Quantum Standing Waves and Tunneling Through a Finite Range Potential
- 1
Abstract
We consider a time independent one dimensional finite range and repulsive constant potential barrier between two impenetrable walls. For a nonrelativistic massive particle projected towards the potential with energies less than the barrier and irrespective of the spatial positioning of the potential allowing for quantum tunneling, analytically we solve the corresponding Schrodinger equation. For a set of suitable parameters utilizing <i>Mathematica</i> we display the wave functions along with their associated probabilities for the entire region. We investigate the sensitivity of the probability distributions as a function of the potential range and display a gallery of our analysis. We extend our analysis for bound state particles confined within constant attractive potentials.
- R. W. Gurney and E. U. Condon, “Quantum Mechanics and Radioactive Disintegration,” Physical Review, Vol. 33, No. 2, 1929, pp. 127-140. doi:10.1103/PhysRev.33.127
- L. Schiff, “Quantum Mechanics,” McGraw-Hill Company, Boston, 1968.
- G. Baym, “Lectures on Quantum Mechanics,” W. A. Benjamin, Inc., New York, 1976.
- A. S. Davydov, “Quantum Mechanics,” 2nd Edition, Pergamon Pr., Mesland, 1976.
- E. Merzbacher, “Quantum Mechanics,” 3rd Edition, Wiley, Hoboken, 1997.
- http://www.youtube/QM5.1,QM5.4,QM6.1 and QM6.3
- J. M. Feagin, “Quantum Methods with Mathematica,” Springer-Verlag, Berlin, 1994.
- Wolfram demonstration Projects, April 2011. http://demonstrations.wolfram.com.
- S. Wolfram, “The Mathematica Book,” 4th Edition, Cambridge University Press, Cambridge, 1999 and “MathematicaTM” software V8.0, 2010.
- N. J. Giordano and H. Nakanishi, “Computational Physics,” 2nd Edition, Pearson, Prentice Hall, Upper Saddle River, 2006.