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Understanding Bound States in Quantum Mechanics: The Square Well Potential

This brief overview explores bound states in quantum mechanics, specifically focusing on the square well potential. We discuss the potential characteristics with depth V0 and width 2a, examining regions where solutions exist. For energies greater than and less than zero, we derive the general solutions in different regions and analyze the implications of the potential as an even function. The treatment includes cases for infinite and finite well potentials, showcasing how energy levels vary and the significance of bound states in both deep and shallow wells.

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Understanding Bound States in Quantum Mechanics: The Square Well Potential

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  1. Source: D. Griffiths, Introduction to Quantum Mechanics (Prentice Hall, 2004) R. Scherrer, Quantum Mechanics An Accessible Introduction (Pearson Int’l Ed., 2006) R. Eisberg & R. Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles (Wiley, 1974) Lecture 5

  2. Topics Today Bound States of Square Well E > 0. Bound States of Square Well E < 0.

  3. Bound States of the Square Well E > 0 -Vo • Consider the potential • This is a "square" well potential of width 2a and depth V0.

  4. Bound States of the Square Well E > 0 Region I and III: -Vo Region II:

  5. The solutions are, in general: • Region I: • Region II: • Region III: • In these the A's, B's, C’s and D's are constants.

  6. Bound States of the Square Well, E > 0 Allowed energy for infinite square well

  7. Bound States of the Square Well E < 0 -Vo • Consider the potential • This is a "square" well potential of width 2a and depth V0.

  8. Bound States of the Square Well Region I and III: E < 0 -Vo Region II:

  9. The solutions are, in general: • Region I: • Region II: • Region III: • In these the A's, B's and C's are constants. Potential is even function, therefore the solutions can be even or odd. For even solution: Κ and l are functions of E, so is formula for allowed energies.

  10. Bound States of the Square Well

  11. Wide, Deep Well If z is very large, Infinite square well energies for well width of 2a. This is half the energy, the others come from the odd wave functions.

  12. Shallow Narrow Well As zo decreases, fewer bound states, until finally, )for zo < p/2, the lowest odd state disappears) only one bound state remains. There is one bound state no matter how weak the well becomes.

  13. Finite Well Energy Levels

  14. Particle in Finite-Walled Box

  15. PROBLEM 1

  16. PROBLEM 2

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