1. A three-dimensional wave function for an
ideal, finite potential quantum well.
Dr. A. Zahedi
Prof. R.E. Morrison
Francis K. Rault
Abstract: The quantum well has been studied for many years and the understandings have proven to be very beneficial for society. The undertaking of this paper was to extend the knowledge and understanding of the well in three dimensions. This paper successfully attempts to find a solution, for an ideal finite three-dimensional well. The solution is based on an existing one-dimensional solution. Using a proposed one-dimensional solution and extending it to three dimensions, a satisfactory model was developed. The model is purely mathematical and satisfies the fundamental criteria. That is, it is a solution of the Schrodinger wave equation and the wave function and its first derivatives are continuous at the boundary regions. The solution also takes into account degenerate energy levels.
The constants can be determined using appropriate boundary conditions.
Theoretical Approach:
Results:
For a three-dimensional solution one must solve Schrodinger’s Wave equation:
Begin with Aronstien and Stroud 1-D definition:
Wave functions:
Region 1: Lower Well Region.
Region 2: Inner Well Region.
Region 3: Upper Well Region.
Cubic 10 nm quantum well.
Particle in energy level 1.
Likely location of finding
the particle within the well: Centre of well.
Transcendental Equation Relationship:
Shows relationship between Well length (L), Energy (E) and Well Potential (Vo).
Takes into account degenerate energy levels.
n is principal quantum number.
Complete wave-functions for the respective regions:
Making the assumptions:
1- The three dimensions are independent to one another.
2- The complete wave function T (x, y, z) for the individual regions is the super position of
the three wave function in the three respective dimension. T (x, y, z)= n (x). n (y). n (z)
3- The total energy E is equal to the summation of the x,y,z energy levels. E = Ex +Ey + Ez .
The total potential of the well is the summation of the individual potentials in the respective dimensions.
Vo(x,y,z)= Vo(x) + Vo(y) + Vo(z)
N.B: Energy is a scalar quantity.
4- m is the effective mass of the particle within the quantum well.
Both Upper and Lower region probability density functions show a slight chance of finding the particle outside the well; due to tunneling effect.
Cubic well. Length 90 nm
Particle in energy level 2.
Energy Level Predictions:
Likelihood of finding particle is at point (3,3,3).
Conclusion:
Electrical and Computer Systems Engineering
Postgraduate Student Research Forum 2001