Studying Quantum Systems: One and two dimensional systems

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Main headlines

Realistic versus model potentials: Eckart barrier Confined particles and quantization: particle

in box Correspondence principle and the classical

limit Zero-point energies Degeneracy and symmetry

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Model versus realistic potentials

• For most potentials Schrödinger equation can not be solved “analytically”.

• Analytical solution is a series with a finite number of terms

• Model potentials are employed since there are analytical solution for corresponding Schrödinger equation

• In some rare cases analytical solutions may be found also for realistic potentials

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Eckart potential barrier: A realistic potential

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Transmission coefficients

• Tunneling and anti-

tunneling are both seen in this case

• Curve 2 corresponds to a larger mass than curve 1

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Bound systems: Confining potentials

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Particle in an infinite well (particle in box): Simplest model potential

• Since the particle is confined, the energies are quantized

• The boundary condition determines which energies are permitted

• Since the potential energy is infinite out of the box, no penetration is possible

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Classical view

• No forces acts inside the box so the particle moves with a constant speed inside the well

• When particle collides with the wall an infinite force is exerted on it and the direction of motion is reversed instantaneously

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Solution: Basics

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Solution: Details

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Contiuned…

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Continued…

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Graphical depiction of solutions

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Nodes

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Zero point energy

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Large length and mass: correspondence principle

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Particle in a two-dimensional infinite well

• The boundary conditions are that the wavefunction must vanish at all four walls

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Solution

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Continued…

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Degeneracy

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The Origin of degeneracy

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An example

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Let’s Consider other potentials in Next

session!

Photon anti-correlation

Interference with single photons

The result of experiment

Theoretical justification