Question

The energy, E, of a quantum particle confined in a ”quantum dot” depends on the linear size of the dot, L, in the following way: E = α mL2 In this equation, the size L is expressed in nanometers (nm), the energy E is expressed in electron-volts (eV), the constant α is given by α = 0.1 ± 0.005 eV-nm2 , and m is a dimensionless parameter representing the mass of the quantum particle.

A series of spectroscopic measurements are made on quantum dot samples of varying size to determine the dependence of the energy upon the linear size of the dot. The following data was obtained, written in ordered pairs (L, E) where L is in nm and E is in eV: (0.800, 3.32), (1.20, 1.42), (1.60, 0.931), (2.00, 0.557), (2.40, 0.448).

a) verify the relationship between E and L in the present problem by making a linear graph using Excel. Make sure to label your axes including units.

(b) Use LINEST to find the slope, s, of that line. From the slope and the parameter α, determine the dimensionless mass of the quantum particle, m. (For your information, the parameter m is the fraction of an electron mass. For instance, m = 0.08 would mean that the mass of the quantum particle is 0.08me where me is the mass of the electron.) 2 2. (continued)

(c) Using LINEST, find the uncertainty in the slope, σs. Using the uncertainty in the slope and the uncertainty in α, find the uncertainty in m.

Answer 1

a) Here,

If we draw a graph of E versus 1/L^2, it will be a straight line.

The graph is plotted on excel and the plot is given below.

The graph is a straight line.

b) Using linset function, the slope was found to be 2.0692

We have

Alpha = 0.1

So,

c) The uncertainty in slope is given by

dm = 0.0552

So, the uncertainty in m is given by