big(\psi_1(x)+i\,\psi_3(x)\big),

with

\psi_n(x)=\sqrt{2/L}\sin(n\pi x/L)

the energy eigenfunctions. ((a)) Read off the energy-representation amplitudes

c_1,c_3

and give

P(E_1)

P(E_3)

. ((b)) Using the position description, evaluate the density

|\Psi(L/2,0)|^2

at the centre of the well in

\mathrm{m^{-1}}

. Compare it with the centre density of

(\psi_1+\psi_3)/\sqrt2

, which has the same

P(E_1)

and

P(E_3)

, and say what in the two states makes the difference.